Biometry

Biometry
FOR
POST GRADUATES

MANDEFRO NIGUSSIE
Ethiopian Institute of Agricultural Research

Prepared for Haramaya University (Biometry PLAG511) Course

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PREFACE

Most professionals in a chosen area have been successful due to partly their
understanding of statistics. We can see that most scientists that went up the ladder in
their career understand statistics very well. This does not mean that a brilliant mind in
statistics is a necessary condition for being a brilliant geneticist, breeder, pathologist,
soil scientist, chemist, engineer or any other profession. But it is necessary at least to
design an experiment, present the results of the experiment and understand the real
meaning of any scientific information that backups the results. The fact most
university degrees in natural sciences include a compulsory statistic course is simply
a recognition of this.

Statistics as a science in its own can be very complicated. The statistical methods we
need to use in doing research are only a small and fairly straightforward subset of
this. A general understanding of the basic concepts and principles will be a great
asset in interpreting the experimental results of other people. Understanding this can
give postgraduate students statistical and scientific skills beyond those of the general
public.

There are so many statistical techniques available for different purposes but it is
impossible to master them all in a short period of time. We have therefore selected in
this book of statistical methods that we consider most relevant and useful in
conducting experiments with careful planning, and proper interpretation.

The book is intended for postgraduate students and scientists who have at least a
little background in basic statistics; but seek to learn advanced ideas of statistics and
their application in a variety of practical situations. The core materials of this book are
lecture notes of the course Advanced Statistical Methods (AGR 5201) at Universiti
Putra Malaysia (UPM) and is designed to be covered well within a semester.

We understand that memorizing formulae is of very little practical use and most
computations are carried out by computers these days. However, computers do not
generally tell whether we are carrying out the right computations or not. Here, this
book has a part to play in presenting the necessary procedures needed to do the
analyses either manually or using the software Statistical Analysis System (SAS). We
also presented fundamental principles of experimentation, methods of sampling, data
analysis and proper interpretation of the results that every post graduate student
should possess. In general, the book will provide the important tools needed to be an
effective user of statistics. We hope that it will provide users with most of the
information required for academic as well as for scientific studies in the research
career.

Mandefro Nigussie

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TABLE OF CONTENTS

1. INTRODUCTION
1.1. Importance and Organization
1.2. Basic Statistical Terms and Concepts

2. EXPERIMENTS
2.1. Introduction
2.2. Experimental Designs
2.2.1. Importance of experimental designs
2.2.2. Planning and execution of experiments
2.2.3. Experimental Error
2.2.3.1. Methods of error estimation
2.2.3.2. Methods of controlling error
2.3. Categories of Experiments
2.4. Hypothesis

3. ORGANIZATION AND DESCRIPTION OF DATA

3.1. Variable
3.2. Sources of Data
3.3. Sampling Methods
3.3.1. Simple random sampling
3.3.2. Systematic sampling
3.3.3. Stratified random sampling
3.3.4. Multistage random sampling
3.3.5. Stratified multistage random sampling
3.3.6. Cluster sampling
3.3.7. Quota sampling
3.4. Sample Size
3.5. Data Description
3.5.1. Frequency distribution
3.5.2. Histogram
3.5.3. Frequency polygon
3.5.4. Measure of central tendency
3.5.6. Measure of dispersion

4. ANALYSIS OF VARIANCE – ONE FACTOR

4.1. Introduction
4.2. Completely Randomized Design
4.2.1. Layout
4.2.2. Analysis
4.2.3. Standard errors
4.2.4. Mean separation

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4.2.6. Numerical examples

4.3. Randomized Complete Block Design
4.3.1. Introduction
4.3.2. Layout
4.3.3. Analysis
4.3.4. Numerical examples
4.4. Latin Square Design
4.4.1. Introduction
4.4.2. Layout
4.4.3. Analysis
4.4.4. Numerical examples
4.5. Augmented Design
4.5.1. Introduction
4.5.2. Layout
4.5.3. Analysis
4.5.4. Numerical example
4.6. Incomplete Block Design
4.6.1. Introduction
4.6.2. Balanced lattice
4.6.2.1. Layout
4.6.2.2. Analysis
4.6.2.3. Numerical example
4.6.3. Partially balanced lattice
4.6.3.1. Introduction
4.6.3.2. Simple lattice
4.6.3.2.1. Layout
4.6.3.2.2. Analysis
4.6.3.2.3. Numerical example

5. ANALYSIS OF VARIANCE – TWO OR MORE FACTORS

5.1. Split-plot
5.1.1. Introduction
5.1.2. Lay-out
5.1.3. Analysis
5.2.4. Numerical examples
5.2. Split-block (Strip-plot)
5.2.1. Introduction
5.2.2. Lay-out
5.2.3. Analysis
5.2.4. Numerical example

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6. FACTORIAL EXPERIMENTS
6.1. Introduction
6.2. Two Factor Experiment
6.2.1. Introduction
6.2.2. Layout
6.2.3. Analysis
6.2.4. Numerical example
6.4. Fractional factorial experiment

7. ANALYSIS OF EXPERIMENTS OVER TIME AND SPACE

7.1. Introduction
7.2. The principle behind analysis of several experiments
7.3. Analysis
7.4. The same experiment across environments
7.5. Summary

8. ANALYSIS OF COVARIANCE
8.1. Introduction
8.2. Covariance Analysis and Blocking
8.3. Numerical example

9. CORRELATION
9.1. Introduction
9.2. Computing r Values
9.3. Correlation and Causation

10. REGRESSION
10.1. Introduction
10.2. Determining a Linear Regression Line with a Single Predictor
10.3. Numerical Example

11. DATA TRANSFORMATION

12. REFERENCES

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1. INTRODUCTION
1.1. Importance and Organization
The core concept underlying all research is its methodology. The research tools are
key elements of the methodology that takes to the ultimate goal of research to
discover something, which was unknown. The physical tools some researchers use
to achieve their goals are distinctly different from those of others, depending on the
discipline. However, the general tools of research that most researchers use to derive
meaningful conclusions are similar. One of the very few important tools of research,
which is common to almost all kinds of research, is statistics.

Statistics play a vital role in virtually all professions. Some familiarity is an essential
component of any higher learning institutions and research organizations. We
understand that all post graduate students and scientific personnel have a good
scientific background in their respective disciplines but their statistical knowledge
ranges from very little to long years of practical experience. Most researchers had
basic statistical courses but they usually face difficulty in applying to their own
practical situations specially in designing sophisticated research work. Hence, this
manual was written to provide researchers with the powerful ideas of modern
statistical methods.

The manual provides basic principles and advanced procedures on the relative
importance of components of variations, the methods to estimate, and means to
control them, in relation to specific experimental designs. The principles and
statistical methodologies illustrated in this manual, provides the type and amount of
data to be collected, the method of organizing data, data analysis and proper
interpretation of the results obtained. It also provides means to draw conclusions,
assess the strength of the conclusions and gauge uncertainties. Examples are drawn
from a wide range of applications to help develop an appreciation of various
statistical methods, their potential uses and their vulnerabilities to misuse.

Statistical concepts have also been explained as an essential component during the
planning stage of an experiment when decisions must be made as to the mode and
extent of the sampling process. This is done because a good design for the process
of data collection permits efficient inferences to be made, with straightforward
analysis.

The manual is organized into 13 sections. In Section 1 introductory points are

presented. In section two, experiments in the widest context are presented. In
Section 3, methods of organizing and describing a given data set are explained.
From Section 4 to 7, different types of experimental designs, situations to use them,
the layout, method of analysis, mean comparison and proper interpretation of the
results are presented. In Section 8, covariance analysis is illustrated. In Sections 9
and 10 correlation and regression analysis are presented. Finally, data
transformation, references and important statistical tables are given in Sections 11,
12 and 13, respectively.

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1.2. Basic Statistical Terms and Concepts

The approach in presenting the basic concepts and principles is based on brief
definition, followed by description of the most commonly used terms in this manual.
These definitions are important to understand the statistical principles and techniques
in the analysis of variance and proper interpretation of the results.

What is statistics? A statistical expression that says, ‘we calculate statistics from
statistics by statistics’ is a good example to understand the different uses of statistics.
It is necessary to know the context within which these three uses of the term are
operating. ‘We calculate statistics…’ what it means here is the results of
computations: the means, variances, graphs, tables, correlation coefficients,
regression lines and so on. ‘…from statistics…’ the term statistics here refers to the
raw data to be analyzed. ‘…by statistics.’ The last use of the term is the process or
method that we use to get the desired results and that is the focus of this manual.

The field of statistics has traditionally been divided into two broad categories defined
as: descriptive and inferential. However, these two categories are not mutually
exclusive and that there is a great deal of overlap in what has been labeled as
descriptive may be labeled inferential. Descriptive statistics can be defined as
those methods involving the collection, presentation and characterization of a set of
data in order to properly describe the various features of that set of data (Caulcutt,
1991). The purpose of a descriptive statistic is to tell us something about a particular
group of observations.

Inferential statistics can be defined as those methods that make possible the
estimation of a characteristic of a population or the making of a decision concerning a
population based only on sample results (Walpole et al., 2002). The task of inferential
statistics is to draw inferences or make predictions concerning the population on the
basis of the data from a smaller sample.

The fundamental concepts of statistical inference consist of two major areas known
as parameter estimation and hypothesis testing. Some of the concepts related to
these two areas will be directly used in the development of statistical techniques for
studying experimental designs and the corresponding experimental results. Other
concepts are important prerequisites for a better understanding of the role and scope
of statistical inference in the analysis of experiments.

The point to be emphasized is that whether a given statistics is descriptive or

inferential depends on the purpose for which it is intended. If a group of observations
were used merely to describe an event, the statistics calculated from these
observations would be descriptive. If, on the other hand, a sample were selected with
the intent of predicting what the larger population looks like, the statistics would be
inferential. In this manual more emphasis is given to the inferential statistics.

A population is the whole set of measurements or counts about which we want to

draw conclusion. The population is always carefully defined by specifying as

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completely as possible those critical attributes that distinguish its members from any
and all other entities (Orr, 1995). This step is essential to designate the data or apply
the result. Populations are described by characteristics called parameters and
parameters are fixed values. If only one variable is of interest, the population can be
called as a univariate. For example, the heights of all male students in universities
form a univariate population. Notice that a population is a set of measurements, not
the individuals or objects on which the measurements or counts are made.

A sample is a sub set of a population, a set of some of the measurements or counts

that comprises the population. Samples are described by sample statistics. We
calculate sample statistics to estimate population parameters. The essential nature of
a sample is that it should be representative. This means that a sample should be a
small-scale replica of the population that might affect the conclusion of the study.
Sampling is dealt in depth in Section 3.3.

Accuracy and precision are used synonymously in everyday speech, but in

statistics they are defined more rigorously. Precision is the closeness of repeated
measurements and accuracy is the closeness of a measured or computed value to its
true value (Sokal and Rohlf, 1987). Based on this definition, a biased but sensitive
scale might yield inaccurate but precise weight, as precision is the closeness of
repeated measurement. By chance, an insensitive scale might result in an accurate
reading, which would however, be imprecise, since a repeated weighing would be
unlikely to yield an equal accurate weight.

Hypothesis is an assertion or conjecture concerning one or more populations. The

truth or falsity of a statistical hypothesis is never known with absolute certainty unless
the entire population is examined. This would be impractical in most situations.
Instead, random samples are taken from the population of interest and the data
contained in the samples are used to provide evidences that either support or do not
support the hypothesis.

The structure of the hypothesis testing will be formulated with the use of the term null
hypothesis. This refers to any hypothesis to be tested and is denoted by H0. The
rejection of H0 leads to the acceptance of an alternate hypothesis, denoted by H1 or
HA. While testing the hypothesis, the critical value is used as a reference point to
differentiate the critical region and the acceptance region. Rejection of the null
hypothesis when it is true is called a Type-I error. Acceptance of the null hypothesis
when it is false is called a Type-II error (Walpole et al., 2002) (See Section 2.4).

Experimental error is the variation between experimental units (plots) treated alike.
There are a number of sources of variations in field experiments, and these sources
contribute to experimental error. These variations could be inherent (natural) or due
to lack of uniformity in the physical conduct of the experiment. One of the goals of
modern experimental designs is to provide a measure of the experimental error and a
means to control the error.

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The experimental error can never be eliminated completely, but can be minimized.
The experimental error can be estimated through replication and randomization and it
can be controlled through blocking, proper plot techniques and application of
appropriate experimental designs. Reduction of the experimental error as much as
possible within the limitations of resources that can be committed to the experiment is
very important for valid comparison of treatment means. The basic idea behind the
reduction of experimental error is that we can detect small differences between
treatment means. Details are given in Section 2.2.3.

Replication is repeating treatments in more than one experimental unit. The

objectives of replications are to: estimate experimental error, increase precision,
broaden the scope of inferences and effect the control of error. Hence, replication is
one of the basic features of experimental designs. Details are given in Section
2.2.3.1.

Randomization is the process of assigning treatments to experimental units at

random. With proper randomization any treatment is equally likely to be assigned to
any experimental unit and in any design. The main objectives of randomization are to
eliminate bias and ensure independence among observations, which is one of the
basic assumptions of the analysis of variance. Details are given in Section 2.2.3.1.
Blocking is a one of the procedures of the refined techniques where by experimental
units are grouped into blocks of homogeneous units. The main objectives are to
minimize within block variations and maximize between block variations in order to
eliminate them from the experimental error. By doing so blocking helps to increase
precision and information and there by ensures uniform comparison of treatment
means. Details are given in section 2.2.3.2.

Analysis of variance (ANOVA) is a procedure that can be used to analyze the

results from both simple and complex experiments (Cochran and Cox, 1957). It is
one of the most important statistical techniques available and provides a link between
the design of experiments and data analysis. Detail discussions are presented in
Sections 4 to 7.

Assumptions of analysis of variance include randomness (sampling of individuals be

at random), independence (the errors are independently distributed), normality (the
errors are randomly, independently and normally distributed), and homogeneity (the
error terms are homogeneous/equal variance).

Treatment means can be compared in five different ways: preplanned t-test,

preplanned discriminate (multiple comparison), preplanned contrast, preplanned
comparison to a check/control and linear and curvilinear trends for quantitative
factors. Preplanned t-test refers to comparison of two treatment means. Multiple
comparison refers to comparing two or more treatment means taking two at a time
and it involves the use of reference point: critical difference/Least Significant
Difference (LSD), Duncan’s New Multiple Range Test (DNMRT), Tukey’s Honestly

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Significant Difference (HSD), Student-Newman-Keuls Test (SNK) and Waller and

Duncan’s Bayes LSD (BLSD).

Correlation analysis attempts to measure the strength of relationships between two

variables by means of a single number called a correlation coefficient (Trumbo,
2002). It is important for the reader to understand the physical interpretation of this
correlation coefficient and the distinction between correlation and regression.
Correlation coefficients close to +1 or –1 indicate a close fit to a straight line (strong
correlation) and values closer to zero indicate a very poor fit to a straight line or no
correlation. There is no convention as to what values of correlation should be
described as strong or weak. The negative correlation values tell that the values of
one variable tend to get larger as the values of the variable get smaller and vice
versa. Details are given in Section 9.

Regression is similar to correlation in that testing for a linear relationship between

two types of measurements is made on the same individuals. However, regression
goes further in that we can also produce an equation describing the line of best fit
through the points on the graph. Regression analysis concerns the study of the
relationships between variables with the objective of identifying, estimating and
validating the relationship (Trumbo, 2002). When using regression analysis, unlike in
correlation, the two variables have different roles. Regression is used when the value
of one of the variables is considered to be dependent on the other, or at least reliably
predicted from the other. In correlation, we take measurement on individuals at
random for both variables, but in regression we usually choose a set of fixed values
for the independent variable (the one controlling the other). Details are given in
Section 10

The covariance between two random variables is a measurement of the nature of

the association between the two. The sign of the covariance indicates whether the
relationship between two dependent random variables is positive or negative. When
the two variables are statistically independent, it can be shown that the covariance is
zero but the converse is not always true (Walpole et al., 2002). Details are given in
Section 8.

In summary, the subject of modern statistics encompasses the collection,

presentation, and characterization of information to assist in both data analysis and
decision-making process. The use of statistical methods in many areas involves the
gathering of information or scientific data. The data are then summarized and
reported and may be stored for later use. However, there is a profound distinction
between collection of scientific information and inferential statistics. The inferential
statistical methods are designed to contribute to the process of making scientific
judgments in the face of uncertainty and variation. The process of making scientific
judgment requires information from properly planned experiments.

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2. EXPERIMENTS
2.1. Introduction
There are several definitions of the term experiment. Steel and Torrie (1980) defined
experiment as a planned inquiry to obtain new facts or to confirm or deny the results
of the previous experiments. An experiment will always start with an idea, which has
to be formulated into objectives. Objectives are formulated into hypothesis to allow
statistical analysis that will support any conclusion drawn.

Experiment is an important tool for research and should have the following important
characteristics: simplicity, measuring differences at higher precision, absence of
systematic error, wider ranges of conclusion and calculation of degree of uncertainty
(Rees, 1995). The results of experiments are used for decision making like giving
recommendations to use a new product/technology, provided that the new product is
economically more useful than the old version. The new technology (research result)
could vary depending on the field of study. In agriculture, it could be new input
(variety, fertilizer rate/type, pesticide, herbicide, media, feed, vaccination, tree, etc.)
agronomic practices and post harvest management. In education, it could be
teaching method, evaluation technique and so on.

Experimental units and treatments are very general terms. An experimental unit may
be an animal, many animals, a plant, a leaf and so on. Accordingly, a treatment may
be a standard ration, inoculation, and a spraying rate/spraying schedule. In the first
case, the experimenter may compare different rations to identify the best for
recommendation. In the second case, the experimenter may compare different
varieties a crop for disease resistance, and categorize varieties as resistance,
moderately resistance, and susceptible. In the third case the experimenter compares
different rates of herbicide to identify the most effective as well as economical rate
and the right time to apply. In running such experiments, efficient experimental
designs that can provide meaningful answers related to the objectives of the
experiment are needed.

2.2. Experimental Designs

Experimental design is the area of statistics concerned with designing an
investigation to best meet the study objectives, as well as the assumptions for
statistical inferences. Fundamentally, the steps involved in experimentation are those
of the scientific methods. The essential addition, however, is that the design involves
the ability to control what goes on and to manipulate one or more of the factors
bearing on the outcomes (Orr, 1995).

2.2.1. Importance of experimental designs

A research programme can have a number of objectives. To meet these objectives
scientists conduct experiments of one type or another. In most instances, data is
collected from experimental units (plots). Since research resources are limited, it is
essential that the research be conducted in such a way as to minimize cost and
maximize the information generated for resources used. Such information can only

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be obtained from designed experiments that involve statistical procedures for valid
conclusions.

Experiments are designed to: provide an estimate of error, increase precision by

planned grouping that can eliminate differences from experimental error, provide
information needed to perform tests of significance and to construct interval
estimates, and facilitate the application of treatments and cultural practices. In this
manual, attempts are made to introduce advanced concepts and principles of
experimental designs as they apply to laboratory and field experiments.

In general, experimental design encompasses the rules and procedures used in

initiating, organizing and analyzing experiments. There are two considerations in
experimental designs: planning and execution of experiments and error control.
Detail explanations of the two considerations are given below.

2.2.2.Planning and execution of experiments

Planning is the most important aspect of any experiment. The thing that separates
competent investigator from incompetent ones, in terms of statistical skill, is nothing,
but careful planning. The idea of design should come at the planning phase of the
experiment. For example, by the time a student sits down at the keyboard with the
data, mistakes might have been done already. Upon thinking to start the statistical
part of the project, the part in statistics is really coming to an end. If the planning has
been done carefully, forming a clear idea of what is to be investigated, following the
layout of appropriate design and conducting the experiment accordingly, the analysis
and interpretation will be easier.

In addition to replication, randomization and blocking, a few other points should be

given considerations in the planning phase. These include: statement of the problem,
statement of objectives, choice of experimental site, establishing experimental
procedures, defining measurements to be taken, preparing data collection and
summary sheet, outlining data analysis, preparing work schedule, analyzing scope
and cost of the experiment, execution of the experiment, data analysis and report
writing.

Statement of the problem: In defining the problem, the most important thing is to
describe the scope of the problem, state the major and specific objectives of the
experiment and major assumptions clearly and concisely. It is also necessary to
review what has been done (information or principles established) in the form of past
research achievements, what is desired (goal/purpose of doing the research) and
then analyze the gap (the deference between future desire and past achievements).
Statement of objectives: It takes time and effort to define objectives. Objectives
should be realistic, measurable and achievable. Defining the objectives can be done
in the form of questions to be answered, hypothesis to be tested or effects to be
estimated (Cochran and Cox, 1957; Steel and Torrie, 1980). Some experiments may
have only one objective but others may have a series of specific objectives and in

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such cases they should be written in the order of priority, i.e. the most important one
first, followed by the second most important, and so on.

Choice of experimental site: The choice of experimental site affects the conduct of
an experiment and the interpretation of the results in several ways. It determines the
population to which the results of experiment will apply. Thus the population about
which information is wanted must be defined carefully and the site must be one that
represents the population. For example, if the results are to be used in acid soil
areas, then the experiment should be conducted in a place where soil acidity is a
problem, but not in alkaline soils.

The characteristics of the site also determine the type of experimental design to be
used, the placement of the experiment, the necessity of formation of blocks, and size,
shape and orientation of plots. If the experiment is to be conducted out in the field,
the site could have variation in soil physical and chemical properties, topography, and
so on. The experiment should be planned in such a way that the effects of these
sources of variability are minimized.

Experimental procedure: Experimental procedure refers to selection of the

appropriate materials (treatments) available for testing, explaining why those
treatments have been chosen, what biological control should be used, the number of
experimental units to be used, the number of replications to be employed, and
selection of proper experimental design. In general, the success of the experiment
rests on careful selection of treatments, to properly meet the objectives set.

Measurements: It is very important to decide on the variable of interest in such a

way that the data can properly evaluate the treatment effects in line with the
objectives and can answer the question ‘why treatment perform the way they do?’ In
short, it is important to have the appropriate measurement taken at the right time.

Preparing data recording and summarizing sheet: It is self-explanatory that we

need to have data collection forms to record our observations, and data summary
sheet that can suit statistical analysis. Here, use of relevant facilities that are secured
from whether damage is advisable.

Outlining data analysis: It is essential to prepare a table containing sources of

variation, the associated degrees of freedom in the analysis of variance and
expected mean squares, depending on the design used. At this point, it is important
to consult statisticians to obtain important points that the experimenter might
overlooked. This kind of adjustment in the planning phase can greatly enrich the
experiment.

Work plan: The easiest and most appropriate form of work plan is the use of a three-
column table that contains type of work in the first column, duration in the second
column and cost attached to each operation in the third column.

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Scope and cost of the experiment: During the planning phase, it is necessary to
consider the scope (local, regional or international) of the experiment, and the cost
attached to it since research costs are usually limited.

Execution of the experiment: During execution of an experiment it is very important

to use procedures that are free from personal biases. It is also important to collect
accurate and precise data so that differences among treatments with the order of
collection can be removed from experimental error.

When running the experiment, it is vital to monitor the process carefully to ensure that
every thing is being done as planned. Error in experimental procedure at this stage
will usually destroy experimental validity. Up-front planning is crucial to success. It is
easy to underestimate the logistics and planning aspects of running a designed
experiment in a complex and well-organized research system.

Coleman and Montgomery (1993) suggested that prior to conducting the actual
experiment; a few trial runs or pilot runs are often helpful. These runs provide
information about consistency of experimental material, a check on measurement
system, a rough idea of experimental error, and a chance to practice the overall
experimental techniques. This also provides an opportunity to revisit the decision on
the choice of treatments, experimental design and the variable of interest.

Data analysis: Statistical methods should be used to analyze the data so that the
results and conclusions are objective rather than subjective judgmental in nature.
There are many excellent software packages designed to assist in data analysis.
Often we find that simple graphical methods (trend analysis) play an important role in
data analysis and interpretation. Hypothesis testing and confidence interval
estimation procedures are very useful in analyzing data from designed experiments
because many of the questions the researcher wants to answer can be casted into a
hypothesis-testing framework.

All data analysis should be done as planned and the results interpreted in line with
the experimental conditions and the relations of the results to previous findings.
Here, it should be noted that statistics couldn’t improve the variation during the
physical conduct of the experiment. Therefore, consider the consequence of making
incorrect decisions and be reserved to give conclusions against established facts, if
the conclusions appear out of line with previously research results.

The primary advantage of statistical methods is that they add objectivity to the
decision-making process. Statistical techniques, coupled with good scientific out look
(process of knowledge) and common sense will usually lead to a sound conclusion.

Report writing: A complete readable and correct report of the research should be
prepared. Once the data have been analyzed, the researcher must present precise
results and draw practical conclusions about the results and recommend the course

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of action. A graphical method is often useful at this stage, particularly in presenting

the results to scientific community.

Throughout the entire process of experimentation, it is important to keep that

experimentation is an important part of the learning process, where we tentatively
formulate hypotheses about a system, conduct experiments to investigate these
hypotheses, and formulate new hypotheses on the basis of the results, and so on.
This suggests that experimentation is iterative.

In summary, a successful experiment requires knowledge of the important factors,

the ranges over which these factors should be varied, the appropriate number of
levels to use and the proper units of measurement for these variables.

2.2.3. Experimental Error

The primary function of experimental design is error control. Experimental error
measures the variations that exist among observations in experimental units treated
alike (Cochran and Cox, 1957). This is not to imply that mistakes have been made in
running the experiment. Rather it is a term used to describe the variation among plots
subjected to the same treatment. This variation may be inherent plant-to-plant
variation or slight variation in the conduct of experiment, applying treatments, taking
measurements and so on. Although experimental error can never be eliminated
completely, it can be minimized.

Steel and Torrie (1980) explained the two sources of variations as inherent variation
and variation due to lack of uniformity in the physical conduct of the experiment.
Inherent variation is due to inconsistency within the population of experimental units.
It cannot be avoided but can be minimized by experimental designs. In general, the
sources of inherent variability in field experiments are plant variability, seasonal
variability and soil variability. Detail examples of the inherent variations include
difference in soil fertility, slope, pH, temperature, sunshine, wind, etc.

Variation due to inconsistency or lack of uniformity in the physical conduct of an

experiment cannot be controlled by experimental design (statistics). This kind of
variation may be due to mistakes and sloppiness, ignorance in the characteristics of
experimental units or proper experimental design, and non-random outcomes like
flooding, insect outbreak, etc.

It is important that every possible effort be made to reduce the experimental error in
order to improve the power of a test, and to decrease the size of confidence intervals.
This can be accomplished by reducing the two sources of experimental error that is,
reducing the effect of the inherent variability and refining the experimental
techniques. One of the goals of modern experimental design is to provide a measure
of experimental error and to reduce it as much as possible within the limitations of the
resources that can be committed to the experiment.

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In general, a good experimental design minimizes error, so that the treatment effect
can be detected more readily.

2.2.3.1. Methods of error estimation

Modern experimental procedures include a number of features that permit the
scientist to measure and control experimental error. These features include
replication, randomization and blocking.

I. Replication
To obtain an estimate of experimental error, the scientist must repeat each treatment
two or more times. This repetition is called replication, which is each treatment must
appear more than once in an experiment. The precision of an experiment can always
be increased by having additional replications. Little and Hills (1978) reported that, to
double the precision in an experiment with which two means can be separated, 16
replications are needed as compared to an experiment with four replications.

In general, in field research, three to six replications are enough for reasonable
precision, but it all depends on the resource and degree of precision required.
Cochran and Cox (1957) provided a formula which could conveniently estimate the
number of replications required to detect a specified difference. The formula is

r ≥ 2[(c.v.)2/D2](t1+t2),

where c.v. is the coefficient of variation; D is the difference desired to detect,

expressed as a percent of the mean of the experiment; t1 is tabulated t-value for a
specified level of significance at the error degrees of freedom; and t2 is the tabulated
t-value at error degrees of freedom and a probability of (1 - p)/2, where p is the
probability level of detecting significance for a given experiment.

Importance of replicating treatments in an experiment: There are so many

reasons for having replications in an experiment.
a. It provides an estimate of experimental error because it measures observations
on several plots receiving the same treatment. In this way, the variation among
plots treated alike can be measured.
b. It increases precision by reducing standard errors. At this point it is necessary to
introduce the term standard error of a mean, SE(m) which is given by

s2
SE(m) =
n
2
where s is the sample variance and n is the number of observations (replications)
on which the mean is based. Therefore, as the number of replications (n) is
increased, the standard error of a mean decreases, and as a result, precision
increases.
c. It can broaden the base for making inferences. As replication is increased, a wider
variety of plots can be brought into the experiment. This can lead to a greater
range of conditions over which the results of the experiment can be applied.

16
 Biometry

d. It effects the control of error. If replication is meant to estimate error, it can be

controlled through measurements, which include:

What are the factors affecting the number of replications? There are several
factors that affect the number of replications.
a. Pattern and magnitude of variability in the experimental units: If the experimental
units are more uniform, lower number of replications is required, and vice versa.
b. Size and differences to be detected: In order to visualize minor differences, more
number of replications is required to maximize the degree of freedom for error and
minimize the error mean squares, which in the final analysis, increases the
calculated F-value.
c. Required level of precision: As the number of replications increases, the precision
also increases up to a certain level, any further increase in number of replication
does not add to increase in precision any more.
d. Number of treatments: As the number of treatments is high, the number of
replications can be reduced, in order to have manageable experimental area. This
is true in areas of plant breeding, at the initial stage of progeny generation. Here,
the target is to narrow down the number of genotypes to a manageable size. For
example, we may use two replications to evaluate 200 progenies out in the field.
e. Availability of resources devoted to the experiment: As the number of replication
increases, other resources used in terms of cost, labor and time also increase.
f. Experimental design used: Certain designs require a fixed number of replications.
For example, in a Latin square design with five treatments, it is necessary to use
five replications to analyze the data as a Latin square experiment. The same is
true with a balanced lattice design, which will be explained in more detail in
Section 4.6.2. Other designs like randomized complete blocks can take any
number of replications, but each treatment should appear at least once in each
block. On the other hand, completely randomized designs, any number of
replications can be used and it is not necessary to have all treatments in each
block, i.e. less restricted.

II. Randomization
Randomization is an underlying the use of statistical methods in experimental design
(Snedecor and Cochran, 1980). The major function of randomization is to provide a
valid or unbiased estimate of experimental error and treatment means by ensuring
random assignment of every treatment to any experimental unit. It is one of the few
characteristics of experimental design and involves the use of some chance device
such as use of random number tables. It is also a precaution against disturbances
that may occur while conducting the experiment (like flooding, shade, etc) and it is
done to ensure independence among observations, which is one of the assumptions
for the analysis of variance.

Systematic application of treatments to experimental units in a non-random but

selected way, results in either underestimation or overestimation of experimental
error. They can also result in inequality of precision in the various comparisons
among the treatment means (Steel and Torrie, 1980). This is true in field experiments

17
 Biometry

where adjacent plots tend to be more alike in productivity than plots which are apart.
Such plots are said to give correlated error components or residuals. Hence, the
precision of comparisons among treatments, which are physically close, is greater
than those among treatments, which are some distance apart. Randomization tends
to destroy the correlation among errors and make valid tests of significance (Gomez
and Gomez, 1984).

With proper randomization, any plot is equally likely to be assigned to any treatment.
In some experimental designs, randomization is restricted in certain ways, but in no
design it is completely eliminated. There are two main reasons for randomization: to
eliminate bias and to ensure independence. Randomization assures that no
treatment is favored or handicapped while assigning to the plots.

In summary, the objective of experiments is to determine if there are real differences

among treatment means and to estimate the magnitude of such differences. A
statistical inference about such differences involves the assignment of a measure of
probability to the inference. Hence, it is necessary to introduce randomization and
replication into the experiment appropriately. Replication ensures means of
computing experimental error and randomization ensures a valid measure of
experimental error for valid test of significance and interval estimates.

2.2.3.2. Methods of controlling error

a. Blocking
Blocking is a designed technique used to improve the precision with which
comparisons among the factors of interest are made. Blocking maximizes variations
among blocks and minimizes variations within blocks. The procedure used is to group
the experimental units into block or groups of homogenous units. Treatments are
then assigned at random to the experimental units within the blocks. The basic goal
of blocking is to remove the block-to-block variation from experimental error. Blocking
increases precision by reducing error, increasing information and making faire
comparison as a result of uniformity in a block (Walpole et al., 2002). Blocks and
replications are not necessarily the same. A block is simply a group of plots, whereas
a replication is a repetition of treatments in an experiment.
b. Refined techniques
This is maintaining uniform application of treatments (like weeding, fertilizer
application, pesticide application, etc), control of external influences (gradients), and
unbiased subjective judgments. The importance of careful technique in the conduct
of an experiment is to ensure uniformity throughout the experiment because no
statistical analysis can improve data from poorly performed experiments. Variation
resulting from carelessness is not a random variation and not subjected to law of
chance on which statistical inferences are based. This variation may be termed as
inaccuracy, in contrast to lack of precision, because precision is concerned with
random variation (Cochran and Cox, 1957).

Faulty techniques may increase the experimental error in two ways (Steel and Torrie,
1980). It may introduce additional fluctuations in a more or less random nature, and

18
 Biometry

possibly subject to the law of chance. Such fluctuations reveal themselves in the
estimate of the experimental error, possibly by the coefficient of variation. If
experimenters find that their estimates of experimental error are consistently higher
than those of others in the same field, they should carefully scrutinize their technique
to determine the origin of the errors. The other way in which faulty techniques may
increase experimental error is through nonrandom mistakes. These are not subject to
the law of chance and may not always be detected by observation of individual
measurements. Faulty technique may also result in measurements that are
consistently biased. This does not affect experimental error or differences among
treatment means, but does affect the values of treatment means. Experimental error
cannot detect bias. The experimental error estimates precision or repeatability, not
the accuracy, of measurements.

Cochran and Cox (1957) pointed out that the principal objectives of a good technique
are: to secure uniformity in the application of treatments, to exercise sufficient control
over external influences so that every treatment produces its effects under
comparable and desired conditions, to devise suitable unbiased measures of the
effects of the treatments, and to prevent gross errors, from which no type of
experimentation seems to be entirely free.

c. Choice of experimental design

The use of proper design which is simple but that can give the required precision is
recommended. Control of experimental error consists of designing an experiment so
that some of the natural variation among the set of units is physically handled so as
to contribute nothing to differences among treatment means. Common sense and
acuity in recognize sources of variation through uniformity trials are seen to be basic
in choosing a design (Steel and Torrie, 1980). Different experimental designs are
available to suit different conditions and detail discussions on experimental designs
are given in Sections 4 to 7.

d. Concomitant measurements
Whenever inconsistency of the treatment over replications occurs, there is a need to
use concomitant observations. One of the techniques for reducing experimental error
is to remove the variability in the dependent variable (Y) associated with some
independent variable (X). This technique is called covariance. It is the use of the
independent variable as a covariate to adjust the treatment means and the
observation is taken simultaneously with the main observation, but not affected by
the treatment.

A good example in plant breeding may be the evaluation of 10 varieties for maize
grain yield, where the varieties varied in maturity from 70 to 120 days. Here, it is not
faire to compare a variety that can mature within 70 days with another variety having
maturity of 120 days, because maturity has direct relation with yield. The probability
that the late variety would be better in yield than the early one is high. Hence, the
better way of comparison is to consider maturity as covariate and analyze the data to
identify the variety that can give the best grain yield per unit time or resource use.

19
 Biometry

The usefulness of covariance analysis can also be done in animal nutrition

experiments. Suppose a researcher would like to compare different formulations of
animal feeds, assuming that there is variation in the initial weights of the animals to
be fed on the new rations. Adjustment can be done to increase the precision in
measuring the treatment effects through the initial weight as a covariate. The
procedures for covariance analysis are dealt in depth in Section 8.

e. Varying size and shape of the experimental units and blocks

As a rule, large experimental units show less variation than small units. However, an
increase in the size of the experimental units often results in the decrease in the
number of replications that can be run; due to a limited amount of experimental
material is usually available for any given experiment.

In field experiments, long and narrow plots and square blocks provide the greatest
precision (Steel and Torrie, 1980). Little and Hills (1978) also reported that
rectangular plots are most efficient in overcoming soil heterogeneity when their long
axis are in the direction of greatest soil variation.

In general, the design used in conducting the experiment determines the size and
shape of plots and blocks. For some types of experiment, the experimental units are
carefully selected to be as uniform as possible to reduce the experimental error. For
example, in studying the effect of fertilizer, herbicide, fungicide or insecticide, it is
more useful to determine how the experimental units respond to increasing doses of
the treatment than to decide whether or not two succeeding doses are significantly
different.

f. Preventing gross errors and mistakes

No design can control gross errors and mistakes committed by the experimenter.
Faulty technique may exaggerate the real experimental error in two ways. It may
introduce additional sources of variation. Such variations can reveal themselves in
the estimate of error on computing the components. A good way of checking such
errors is to cross check with previous research results. If any difference is detected, it
is better to explain it in terms of technical errors.

Faulty techniques could also result from measurements that are consistently biased.
Care and skill in managing measuring devices are important safeguards against such
biases. On top of that, simplified techniques without sacrificing accuracy are a
consideration from time to time.

2.3. Categories of Experiments

In general, there are three categories of experiments. These are preliminary,
demonstrational and critical. Preliminary experiment is meant for testing a large
number of treatments to select the top ranking ones for further investigation. A good
example is evaluation of genotypes generated from plant breeding programmes
planted for observations to narrow down the number of genotypes to be evaluated

20
 Biometry

across locations and/or years. In this kind of experiment treatment, treatments are
unreplicated. Another example is a uniformity trial planted with the same variety to
understand the direction of soil variability in field experiments.

In preliminary experiments, there is no proper experimental design used except for

augmented design presented in Section 4.5, which is used with the help of some
checks at the early stages of a breeding programme.

Experiments that involve demonstrations are performed when a new treatment or

treatments are compared with controls (checks). The treatments are arranged
systematically for ease of comparison. The intention here is to demonstrate end
users or front line extension agents so that they can make use of the new technology
or knowledge in their respective farms.

In a critical experiment, the use of a randomized design is necessary but the choice
of the type of design depends on the objective of the experiment, experimental area
and the number of treatments. In this experiment, treatments are compared and
meaningful differences are detected based on the observations of the responses.
Hence, precision and accuracy are very important elements of critical experiments,
and these are achieved through the use of appropriate experimental designs, which
are illustrated in the sections ahead.

2.4. Hypothesis
Statistical tests start with a statement called a null hypothesis, which is always along
the line ‘there is no difference between treatments’ or ‘there is no relationship
between the measurements’. The structure of the hypothesis testing will be
formulated with the use of the term null hypothesis. The null hypothesis is given a
symbol H0. To test whether the null hypothesis is true or not depends on the
probability level (p value). The p value given by the test tells us the probability of
getting such a result if the null hypothesis is true. A high p value indicates that the
hypothesis could be easily being true and as a result we should not conclude that
there is a difference. A low p value indicates that there is a difference.

The alternative hypothesis is a statement which usually takes the form like ‘there is a
difference between treatments’ and is given the symbol H1 or HA. The two
hypotheses cover all eventualities. A good example is to investigate whether
supplementary irrigation can improve grain yield of maize in a given environment. In
this example, there is a possibility that supplementary irrigation could reduce yield
(like by water logging if there is excess water), otherwise the result of the test will
bias in favor of finding what we hope to find. Therefore, the test should allow for both
possibilities, called two-tailed test (with the two-sided alternative hypothesis). Hence,
the alternative hypothesis is that supplementary irrigation can bring about difference
in grain yield of maize (either increase or decrease). In short, the two hypotheses can
be written as:

H0: null hypothesis

21
 Biometry

HA: alternative hypothesis

In some cases, the only possibilities are: there is no difference (H0) and one
treatment is greater than the other (HA). Taking the same example, adding
supplementary irrigation in area where the rainfall is not sufficient, always improves
grain yield because adding water where there is water stress does not reduce grain
yield of maize. In such cases we use a one-tailed test (with one-sided alternative
hypothesis).

Hence the alternative hypothesis is that supplementary irrigation increase grain yield
of maize. It is not common to use a one-tailed test in biological science. The most
important thing is that null hypothesis does not change but it is only the alternative
hypothesis that changes into either one- or two-tailed test.

While testing the hypothesis, the critical value is used as a reference point to
differentiate the critical region and the acceptance region. The rejection of H0 leads to
the acceptance of an alternate hypothesis. The error committed in the case of
rejecting the null hypothesis when it is true is called a Type-I error. Error committed
in accepting the null hypothesis when it is false is called a Type-II error (Walpole et
al., 2002).

4. ORGANIZATION AND DESCRIPTION OF

DATA
3.1.
3.1 Variable
In the planning stage of an experiment it should be decided which variables will be
measured, how the measurement will be made and what method will be used to
record the effects of treatments and to meet the objectives of the experiment
regardless of the resources. It is necessary to decide at the outset the main response
variable. This is determined by the main objective. This main variable must be
capable of answering the original question and may be derived from a series of
measured variables. Other response variables can be measured to meet specific
objectives and this can also be a derived variable. Variables that are used as part of
the calculations may not need separate analysis, as they could be meaningless on
their own. The most important variables are those that are affected by the treatments
applied in the experiment (Walley, 1991).

After choosing the variables to be measured, it is very important to determine the

frequency and the time of measurements. Measurements should be taken so that the
results obtained will be as widely applicable as possible. The objectives determine
the frequency and time to take the measurements but there are cases where the
timing and frequency of the measurements is determined by the variables
themselves.

22
 Biometry

A data set may range from a few entries to hundreds of or even thousands of them.
Each entry corresponds to the observation of a specified characteristic of a sampling
unit. For example, if an animal breeder would like to compare different breeds of
dairy cows and record their milk yield per day, then the dairy cows are sampling
units, and the data set would consist of measurements of milk yield per day from the
animals as experimental units. Once the data are collected, a primary step is to
organize the information and extract descriptive summary that highlights its salient
features. The following sub-sections will cover steps to organize and describe a set of
data by means of tables, graphs and calculation of some numerical summary
measures.

3.2. Sources of Data

There are many methods by which researchers can get the required data set. Firstly,
they may seek data already published by governmental organizations (ministries,
departments, agencies, etc.) or by non-governmental organization (international
research and development organizations, regional networks, private companies,
etc.). Such sources of data are categorized as secondary sources.

A second method of obtaining data is through designed experiments. In the

experiments, strict control is exercised over the treatments applied to experimental
units (plots). Proper experimental designs are in most cases the subject matter of this
manual and it involves sophisticated statistical procedures but the authors have tried
their best to make them simpler.

A third method of obtaining data is by conducting surveys. Here, no control is

exercised over the entities (behavior of people, disease occurrence, distribution and
intensity on a certain crop, a certain grass weed distribution, insect pest occurrence
on a given crop at a specific location, and so on) being surveyed. They are merely
surveyed and then characterized based on pre-determined classes.

A fourth method of obtaining data is through observations. The researcher makes

observations on a given material in its natural setting. The observations are recorded
usually against a standard reference. This method is mostly done when surveys and
designed experiments are impractical due to either physical impossibility of other
methods (like in animal behavior, astronomy, geology, etc.) or limitation in resources
(early stages of progeny generation in plant breeding where thousands of progenies
are to be evaluated for major pests).

Qualitative data: In discussing the methods for providing summary descriptions of

data, it helps to distinguish between the two basic types: qualitative (categorical) data
and quantitative (numerical/measurement) data. When the characteristic under study
concerns a qualitative trait that is only classified in categories and not numerically
measured, the resulting data are called categorical data. Color, employment status
and blood types are few examples.
Quantitative data: If a characteristic is measured on a numerical scale, the resulting
data consist of a set of numbers and are called measurement data. The term

23
 Biometry

‘variable’ will be used to refer to a characteristic that is measured on numerical scale.

A few examples of numerically valued variables are height, weight and yield. The
variables that can only take integers are called discrete variables. The name discrete
is drawn from the fact that the scale is made up of distinct numbers with gaps. On the
other hand, variables that can take any value in an interval are called continuous
variables.

Discrete quantitative data are numerical responses, which arise from a counting
process, while continuous quantitative data are numerical responses, which arise
from a measuring process.

Measurement is the assignment of numbers to objects or events according to certain

prescribed rules. A number is a symbol that stands for a concept. The number is
intimately related to an idea and thinking of the normal human activity. A concept, in
turn is a collection of critical attributes, that is, a collection of specific characteristics
that serve to distinguish that concept from other similar ones(Orr, 1995).

According to Berenson et al. (1988), there are four kinds of scales with different
levels of precision in treating numbers. These are, in increasing order of precision:
nominal, ordinal, interval and ratio scales.

Nominal scale is the simplest and most elementary type of measurement where
numbers are assigned for the sole purpose of differentiating one object from another.
When numbers are used in a nominal scale, it cannot be added them together, or it is
not possible to calculate an average, because the scale does not have the necessary
properties to do so. Examples of nominal scale can be names of highways like High
way 45, Street 54, etc.

Ordinal scale implies the measurement that has the property of order. Here one
object can be differentiated from the other and the direction of the difference can also
be specified. Statements like ‘more than’ or ‘less than’ can be used because the
measuring system has the property of order. The objects or events can be placed on
a continuum in terms of some measurable characteristics. The ranking of objects or
events can be a good example of an ordinal scale. For example, third year students
of Bioindustry: Iden, Nik and Nasir might be ranked as 1, 2 and 3, respectively, based
on their CGPA, but how much difference that exists among them may not be known.

Interval scale is known for its character to have equality of units. There are equal
distances between observation points on the scale. This scale specifies not only the
direction of the difference, as in the ordinal scale, but also indicates the amount of
the difference as well. The measurement that characterizes education and behavioral
science are of the interval type. For example, most test scores, in terms of number of
items corrected, are based on interval measures.
Ratio scale has all the characteristics of interval scale plus an absolute zero. With an
absolute zero point, statements can be made on ratios of two observations, such as
‘twice as long’ or ‘half as fast’. Most physical scales such as time, length and weight

24
 Biometry

are ratio scales. For example, if Halima can run 5 kilometers in 30 minutes, but it
takes Siti 60 minutes, then it is known that Halima can run twice as fast.

In summary, it is important to be aware of what sort of scale a given body of

measurement represents and the use of statistical procedures that are appropriate
for that particular scale in a given experiment. For example, data obtained from a
qualitative variable are said to have been measured either on a nominal scale or an
ordinal scale. However, interval scale is an ordered scale in which the difference
between measurements is a meaningful quantity, while the ratio scale is meaningful
and equal at all points on a scale that makes it a sensible measurement.

3.3. Sampling Methods

Complete information would emerge only if data were collected from every individual
in the population, which are undoubtedly a monumental if not an impossible task. To
collect data of destructive type can lead to all individual to be eliminated. Thus, the
limitation of time, resources, and facilities, and sometimes the destructive nature of
the study that leads to incomplete information for the fact that the data are collected
in the course of conducting the experiment necessitate sampling.

Sampling is the taking or measuring of more than one observation per experimental
unit. It is not a design but it is an aspect of experimental design. Sampling error
occurs during sample measurements. The sampling error is the variation among
observations taken within experimental units. The main reason for sampling is to
control this variation. It can be done in green houses, under filed conditions, in
laboratories, on life plants like perennial crops, on animals, and so on. Sampling can
be done in one run, at two stages, three stages or so.

The main reason for sampling is to save resources (time, money and efforts). The
second reason for sampling is that, even though part of all the information about the
population is there, the sample data can be useful in drawing conclusions about the
population, with appropriate sampling method and sample size. The third reason for
sampling applies to the special case where the act of measuring the variable
destroys the individual, such as in destructive sampling. Clearly, testing a whole
batch of explosives would be inappropriate, for example.

In some studies, experimental units could be large. In such cases, an appropriate

sample is the one that provides a sample value close to the value that would have
been obtained had all entities in the experimental units (plots) been measured. The
difference between the sample value and the plot value constitutes the sampling
error. A good sampling technique is one that gives a small sampling error.

There are many ways of selecting a sample from a population, but the most important
is random sampling. Indeed, the methods of statistical inference used in this manual
apply only to cases in which the sampling method is random. However, there are
other sampling methods, which are worth mentioning for reasons stated below.

25
 Biometry

3.3.1. Simple random sampling

Simple random sampling is defined as one for which each measurement or count in
the population has the same chance (probability) of being selected. A sample
selected with the probabilities of not getting representative sample for each
measurement or count is said to be a biased sample. For the population of male
student heights, selecting only those who are members of the basketball teams might
result in a biased sample since they would tend to be taller than the average male
students. Random sampling requires that we can identify all the individuals or objects
which comprise the population, and that each measurement or count to be included
in the sample is chosen using some method which ensures equal probability, for
example by the use of random number tables like the one in Appendix 1. Random
numbers are also available on most scientific calculators and on computers.

Where the individuals are objects, which are fixed in a given location, the method of
random sampling can also be used. There are many examples in most fields of study
where such samples are required. One method of obtaining a random sample is to
over lay a plan of the area to be investigated with a rectangular grid, which includes
all points of interest in the area.

3.3.2. Systematic sampling

Systematic random sampling may be used to cut down the time taken in selecting a
random sample. In the student height example, if a 1% sample from 80000 students
was required, then one number can be selected in the range 1 to 100 and derive all
other student numbers to be included in the sample by adding 100, 200, 300, and so
on, giving a sample size of 800 in all. Systematic sampling is suitable in that it
provides a quasi-random sample, so long as there is no periodicity in the population
list or geographical arrangement, which coincides with the selected numbers.

3.3.3. Stratified random sampling

It is a method of sampling where there is a possibility of dividing the populations to be
sampled into homogeneous strata so that a small sample from each stratum will
provide a good estimate of the quantity of interest for the whole of that stratum. The
sample means from each stratum can then be combined to give a good estimate of
the mean of the whole population.

The strata may have quite different distribution for the variable of interest. For
example, for the population of heights of maize varieties, it is necessary to separate
early, intermediate and late varieties.

Stratifying populations is worthwhile if the quantity being analyzed shows consistent

differences between the various strata. In a plot, of a certain trait (yield) of the plants
in the various strata within the plot is only justified if the proportion of the plants in the
various strata represents the proportions the total measurement of the whole plot is
the quantity of the direct interest. The existence of difference between strata within
plots may suggest that the different treatments could have different effects on the

26
 Biometry

plants of different strata and this may justify an investigation of possible treatment
stratum interactions.

3.3.4. Multistage random sampling

Multistage random sampling is characterized by a series of sampling stages and is
suitable for cases in which the sampling unit and unit of measurement are different.
At each stage, a unique sampling unit is used. If for example, a sugar cane
researcher plans to measure internode length, the sampling unit would be the plant.
A multistage random sampling method could be used to identify sample internodes to
under take measurement. A two stage sampling method, each with simple random
sampling is used. The first sampling helps to identify the hills to be used and the
second sampling is to take measurements for the internode length. A simple way of
doing this is to have the plot divided into n number of hills and employ simple random
sampling to get n1 number of hills taken from the plot, and at random again n2
samples would be taken for inter-node length from the selected (n1) hills that will give
the product of the two sample sizes (n1 x n2) per plot.

3.3.5. Stratified multistage random sampling

This sampling method is a combination of stratified random sampling and multistage
random sampling. In this method, first multistage random sampling is applied,
followed by application of stratified sampling on the selected sampling stages. The
sugarcane researcher, given as an example in multistage random sampling,
measured the mean internode length by employing a two-stage sampling method
(hills as primary and internode length as secondary sampling units). There is a great
variation in the internode length, depending on the position of the stem (bottom,
middle and top) within the same hill that would cause the samples estimate less
precise. A good statistical remedy is to use the stratification technique by dividing the
stem into k strata, based on the relative position before taking internode
measurements.

Let the stems in each selected hills are divided into three (bottom, middle and top)
strata based on the internode lengths on the stems, that is, shortest for the bottom,
intermediate for the top and longest at the middle strata and a random sample of
three internodes will be taken from each stratum, then the total number of sample per
plot will be 3 x 3 x number of hills. In this example stratification was applied on the
secondary unit of the two-stage sampling method, but the stratification can be
applied on the primary unit (stratifying the hills depending on a certain gradient) and
on both primary and secondary units.

3.3.6. Cluster sampling

Cluster sampling is a sequential sampling procedure in which a researcher samples a
map and then sample what is in the map (farmers, crops, forests, animals, etc). The
study area is divided into sub categories and sub-sub-categories of different sizes,
with a random number used to pick specific parts of the study area. Each specific
part of the study area has the chance of being selected proportionately to its size.

27
 Biometry

The sequential nature of the sampling process makes it unnecessary to list every
element in the study area. Yet, the random selection at each stage means that every
element has an equal chance of being included in the sample.

3.3.7. Quota sampling

There are so many cases in which a true random sample is difficult to undertake.
One of the alternatives is to do quota sampling, in which a sample is constructed by
filling quotas of certain characteristics that are thought to be true of the population as
a whole. Based on surveys done in the past on the agricultural input requirement, for
example, the Ministry of Agriculture and Rural Development can estimate the number
of farmers who are in various economic categories. Then, input distribution can be
done to farmers who are in the same category at each district of a given state. It can
also be done to each state and district on quota basis. Quota sampling avoids the
drudgery of tracking down a random sample, but gives the person in charge too
much discretion. One mechanical problem is in the verification of the categories, to
draw the line between rich and poor farmers, for example.

3.4. Sample Size

In the planning phase, it is necessary to decide what kind of measurements to be
taken and which sampling techniques are going to be used. The next step is to
decide how many measurements to take to get representative of the population being
dealt with, the sample size. A sampling size or unit is the unit on which actual
measurement is made. Each plot is considered as population and the sampling unit
must be smaller than the plot. The sampling unit varies depending on the
experimental entities (living entities like crops, animals, micro organisms or non-living
things like soils, climate, mechanical bodies, medicine, computer, engine, etc.),
characters to be measured and among cultural practices.

The most common question asked by every investigator who whishes to collect and
analyze data is, ‘how much data should be collected?’ Here, it refers to the number of
observations to be included in the sample as the sample size, so the investigator
should be asking, ‘what sample size should be chosen?’ A sample size of 30 sounds
enough, depending on conditions.

If the mean height of a population of crop variety in an evaluation experiment is to be

estimated then, the required sample size should depend on two factors. First, the
precision required for the estimate, which the investigator must specify. Knowing the
more precision he/she requires, the larger the sample size must be. Secondly, the
variability of height, as measured by its standard deviation is important. The larger
the standard deviation of the height, the larger will be the required sample size.
However, the standard deviation can only be determined when some data are
available. At this point, it is possible to take preliminary data to estimate the standard
deviation.

28
 Biometry

Mead et al. (1993) suggested three possible sample sizes: the first suggestion is to
take 47 units equally from each stratum. The second suggestion is to take 20% of the
units from each stratum (this is called proportional sampling) and the third one is to
take optimal samples from each stratum. The third method would have sample size
proportional to number of samples x standard deviations (nisi). For example, if the
plot to be sampled is stratified into 3 strata, having 200, 400 and 800 number of
samples, then for method-I (equal sampling) 47, 47 and 47 samples will be taken for
each stratum. For method-II (proportional sampling), 40, 80 and 160 will be taken for
each stratum and for method-III (optimal sampling), 40, 160, 80 will be taken for each
stratum. The third method was criticized by Townend (2002) who suggested that the
effect of changing the sample size is to alter the sensitivity or accuracy of statistical
method.

If the cost or effort in sampling is different for the strata then it can be shown that the
sample size should be inversely proportional to the square root of the costs or efforts
involved, as well as to the product of the stratum size and the stratum standard
deviation (Mead et al., 1993).

In designed experiments, simple random sampling is considered and the formula for
the variance to estimate, for example, the total yield of the plot only measures the
sampling precision for the plot. It cannot be used to decide on the number of plants to
be sampled from the plot or it cannot be used to calculate standard errors for
comparing treatment means. These are calculated from analysis of variance of the
estimates of the yield for each plot.

Precision of a sample estimate generally increases with the sample size, the number
of sampling units per plot and the complexity of the sampling design used. But the
increase in any of these three sampling components entails cost that necessitates
the choice of appropriate sampling technique to maintain the balance between
sampling size, size of sampling unit and sampling method to achieve minimum cost.

There is no formula to get the proper combination of these sampling components but
it all depends on the required information, the variability of the character of interest
and the degree of precision to be estimated. In general, data from previous
experiments, ongoing experiments and data from specifically planned sampling
studies can be used as sources of information. In using methods that have been
used in similar way by other workers, a much simpler approach is to follow published
study and draw on their experience. If the intension is to achieve higher precision of a
given results than what has been done, the use of larger sample size is required.

3.5. Data Description

It is often useful to be able to characterize a population in terms of a few well-chosen
statistics. These allow us to summarize possible large numbers of measurements in
order to present results and also to compare population with one another. Under this
Section a very brief summary of commonly used statistics are presented because the

29
 Biometry

descriptive statistics have been covered in the basic statistical courses and are also
commonly available in any text.

3.5.1.Frequency distribution
Basically, the frequency distribution is simply a table constructed to show how many
times a given score or group of scores occurred. We can set up a table where the
highest score is at the top and the lowest at the bottom, with all possible scores in
between, and indicate how often each score occurred. The most common form of the
frequency distribution is the grouped frequency distribution (Table 3.1).

Table 3.1. Group frequency distribution of plant height of a sweet corn population
(BC2-10 MSC1)
___________________________________
Class frequency
___________________________________
156-160 10
161-165 80
166-170 235
171-175 370
176-180 220
181-185 80
186-190 5
___________________________________
Source: Mandefro (2004)

Real limits and apparent limits: The apparent limit is like the one written as 156-
160, 161-165, and so on. To preserve the continuity of the measuring system for
calculations to be discussed later on, there should not be a gap between 160 (the top
score in the 156-160 interval) and 156 (the lowest score in the 156-160 interval). For
this reason, it is understood that the real limits of any interval extended from 1/2 unit
below the apparent limit to 1/2 unit above the apparent upper limit. Thus, the real
limits of the 156-160 intervals are 155.5 and 160.5. The real lower limit is designated
L and the real upper limit is U.

Mid point is the exact center of any interval. The mid point of any interval is found by
adding the apparent upper limit to the apparent lower limit and then dividing by 2. For
156-160 interval, the mid point is (156+160)/2 = 158. Interval size is the distance
between the real lower limit and the real upper limit. In other words, it is the distance
between L and U. For the interval 165-160 for example, i is 160.5-155.5 = 5.

The first step in constructing a frequency distribution from any group of data is to
locate the highest and the lowest scores. The distance between the highest and
lowest score is the range. The next step is to determine i, the size of the interval.
Dividing the range by the number of intervals that will be employed does this. Most
commonly i values of 3, 5, 10, 25, 50, and other multiple of 10 are used (Berenson et
al., 1988). The choice of the number of intervals and the size of the interval is quite

30
 Biometry

arbitrary. The main point is to have the frequency distribution display as much
information as possible concerning the concentrations and patterns of scores. The
top interval should contain the highest score, and the bottom interval begins with a
multiple of interval size (Caulcutt, 1991).

After determining the size i and the number of intervals to use, then it is a matter of
placing intervals in a column labeled scores at the left side and then begin going
through the data, placing a tally mark by the interval in which each score lies. Each
entry in the frequency column is the addition of these tally marks for each interval.

Many quantitative measurements (height, weight, yield, etc.) are normally distributed
in the population. A normal curve has a bell-shaped symmetrical curve. There will be
times when a graph of the frequency distribution will not have the typical, symmetric
bell-shaped with the majority of scores concentrated at the center of the distribution.
Instead the majority of the scores are clustered at either the high end or the low end
of the distribution. This concentration of scores or measurements at one end or the
other of the distribution is called skewness (Caulcutt, 1991).

If the measurements are concentrated at the upper end of the distribution so that the
tail of the curve skews to the left, we say that the curve is negatively skewed. If the
measurements are clustered at the lower end of the distribution so that the tail of the
curve skews to the right, then the curve is positively skewed. Positive skewness is
characterized by a preponderance of low score, such as plants grown in stressed
environments. Another property of a frequency distribution besides the amount of
symmetry is its kurtosis. Kurtosis is the peakdness or flatness of a frequency
polygon. If the curve has a very sharp peak, it indicates an extreme concentration of
scores about the center, it is said to be leptokurtic. If the curve is quite flat, it would
tell that while there is some degree of concentrations at the center, there are quite a
few measurements that are dispersed away from the middle of the distribution, the
curve is called platykurtic. The curve that represents a medium kurtosis is called a
mesokurtic curve.

3.5.2.Histogram
Histograms are vertical bar charts in which the rectangular bars are constructed at
the boundaries of each class. The steps in graphing frequency distribution are: to
layout an area on a paper having the height of the graph about three-fourths of the
width. The horizontal line, called the x-axis or the abscissa, is drawn long enough to
include all of the scores plus a little unused space at each end. This axis is labeled
and put a number of scores at appropriate intervals. At the left end of the x-axis a
vertical line (the ordinate or x-axis) is drawn. Then the x-axis is divided into units so
that the largest frequency will not reach the top of the graph.

Simply, drawing lines parallel to the x-axis at the height of the frequency for each
interval and connecting the lines to the x-axis by vertical lines to the real limits of the
intervals complete the histogram. The histogram is given a title below the figure and
the title should be a clear statement of what the histogram is representing. When

31
 Biometry

comparing two or more sets of data, the various histograms cannot be constructed on
the same graph because superimposing vertical bars of one on another would cause
difficulty in interpretations. For such cases, it is necessary to construct relative
frequency polygons.

3.5.3.Frequency polygon
The polygon is formed by letting the mid point of each class that represent the data in
that class and then connecting the sequence of midpoints. The steps in graphing a
frequency polygon are as follows. First, the area is determined for the graph with the
proper proportions, the x-axis and y-axis are labeled as in histogram. Secondly, a dot
is placed above the mid-point of each interval. After placing the points for all intervals,
the points are connected with straight lines. Thirdly, the curve connecting the points
drops down to the base line at the extreme ends of distribution because it is not
impressive to leave the polygon suspended in midair. The curve touches the base
line at the mid-point of the adjacent interval, whose frequency is zero.

Polygons provides with a useful aid for comparing two or more sets of data. The
properties of central tendency, dispersion and shape can be depicted by comparing
the polygons of particular distributions of data, like normal distribution. Central
tendency and dispersion will be discussed in sections ahead.

3.5.4. Measure of central tendency

A number describing the location of a set of values is a measure of central tendency
or measure of location. In the previous session, frequency distribution along with the
histogram and frequency polygon, are valuable devices that enable us to extract
meaning from a mass of data. The intension here is to have more efficient ways of
expressing the results than a mere picture of the distribution as a whole. Specifically,
measure of central tendency is a means to have a statistical method that would yield
a single value, which could tell us something about the entire distribution. One of
such single value is called measure of central tendency. It is a single value that best
describes the performance of the group as a whole.

The central tendency, dispersion and shape are three major properties that describe
a set of numerical quantitative data. In data analysis and interpretation, a variety of
descriptive measures representing the properties of central tendency, dispersion and
shape may be used to extract and summarize the salient features of the data set
(Rees, 1995).

There are a number of measures of central tendency, all designated to give

representative values of some distribution. The three most common measures of
central tendency are the mean, the median and the mode.

Mean is the arithmetic average of the values. To calculate the mean, all
measurements are added and then be divided by the number of observations.

32
 Biometry

Sum of the values

Mean ( x ) =
number of values

=
∑ xi
n
If for example, yield per plot of five treatments is 150, 200, 250 240 and 260 kilogram
(kg), then the mean yield of the five treatments will be 220 kg, i.e., 150 + 200 + 250 +
240 + 260)/5 = 220 kg. Statistically this can be rewritten as:

x =
∑ xi
n
1100 kg
=
5
= 220 kg, where Σxi is the sum of x-values and n is the No. of values.

In short, a value 220 kg is the best single value that describes yield distribution.
Another useful feature of the mean is that it makes it possible to compare an
individual performance with the rest of the group. A single yield per plot that can be
below or above the mean can be determined. Then, how far below or above the
mean can be seen in terms of deviation or distance from the mean. The deviation can
be obtained by subtracting the mean from each value.

In order to determine the level of precision in a group of data, it is important to know

whether the observations are from nominal, an ordinal, an interval or a ratio scale
because knowing the type of data is appropriate for a given statistic. The mean takes
into account the distances between observations, the measurements from which the
mean is calculated must be at least of the interval type. A mean calculated from
ordinal data (e.g. Mean rank) may be misleading. Mostly, the mean is calculated from
the raw data, data that have not been treated in any organized fashion. However, if
the data are already in the form of a single frequency distribution, the process of
calculating the mean can be simplified greatly.

For example, in comparing 10 genotypes, ear height (cm) data were collected from
40 plants for a genotype (10 plants/replication for four replications) and the data were
grouped as in Table 3.2. In the Table 3.2., the first two columns (scores and
frequency) are the usual classes of frequency distribution. The third column, fx, is
simply the result of multiplying each score by its f value. Hence, the mean is equal to
65.425 cm, i.e., the sum of fx divided by the number of observations, n.

Table 3.2. Ear height (cm) of a sweet corn population,

(BC11-10 x Syn-II)
______________________________________
Score (x) frequency (f) fx
______________________________________
70 1 70
69 4 276
68 6 408

33
 Biometry

67 3 201
66 6 396
65 5 325
64 3 192
63 6 378
62 5 310
61 1 61
_____________________________________
Total 40 2617
_____________________________________
Source: Mandefro (2004)

Median is the value that exactly separates the upper half of the distribution from the
lower half. It is a measure of central tendency in that the median is the point located
in such a way that 50% of the scores are lower than the median and the other 50%
are greater than the median. It is important to not that the mean is the exact center of
the deviations of the scores from the mean whereas the median is the exact center of
the scores themselves. The calculation of the median requires arrangement of the set
of scores in the order of magnitude, usually the highest score on the top. The median
is then found by counting up from the bottom (n+1)/2 scores. The procedure is
slightly different depending on whether the number of scores is odd or even.
When n is odd, the median is obtained by counting up from the bottom (n+1)/2. For
example, for 11 observations (11+1)/2 = 6 that is the 6th observation is the median. If
n is even, the same formula can be used, suppose there are eight observations
(8+1)/2 is 4.5. This means that the median is half way between the fourth and the fifth
scores from the bottom. This would be the average of the two values (value 4 + value
5)/2. Here, the median is not of the actual values in the distribution but by definition,
one-half of the observations are above the median and half are below. The median is
a valuable measure of central tendency for measurements that are only of the ordinal
type.

Mode is the most frequent value. It is categorized as a measure of central tendency,

because a glance at a graph of the frequency distribution shows the grouping about a
central point, and the mode is the highest point in the hump or it is the most frequent
score. The mode is easy to obtain, but it is the crudest measure of central tendency
and is not used as often as either the mean or the median. A distribution can have
two modes, principal and secondary. Principal mode is the most frequent value while
secondary mode is the second most frequent value.

In summary, mean is the most stable of the three measures of central tendency. That
is stability in repeated sampling the means of the samples will tend to vary the least
among them. If a constant is added to each observation then mean will be increased
by the amount of the constant. In negative skewness, the mean is the lowest value of
the three measures of central tendency. In positive skewness, the mean is the
highest value of the three measures of central tendency. Therefore, median is the

34
 Biometry

preferred measure of central tendency when there is a marked degree of skewness

in the distribution.

3.5.6. Measure of dispersion

Measure of dispersion is a second important property, which describe a given data
set. Dispersion is the amount of variation or spread in the data. It can also be defined
as the tendency of individual member of a set of numbers to differ from one another
(Orr, 1995). A measure of central tendency does not give information about the
variability of a group of observations. Hence, it is necessary to have measurements
to precisely know the amount of variability that is present in the distribution of scores.
Two sets of data may differ in both central tendency and dispersion; or may have the
same central tendency but do differ in terms of dispersion.

In dealing with numerical data, then, it is insufficient to summarize those data by

merely presenting some descriptive measures of central tendency. The data can also
be characterized in terms of their dispersion or variability. Four of such measures are
the range, the average deviation, the standard deviation and the variance.

Range is one of the simplest and most strait forward measures of variability. It is the
difference between the highest and lowest scores in the distribution. If the highest
weight of sweet corn is 10 kg per plot and the lowest weight is 5 kg per plot, then the
range will be 10 - 5 = 5 kg per plot.

Range has two serious weaknesses. First, one extreme value can greatly alter the
range. These extreme values are called outliers. The outliers could affect the
outcome of our data analysis. The second weakness is that, since the range is based
on only two measures, the highest and the lowest, it tells us nothing about the pattern
of the distribution. It is this last point that is important in attempting to measure
variability in an accurate way.

If variability is defined as the fluctuation of the scores about the central point, a
method that will tell how much those scores are fluctuating about that point, or how
far they are from the mean is required. That will lead to look at the next method that
helps measure the amount of deviation.

Average deviation is a measure of variability that takes into account how far each
score deviate from the mean, the deviation score (x- x ) would give us this
information. If the values of (x- x ) relatively small, it would indicate that there is less
variability and the scores are near the mean. If on the other hand the values of (x- x )
are large then the scores must be scattered farther from the mean. To get a measure
of variability, it would be necessary only to add the (x- x ) values that give an overall
picture of how much variation there exist.

The value of the average deviation would tell that on average each observation
deviates by that value from the mean. Therefore, the average deviation is a sensible,

35
 Biometry

easily understood and accurate measure of variability. Unfortunately, it is hardly used

because all of the statistical techniques are linked together and that average
deviation does not fit into this framework. It is just to emphasize the use of the
deviations from the mean as a way of indicating the amount of variability.

Standard deviation is the most widely used measure of variability (Sokal and Rohlf,
1987). This statistic makes use of the deviation of each score from the mean, but the
calculation, instead of taking the absolute value of each deviation, squares each
deviation to obtain values that are all positive in sign. When each of the deviations
are squared, positive numbers are obtained whether the original deviations were
positive or negative because when two numbers of the same sign are multiplied
together their product is always positive. The formula for standard deviation (s) is

s=
∑ (x - x ) 2
n -1

What does s tell? It tells in a relative fashion how many of the scores in a distribution
deviate from the mean. If s is small, there is little variability, and the majority of the
observations are tightly clustered above and below the mean. If s is large, the scores
are more widely scattered about the mean. One of the primary uses of s is to
compare two or more distributions with respect to their variability. Standard deviation
is sensitive to the pattern of score in a distribution. The scores having s value of 4.2
are more tightly clustered than scores having s value of 6.9 for another group.

Note that adding a constant number to each score does not affect s (remains
unchanged) but it alters the mean value by the amount equal to the constant. In
computing s using the computational formula, a negative number may be obtained
under the square root. In such cases, a mistake was made somewhere and hence it
is necessary to recheck the work.

Variance is the square of the standard deviation, s. Hence,

s2 =
∑ ( xi − x)2 .
n −1

The sample mean ( x ) and population mean (µ) have been introduced earlier. Here,
the symbol σ (sigma, lower case Greek letter) and its square (σ2) represent the
population standard deviation and variance, respectively. The formula for σ2 is given
below.
2 ∑ ( x − µ )2
σ = .
N
For population variance, the denominator n-1, which was used for sample variance
cannot be used for population because all the elements of the population are used in
the computation.

Standard
Standard deviation and the normal curve

36
 Biometry

In a normal curve, the mean is drawn from the baseline that divides the distribution
into two equal parts. In other words 50% of the scores lie below the mean (to the left)
and 50% above the mean (to the right). Vertical lines can also be drawn from the
base line corresponding to the different σ units so that the area under the curve
between 1σ below mean and 1 σ above the mean is approximately 68% of the total
area. Similarly, about 95% of the distribution lie between -2 and +2 σ units from the
mean, and about 99% of the distribution lie between -3 and +3 σ units from the mean
(Wardlaw, 2000).

The z scores is a way of telling how far a score is from the mean in standard
deviation units. The formula to convert any score (x) into its corresponding z score is
( x − x)
z= ,
s
where x is the observed score and s is the standard deviation of the distribution. If for
example, the following observations are given, then z scores can be calculated for
selected observations (Table 3.3).

Table 3.3. Yield (t/ha) of five inbred lines

evaluated at Melkasa
______________________________
Yield (x) (x- x )2
______________________________
10 2.25
8 12.25
12 0.25
15 12.25
13 2.25
11 0.25
_____________________________
Mean = 11.5 Sum = 29.5
_____________________________

s =
∑ (x - x ) 2
n -1
29.5
=
6 -1
= 2.42

Here, for x = 8, what will be the corresponding z score? Using the above data,
( x − x)
z =
s
(8 − 11.5)
=
2.42
= -1.45

37
 Biometry

In the same way, for x = 15, the z score will be z = 1.45. When a z score is positive, it
is located above the mean, and when negative it is below the mean. if the population
parameter µ and σ are known, the z score can also be calculated as:
(x − µ)
z= .
σ

Another reason for using z score is to make comparisons between different

distributions. Let the yield of a new variety be 78 at Bako, 67 at Awasa, and 57 q/ha
at Melkasa upon evaluation with other varieties. These values do not tell us any thing
about the performance of the new variety in relation to the rest of the varieties. If
these variables are normally distributed in the population, we can make direct
comparisons by using the z score approach. Let the over all mean of the varieties at
each location be 75, 77 and 60 q/ha, respectively. And the standard deviations of the
locations were also given: 6, 12 and 10, in that order. To compare the z score for the
three locations, the z score should be calculated for each location.
(x − µ)
Bako: z =
σ
= (78-75)/6
= 0.5,
(x − µ)
Awasa: z =
σ
= (67-77)/12
= 0.83,
(x − µ)
Melkasa: z =
σ
= (72-60)/10
= 1.2.

Therefore, the new variety did best at Melkasa and poorest at Bako.

The z score is sometimes called a standard score, since it is based on standard

deviation units. The minor disadvantages in using the z score are its negative values
for any score below the mean. The mean of the z score distribution is zero, and the
scores are decimal fractions all of which results in a little computational complexity.
Other standard score systems are present that do not have these disadvantages. In
this manual, highlights of the basic statistics are only presented to make a smooth
flow of the subject. Further explanations can be obtained in Sokal and Rohlf (1987),
Berenson et al., (1988), Caulcutt (1991) and Trumbo (2002).

4. ANALYSIS OF VARIANCE – ONE FACTOR

4.1. Introduction

38
 Biometry

The concept of variation is so fundamental to scientific experimentation that it is

virtually important for anyone who comes in contact with the experimental results,
must appreciate the universality of variation. In field experiments, the sources of
variation and the relative importance of the different causes of variation are often of
interest in themselves. In such experiments, variation tends to obscure the effects of
treatments, making comparison difficult. This in turn can lead to mistakes in
interpreting the results of experiments and to wrong conclusion about the best
treatments to be recommended for commercial use.

To handle such sources of variations, earlier scientists introduced the analysis of

variance techniques. The analysis of variance (ANOVA) was introduced by Ronald
Fisher and is essentially an arithmetic process of partitioning a total sum of squares
into components associated with recognized sources of variations (Steel and Torrie,
1980). It has been used in all fields of research where data are measured
quantitatively.

There are basic questions that the answers will help to decide whether an analysis is
worthwhile or not. First, what are the goals of analysis? There are times when very
little is known about the variables and relationships between variables. Secondly, a
vague goal might be stated, such as learning about the variables and about
relationships between these variables. The vague idea can be developed into new
goal being revamped, which in turn can provide the fundamental thinking behind the
experiment. Thirdly, the vague idea becomes rephrased as more formal questions.
For example, the goal of a research program is to increase yield of rice. Here, the
question is narrowed down to how is the crop best managed to obtain the best yield,
more specifically what types and rates of fertilizer can give the best yield of rice at a
given location? Then the objective will be to determine the level of nitrogen fertilizer
that provides maximum yield of a crop on three improved varieties, for example.

The next question is: can the data set be used to meet the goal of the analysis? This
is a question often overlooked by researchers and data analysts. There are cases
when the variables recorded cannot legitimately meet the goals of the research or the
data are so poorly collected that the analysis is worthwhile. We have to be careful on
the collection of data so that it can be useful in analysis, interpretation and writing
results. Recognize the limitation of our data; we can set limits on our interpretations
of analysis.

The following question is: how can a data set are obtained for analysis? Data must
be collected, compiled and organized before going for analysis. Each variable should
be measured and recorded correctly. The collected data should be formatted the way
the programme for analysis requires. Mistakes in inputting data from data forms to a
computer file must be avoided. After proof reading, error checking and formatting
analysis of variance can be done. What is analysis of variance (ANOVA)?

The ANOVA is a method that can be used to analyze data that resulted from both
simple and complex experiments (Cochran and Cox, 1957). It is one of the most

39
 Biometry

important statistical techniques available and provides a link between the design of
experiments and the analysis of experimental data. The ANOVA has its origin in
biology, or at least agriculture, since the methods were specifically developed to deal
with the type of variable response that is common in field experiments but currently it
has a much wider application (Steel and Torrie, 1980).

In a single factor experiment, there are several independent random samples that
can be used to make inferences about the populations from where the samples are
extracted. Here, the concern is in making inferences about means of the populations
sampled.

The objective of this Section is to enable the reader explain the purpose of ANOVA,
identify the assumptions that underline the ANOVA technique, describe the ANOVA
hypothesis-testing procedure and use the ANOVA testing procedure to arrive at
statistical decision about the means of treatments.

4.2. Completely Randomized Design

Completely randomized design (CRD) is the simplest and least restrictive
experimental design. In CRD the treatments are assigned to the experimental units
without restriction. That is, with CRD every plot is equally likely to be assigned to any
treatment.
The CRD is used where there is relatively uniform experimental site, when there are
missing plots during the course of the experiment in other designs, and when the
number of experimental units is limited to get maximum degrees of freedom for error.

There are a number of advantages with the CRD. It is flexible in terms of the number
of treatments in each replication, simple for statistical analysis, provides maximum
degrees of freedom for error and missing data has no effect on the analysis. The
primary disadvantage is that it gives less precise results in heterogeneous
experimental units. Another disadvantage is that all variations other than that of the
treatment are considered as experimental error.

4.2.1. Layout
There is no restriction on the assignment of treatments to experimental units (plots) in
CRD. The only situation is that treatments should be assigned to plots at random. For
example, for t number of treatments to be assigned in n number of plots where, n > t,
there will be tr number of plots, if each treatment is equally repeated (r). If there are
different repeats for different treatments, then n = ∑ti ri.

There are a number of ways to assign treatments to plots at random. Randomization

(the random assignment of the treatments to plots) can be done by lot method or
using random number table like the one in Appendix 1. In this manual, researchers
are encouraged to the use of random number table for randomization.

40
 Biometry

The randomization process can be shown as follows, taking an example of

experiment with three treatments (A, B and C) each repeated four times. Here, there
will be n = t x r = 3 x 4 = 12 plots. The plots can be labeled as in Figure 4.1.

1 2 3 4
5 6 7 8
9 10 11 12
Figure 4.1. Field plot plan for 12 plots in CRD

Then, randomization is applied using the random number table as follows. First, a
starting point is selected anywhere in a table of random numbers (Appendix 1).
Secondly, the first 12 digit numbers are written by moving up or down, left or right.
Here, it is better to use three digit numbers for the reason that they are less likely to
contain the same number than when numbers with fewer digits are used. Thirdly, the
numbers are ranked from smallest to largest as in third column of Table 4.3. These
ranks correspond to plot numbers on the field plot plan. Fourthly, the first treatment is
assigned to the first four plots in sequence, the second treatment to the second four
plots and the third treatment to the remaining plots.

The resulting randomization would look like Table 4.3 and the master field plan would
be as in Figure 4.2.

Table 4.3. Assignment of treatments using random number table

Sequence Random number Rank

(plot number) Treatment
1 481 5 A
2 516 6 A
3 991 12 A
4 062 1 A
5 804 11 B
6 675 9 B
7 154 2 B
8 437 4 B
9 571 7 C
10 769 10 C
11 639 8 C
12 428 3 C

41
 Biometry

(1) (2) (3) (4)

A B C A
(5) (6) (7) (8)
A A C C
(9) (10) (11) (12)
B C B A
Figure 4.2. Master field plan with treatments randomly assigned to plots. () = Plot number

4.2.2. Analysis
The analysis can be started by calculating the total amount of variability in the data,
irrespective of the treatments. The best way to do this is to calculate a sum of
squares (SS) as a measure of variability. That is the total sum of squares (Total SS).

Recalling that a sum of squares (SS) is calculated from ∑(x - x )2, if there is a need to
calculate the Total SS, the mean to be used in this case will be the grand mean of all
of the data. Detailed calculations are shown later. The total variability, measured by
the Total SS, can be partitioned into two components: the explained and the
unexplained variability. The explained variability is that represented by differences
between the treatments. This variability is explained because there is a potential
explanation for it, i.e. the treatments applied to the experimental units were not the
same. The unexplained variability is that within the treatments, i.e. differences
between replicates. This variability is not caused by the experimental treatments.
Hence, it is often called experimental error. Thus, the following important relationship
holds:

Total SS = Explained SS + Unexplained SS

Since the values of two elements in the summation are known, the third value can be
obtained by subtraction. If Total SS = Explained SS + Unexplained SS then
unexplained SS = Total SS - explained SS.

A test statistic is the ratio of explained to unexplained variability, i.e.,

explained/unexplained, but the SS cannot be used directly in the calculation of the
appropriate test statistic. This is because the magnitude of a sum depends, to some
extent, on the number of items summed. This can be corrected by dividing it to the
number of items in the summation. If a SS is divided by n, a variance or mean square
can be obtained. This is where the name of the method comes from. The test statistic
to be calculated is F (named after Fisher who developed the method).

F = Explained MS/Unexplained MS

The terms Treatment and Error are often given other names depending on the
authors’ notation. So it is advisable to provide readers with different terms to mean
the same concept. Treatment is synonymous with factor or between and error is
synonymous with residual or within. There are few basic computations to be done.

42
 Biometry

These are estimating the treatment effects, standard error of the mean or difference
and test for significant differences among the treatments. These can be done
following the procedure of the ANOVA. In this procedure, the components should be
explained by linear model at the beginning.

The linear additive model (L.A.M.) for CRD is Xij = µ + Ti + Eij where, Xij = the jth
observation of ith treatment, µ = the overall mean, Ti = ith treatment effect (µi-µ) and Eij
= the effect of jth observation of ith treatment. j=1…r, I=1…t.

The following assumption must be made concerning the random error in order to use
the one-way ANOVA technique:
a. Each treatment population has the same variance, that is, the random errors
have the same variance in experiments designed to compare treatment
means,
b. The error terms are statistically independent, that is, the magnitude of the error
in one observation is not influenced by the magnitude of the error in an other
observation,
c. The errors are normally distributed. These assumptions can be abbreviated as
follows and the abbreviations will be used here after.
Assumptions
Ti ~ NI (0, σ2t)
Eij ~NIR (0, σ2)

Practical situations occur where one or more basic assumptions of the ANOVA are
violated. For example, the assumption of independence is not true when the errors
are correlated. The most common causes of positive correlation between errors are
time and space trends in experimental materials, environmental variables or tools
(Steel and Torrie, 1980). Negative correlation may also occur in the case of
compensating errors and is seen in data such as those taken for successive batch
yields from multistage process (Cochran and Cox, 1957). The elimination of
correlated errors is best achieved by removing the sources of correlation. This can be
achieved through randomization in sampling to average out the trends and allow us
the ANOVA technique (Sokal and Rohlf, 1987).

In many cases the equal-variance assumption does not hold as a result of correlation
between the mean and the standard deviation. This relationship can be eliminated
and the variance made uniform by a simple data transformation (Cochran and Cox,
1957). Correlation between the mean and standard deviation is frequently found in
samples exhibiting non-normality. If the errors are not normally distributed the true
probability of accepting the hypothesis that there is no treatment effect, is not equal
to the probability given by significant tables.

Most statisticians agree that one or more assumptions are violated to some degree
(Cochran and Cox, 1957; Little and Hills, 1978; Steel and Torrie, 1980; Peterson,
1985 and Sokal and Rohlf, 1987). The violation in a given research setting affects the
validity of inferences under some statistical procedures more severely than others.

43
 Biometry

The primary interest here is the validity of inferences under the ANOVA, because the
assumptions are the basic tool of analyzing data from designed experiments. Thus,
the concern is whether or not the underlying assumptions of the ANOVA
methodology are violated to the extent of appreciably changing the error rates.
Several authors reported on the validity of inferences from the ANOVA and their
report can be summarized as follows:
a. The classical ANOVA F-test is very robust with respect to normality.
b. When there are equal replications, mild departure from homogeneous
variance has little impact on the validity of the ANOVA test.
c. Unequal variances have a greater impact on estimates of treatment effects
and their variances than on tests.

Suppose an experiment was conducted with t number of treatments each replicated r

times. Then the analysis is done by setting up a table of observation, computing
treatment total (Ti), grand total (GT) and grand mean (GM). The summarized data
can be done as in Table 4.4.

Table 4.4. Data summary for an experiment conducted in CRD

Replications
Treatment 1 2 3 …… r Treatment Total (Ti)
1 X11 X12 X13 …… X1r T1
2 X21 X22 X23 …… X2r T2
3 X31 X32 X33 …… X3r T3
. . . . . .
. . . . . .
. . . . . .
t Xt1 Xt2 Xt3 …… Xtr Tt

Grand Total GT

The next step is to set up analysis of variance table as in Table 4.5a to 4.5d. In the
ANOVA tables, the total variation is partitioned into two components: variation due to
treatment and variation within treatment (experimental error/residual). The degree of
freedom for each component is one less than the number of observations. That is t-1
for treatment, tr-1 for total and t(r-1) for error with equal repeats or n-t for unequal
number of repeats.

The computations can be done in the following order:

1. Correction factor, C.F. as CF = (GT) 2/n
2. Total sum of squares, Total SS as Total SS = ∑i∑j Xij2 –C.F.
3. Sum of squares due to treatment, SSt = (∑Ti 2)/rj – C.F.
4. Sum of squares due to error, SSE = Total SS – SSt
5. Mean square of treatment, MSt = SSt/(t-1)
6. Mean square of error, MSE = SSE/(n-t)
7. F-calculated for treatment = MSt/MSE

44
 Biometry

The computations are easier if the treatments are equally replicated. Both conditions
will be explained by taking treatments with equal and unequal replications, for the
reason that CRD with different number of replications is not commonly found in text
books.

Table 4.5a. Analysis of variance for CRD where treatments have different replication

Source of Sum Mean

Variation d.f Squares (SS) Squares F-cal
Treatments t-1 SSt MSt MSt/MSE
Error n-t SSE MSE

Total n-1 Total SS

Table 4.5b. Analysis of variance for CRD where treatments have equal replications

Source of Sum Mean

variation d.f Squares (SS) Squares F-cal
Treatments t-1 SSt MSt MSt/MSE
Error t(r-1) SSE MSE
Total rt-1 Total SS

Table 4.5c. Expected mean square for CRD of equal replications (Random model)

Source of Expected
variation d.f. Mean Squares E(MS)
Treatments t-1 σ2e + rσ2t
Error t(r-1) σ2e

Table 4.5d. Expected mean square for CRD of equal replications (Fixed model)

Source of Expected
variation d.f. Mean Squares E(MS)
Treatments t-1 σ2e + r∑(xi- x ) 2/t-1
Error t(r-1) σ2e

45
 Biometry

After the ANOVA is done, F-calculated is tested against the tabulated F-value at the
desired level of probability. F = MSt/MSE. F-values, unlike t-values, have 2 parts to
the degrees of freedom: n1 = treatment d.f.; n2 = error d.f. F-tables are arranged with
the treatment degrees of freedom horizontally and the error degrees of freedom
vertically as in Appendix 4a and 4b.

If the F-calculated is greater than F-tabulated, then it will be concluded that there is a
significant treatment effect. Comparison between calculated and tabulated F-vales
will lead to either rejecting or accepting the null hypothesis. The hypothesis can be
framed as H0: all treatments have equal means and the alternate hypothesis as H1: at
least one treatment mean is different from other treatment means. On the other hand
the MSE = s2 is the sample estimate of experimental error that can be used to
compute standard errors and interval estimates. The key issue here is that standard
errors are computed and intervals will be estimated, if the null hypothesis (H0,) is
rejected, that is, the Mean Squares of treatment are found to be significant.

4.2.3. Standard errors

There are two standard errors that help estimate the critical or least significant
differences (CD/LSD). These are standard error of the mean, SE(m) and standard
error of the difference, SE(d).
MS E MS E
Standard error of mean, SE(m) = or unequal & equal r, respectively,
ri r
2 MS E 2MS E
Standard error of difference, SE(d) = or and
ri r
Critical difference, CD or LSD = SE(d) x t0.05 at error degree of freedom.

4.2.4. Mean separation

There are different kinds of mean separation procedures. Mean separation
procedures are designed to make statistical inferences concerning a given set of
treatment means. There are two categories of mean separation procedures,
Category I and Category II. The first category consists of a group of studentized
range-based procedures to test all possible pairs of mean differences. This
procedure can be subdivided into multiple comparisons (LSD and Tukey’s test) and
multiple range tests (Student-Newman-Keuls’ test and Dancun’s New Multiple range
Test). Under this category, for an experiment with t number of treatments, t(t-1)/2
possible number of comparisons can be done. In multiple comparison method, a
single critical value is used while in multiple range test procedure two or more critical
values are used.

The purpose of the second category of mean separation is to test mean difference
between any treatment and a specified treatment usually a control or standard check.
In this case, there are only t-1 paired comparisons (Peterson, 1977). This category
includes preplanned contrast and preplanned comparison to a check or control.

a. Least significant difference method

46
 Biometry

The least significant difference (LSD) procedure, also known as Fisher’s LSD,
compares all possible [t(t-1)/n] pairs of standard t-test for t treatment means. The
LSD procedure is only used when the treatment source of variation is found to be
significant by the F-statistic. The purpose of testing each mean difference is to test
the null hypothesis that the corresponding two treatment means are equal.

The critical value for conducting each test is the same for each individual comparison
of paired means, and it depends on the number of degrees of freedom for the error,
as well as the significance level of the test, α. It should be noted that α reflects the
probability of a Type-I error for each individual comparison, and therefore cannot be
applied to the entire group of individual comparison since these are not mutually
independent. Thus, the t-tests of all possible hypotheses are not independent.

The LSD method differs from others in that it has a comparison-wise Type-I error, α,
over all repetitions of the experiments (Steel and Torrie, 1980). This means that the α
risk will be inflated when using the LSD procedure. As the number of treatments
increases, the Type-I error for experiment becomes large. Due this fact, a situation
may arise whereby the F-statistic in the ANOVA is significant, yet the LSD procedure
fails to find any pair of treatment means which differ significantly from one another.
This occurs since the F-statistic is considering all possible comparisons between
treatment means simultaneously, not in a pair-wise manner as the LSD procedure
does (Cochran and Cox, 1957).
2MS E
LSD = t(1- α)/2
r
Assuming two-sided alternatives, the pair of means would be declared significantly
different if the difference between the two means is greater than the LSD value.
2MS E
LSD = (tα/2, error d.f.) for equal replication and
r
1 1
LSD = (tα/2, error d.f.) 2MS E ( + ) for unequal replication.
ri rj

Taking five treatments, for example, the process can be explained as T1, T2, T3, T4
and T5 with mean values of 10, 11, 15, 17 and 22, respectively, with α= 0.05, error
degrees of freedom 20, and MSE for the five treatments = 8.06, then the critical
difference = 4.37. From this statistics one can see that treatment 5 had the highest
mean and two pairs of means that do not significantly differ from each other are T1
and T2 and T3 and T4.
T1 T2 T3 T4 T5
9 10 16 17 22

The overall α risk may be considerably inflated using this method. Specifically, as t
getting larger, the type I error of the experiment becomes larger (Steel and Torrie,
1980).

47
 Biometry

b. Tukey’s Test
Tukey (1953) proposed a procedure for testing hypotheses for which the overall
significance level is exactly α when the sample sizes are equal and at most α when
the sample sizes are unequal. His procedure can also be used to contract confidence
intervals on the differences in all pairs of means. For these intervals, the
simultaneous confidence level is 100(1- α) percent when the sample sizes are equal
and at least 100(1- α) percent when the sample sizes are unequal. This is good
procedure when interest focuses on pairs of means. Tukey’s procedure makes use of
the distribution of studentized range statistic

(Maximum mean minimum mean)

q=
MS E
r

Appendix Table A.8 of Steel and Torrie (1980) contains values of qα (p, d.f.), the
upper α and percentage points of q where d.f. is degrees of freedom for error. For
equal sample size, Tukey’s test declares two means significantly different from each
other, if the absolute value of their sample differences exceeds Tα.

MS E
Tα = qα(t, d.f.) .
r

In the same way, we can construct a set of 100(1- α) percent confidence intervals for
all pairs of means. For the previous example, with means 9, 10, 16, 17 and 22, with
α= 0.05 and error degrees of freedom 20, there will be q0.05 (5, 20) = 4.23 and if the
MSE for the five treatments = 8.06, then
8.06
qα (5, 20) = 5.37.
5

Hence, any pair of treatment means that differ in absolute value by more than 5.37
would imply that the corresponding pair of population means is significantly different.

T1 T2 T3 T4 T5
9 10 16 17 22

c. Duncan’s New Multiple Range Test (DNMRT)

A widely used procedure for comparing all pairs of means is the DNMRT. To apply
DNMRT for equal sample size, the t treatment means will be arranged in ascending
order and the standard error of each mean is determined as

48
 Biometry

MS E
SE(m) = .
r

From Duncan’s Table of significant ranges, Appendix Table A.7 of Steel and Torrie
(1980), rα (p, d.f.) values are obtained for t = 2, 3, …, t, where α is the significant level
and d.f. is error degrees of freedom. This ranges will be converted into a set of t-1
least significant ranges (Rp) for p = 2, 3, …, t by calculating Rp).

Rp = rα(p, d.f.) SE(m) for p = 2, 3, …, t.

Then, the observed differences between means are tested, beginning with the largest
versus smallest, which would be compared with the least significant ranges Rt. Next,
the difference of the largest and second smallest is computed and compared with the
least significant range Rt-1. Such comparisons are continued until all means have
been compared with the largest mean. Finally, the difference between the second
largest mean and the smallest is compared against the least significant range Rt-1.
This process is continued until the difference between all possible n(n-1)/2 pairs of
means have been considered. If the observed difference is greater than the
corresponding least significant range, we conclude that the pair of means in question
is significantly different. To prevent contradictions, no differences between a pair of
means are considered significant if the two means involved fall between two other
means that do not differ significantly.

DNMRT can be applied on the previous example. Recalling that MSE = 8.06, n = 5
and error degrees of freedom = 20, then the treatment means can be ordered in
ascending order as T1=9, T2 = 10, T3 = 16, T4 = 17 and T5 = 22.

MS E
The SE (m) =
r
8.06
=
5
= 1.27.

From the table of significant ranges in Appendix Table A.7 (Steel and Torrie, 1980)
for 20 degrees of freedom and 0.05 level of probability, we obtain r0.05(2, 20) = 2.95,
r0.05(3, 20) = 3.10, r0.05(4, 20) = 3.18 and r0.05(5, 20) = 3.25. Hence, the least
significant ranges are
R2 = r0.05(2, 20)SE(m) = 3.75
R3 = r0.05(3, 20) SE(m) = 3.94
R4 =r0.05(4, 20) SE(m) = 4.04
R5 = r0.05(5, 20) SE(m) = 4.13

The comparison would produce that there are significant differences between all
pairs of means except treatment 1 and 2, and 3 and 4. In this example, DNMRT and

49
 Biometry

the LSD method produced the same result that leads to identical conclusions. DMRT
requires a greater observed differences to detect significantly different pairs of means
included in the group. For two means the critical value of DNMRT and the LSD are
exactly equal. Generally, DNMRT is quite powerful when n is getting larger; that is, it
is very effective at detecting differences between means when real differences exist
and for this reason it is popular.

d. The Newman Keuls Test

Newman the procedure is usually called the Student-Newman-Keuls’ test.
Operationally, the procedure is similar to DNMRT, except that the critical differences
between means are calculated somewhat differently. Specifically, we compute a set
of critical values Kp = qα(p, d.f.) SE(m), p = 2, 3, …, t where qα(p, d.f.) is the upper
the α percentage point of the studentized range for groups of means of size p and
d.f. error degrees of freedom. Once the value of Kp is computed, extreme pairs of
means in groups of size p are compared with Kp exactly as in DNMRT.

Which comparison method is the best?

A logical question at this point is which one of these methods should be used?
Unfortunately, there is no clear-cut answer to this question, and professional
statisticians often disagree over the utility of the various procedures. Carmer and
Swanson (1973) have conducted simulation studies of a number of multiple
comparison procedures, including others not discussed here. They reported that the
least significant difference method is a very effective test for detecting true
differences in means if it is applied only after the F test is significant in the analysis of
variance. They also reported good performance in detecting true differences with
DNMRT.

Some statisticians use Tukey’s method as it does control the overall error rate. The
Newman-Keuls Test is more conservative and the power of the test is less than
DNMRT. There are other multiple comparison procedures and further references can
be obtained in Steel and Torrie (1980) and other texts of experimental designs.

Therefore, the comparison between the test methods can be summarized by

suggesting the use of either LSD or DNMRT suffice the present knowledge. Besides,
latest statistical packages like SAS are programmed to produce results of mean
comparison by these two statistics (LSD and DNMRT).

4.2.6. Numerical examples

Example-I: The same repeats

A company would like putting together a computer system for sale to business. Upon
manufacturing, they do not wish to take time to develop a laser quality printer (LQP)
to go with their system. Hence, they planned to find best reasonably priced LQP
available to subcontract as a system option. To determine difference between
machines, six of the requested models (replications) were taken from three randomly

50
 Biometry

chosen manufacturers (treatments). The data in hours to first failure is given below
(Table 9) for the three models (manufacturers’ printers).

Table4.9. Data on first failure in LQP of three manufacturers

I II III IV V VI Treatment Total (Ti)

Model-I 60 45 72 68 71 52 368
Model-II 102 96 105 99 103 95 600
Model-III 121 132 118 128 131 126 756
Grand Total 1724

In this example, the shortcut methods will be introduced as they are of practical
utilities. The stepwise computations of the different components are as follows.

1. Grand total (GT) = sum of all observations for a particular observations.

GT = 368 + 600 + 756
= 1724
Once, the grand total is obtained, the next step is to compute correction factor.
(GT ) 2
2. Correction factor (C.F.) = where, r = number of replication and
rt
(1724) 2
= t = number of treatment
6 x3
=165120.89

3. Total sum of squares (Total SS) = SS of each observation – C.F.

Total SS = ∑∑Xij 2 –C.F.
= (60)2 +(45)2 + … + (126)2 – 165120.89
= 13547.11

4. Sum squares of treatment (SSt) = (SS of each treatment) – C.F.

SSt =
∑ Ti 2 – C.F.
r
[(368) 2 + (600) 2 + (756) 2 ]
= - 165120.89
6
=12705.78
5. Sum squares of error (SSE) = Total SS - SSt
= 13547.11 - 12705.78
= 841.33

Table 4.10. Analysis of variance for CRD with three treatments and of six repeats

Source of

51
 Biometry

Variation d.f. SS MS F-cal F-tab0.05 0.01

Treatments 2 12705.78 6352.89 113.26** 3.68 6.36
Error 15 841.33 56.09

**, Significant at 0.01 level of probability

There is a significant effect of the treatments (there is significant difference among

the models) at 0.01 level of probability (Table 4.10). Hence, the hypothesis that says
all treatments (models) have the same mean values (hours to first failure) is rejected.
The computation for standard error of means, standard error of difference and the
critical values are the same as in the pervious example.

The formula to be used will be the one with equal number of repeats. A common
standard error of difference will be calculated to serve all treatment means, but in the
previous case where there were different repeats, a standard error of difference for
each treatment means was computed.

6. Standard error of means, SE(m)

MS E
SE(m) = ±
r
56.09
=±
6
= 7.489
7. Standard error of differences, SE(d)
2MS E
SE(d) = ±
r
2x56.09
=±
6
= 10.591
8. Critical difference, CD or Least significant difference (LSD)
CD = SE (d) x t-at 0.05
= 10.591 x 1.753
= 18.566
9. Mean differences
Mean of Mode-l - Mean of Model-II = 61.33 - 100 = - 38.67
Mean of Mode-l - Mean of Model-III = 61.33 - 126 = - 64.67
Mean of Mode-Il - Mean of Model-III = 100 - 126 = - 26
Treatment means can be compared as in Table 4.11.

Table 4.11. Mean comparison

Paired treatments Mean difference CD/LSD values Status

Mode-l and Model-II 38.00 18.566 *

52
 Biometry

Mode-l and Model-III 64.67 18.566 *

Mode-lI and Model-III 26.00 18.566 *

The conclusion from this mean comparison is that all models are statistically different
where the highest mean first failure value was due to Model-III. Hence, the company
can go for Model-I system option to develop a laser quality printer as it has lowest
mean value of first failure.

4.3. Randomized Complete Block Design

4.3.1. Introduction
Randomized complete block design is one of the most widely used designs. The
principle behind this design is that the experimental area is divided into groups called
blocks, each block containing all treatments. The term ‘block design’ originated from
the design of agricultural field experiment, where ‘block’ refers to a group of adjacent
plots. The essence of blocking is to minimize the within block variation, experimental
error. The block simply represents one restriction on complete randomization due to
the environment in which the experiment is conducted. When the experiment is
conducted a uniform technique should be followed for all experimental units within
the block as suggested in section 2.2.3.2b. In field experiments, the shape of the
block is recommended to be nearly square as much as possible. This kind of shape
can help us to get strong correlation among plots.

The principal advantages of RCBD are: it is more precise than CRD when block
effect is significant, any number of treatments and replications can be included, the
statistical analysis is easy, and it provides information on the uniformity of
experimental units. This design is one of the most frequently used designs as it
provides a required precision with limited cost.

In this design, if each treatment is applied to exactly one unit in the block and
comparisons are only drawn between treatment responses from the same block,
extraneous variability should be reduced. This is the concept underlying the
randomized complete block design.

A few typical examples for which the RCBD may be appropriate are performance
evaluation of newly developed varieties against the standard check; a specific
fertilized rate determination trial on a specific crop; animal nutrition experiment to
identify the best ration (using body weight or age as blocks); clinical trials to compare
several competing drugs (using age or severity of symptom as blocks); psychological
experiments comparing several stimuli (using socioeconomic background as a block);
comparison of several techniques for storing fruits or vegetables (using each
incoming shipment as a block); and so on.

4.3.2. Layout
Randomization is a basic part of the RCBD. After the experimental area is grouped
into blocks and plots, the treatments are assigned at random to the plots within each

53
 Biometry

block. The same procedure is repeated with a new randomization for each of the
remaining blocks.

Suppose there are six treatments to be evaluated in RCBD with four replications
under field condition. Then, the field plan can be constructed as follows. Assuming
the gradient as soil fertility that runs from top to bottom, blocking is done
perpendicular to the gradient, in this case left to right as shown in Figure 4.3 and the
master plan can be done as in Table 4.17, respectively.

Soil fertility
High Block-I
Plot 101 102 103 104 105 106
Block-II
Plot 201 202 203 204 205 206
Block-III
Plot 301 302 303 304 305 306
Block-IV
Plot 401 402 403 404 405 406
Low
Figure 4.3: Field plot plan for evaluating six treatments in RCBD

Table 4.17. Master plan for evaluating six treatments

Plot number
Treatment Replication-I Replication-II Replication-III Replication-IV
T1 102 205 303 404
T2 105 203 306 402
T3 103 201 305 401
T4 101 204 301 406
T5 106 202 304 403
T6 104 206 302 405

4.3.3. Analysis
Once the data are obtained, they can be arranged in a two-way table, where rows
represent treatments and columns represent blocks or vice versa. Depending on the
size of the block used, there are two basic types of randomized block designs:
complete block design and incomplete block design. Complete block design refers to
a situation where each block contains all the treatments while in the case of
incomplete block designs at least the size of one block is less than the number of
treatments in the experiment. As the incomplete block designs are illustrated in
Section 4.6, the discussion is limited to RCBD in this Section.

If the measurements that corresponds to treatment i and block j are designated by xij,
the data structure of the RCBD with b block and t treatment is shown in Table 4.18.
The arrangement of the data depends on personal preference. The column can be

54
 Biometry

assigned for the treatments and the rows for the blocks. If the numbers of treatments
are more than the number of replication it is better to use the rows for treatment and
column for blocks.

Table 4.18. Data structure of a RCBD with b blocks and t treatments

Treatment Block Ti Mean

1 2 . . . b
Treatment 1 x11 x12 . . . x1b T1 M1
Treatment 2 x21 x22 . . . x2b T2 M2
Treatment 3 x31 x32 . . . x3b T3 M3
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Treatment t xt1 xt2 . . . xtb Tt Mt
Block total B1 B2 . . . Bb GT GM
Block mean Mb1 Mb2 . . . Mbb

1. Correction factor (C.F.) = (GT)2/rt where, r= number of replication and

t = number of treatment
2. Total sum of squares (Total SS)
Total SS = ∑∑Xij 2 –C.F.
3. Sum of squares due to treatment (SSt)
SSt = (∑Ti 2)/r – C.F.
4. Sum of squares due to block (SSB)
SSB = (∑Bj 2)/t – C.F.
5. Sum of squares due to error (SSE) = Total SS – SSt - SSB
6. Mean squares of block (MSB) = SSB/degrees of freedom for block
7. Mean squares of treatment (MSt) = SSt/degrees of freedom for treatment
8. Mean squares of error (MSE) = SSE/degrees of freedom for error
9. F-calculated (block) = MSB/MSE
10. calculated (treatment) = MSt/MSE (Table 4.19a and 4.19b)
11. F-tabulated (0.05) for block = F-value in Appendix 4a, at (b-1) and (b-1)(t-1)
12. F-tabulated (0.05) for treat. = F-value in Appendix 4a, at (t-1) and (b-1)(t-1)
13. F-tabulated (0.01) for block = F-value in Appendix 4b, at (b-1) and (b-1)(t-1)
14. F-tabulated (0.01) for treat. = F-value in Appendix 4b, at (t-1) and (b-1)(t-1)

Table 4.19a. ANOVA Table for a RCBD with b blocks and t treatments

Source of
variation d.f. SS MS F-cal
Blocks b-1 SSB MSB MSB/MSE
Treatments t-1 SSt MSt MSt/MSE
Error (b-1)(t-1) SSE MSE

55
 Biometry

Table 4.19b. Expected mean square for a RCBD with b blocks and t treatments

Source of
variation d.f. E(MS) Random E(MS) Fixed
Blocks b-1 σ2 + tσR2 σ2 + t(Rj2)/(r-1)
Treatments t-1 σ2 + rσT2 σ2 + r(Ti2)/(t-1)
Error (b-1)(t-1) σ2 σ2

4.3.4. Numerical examples

Example-I. Analysis of variance with short cut method
Six populations of maize were evaluated for 100-grain weight in RCBD in Serdang,
Selangor. Data for 100-grain weight were collected and provided as in Table 4.23. All
the cultural practices were applied uniformly for all the populations following the
recommendation for Serdang and the assumptions for analysis of variance are
fulfilled. Then the data will be analyzed as follows. To make the computation easier,
treatment total (Ti), replication total (Rj), grand total (GT) and grand mean (GM), are
added in Table 4.23.

Table 4.23. Data from six populations of maize evaluated for 100-grain weight in
Serdang, Selangor.

100-grain weight (g)__________ Treatment

Replication Total Treatment
Population I II III IV (Ti) Mean
Suwan-I 27.0 30.1 26.4 31.3 114.8 28.700
Synthetic GxA 30.4 31.1 28.9 27.7 118.1 29.525
Hercules 28.1 29.0 31.0 26.2 114.3 28.575
Red Iron 45 23.1 28.2 24.1 27.0 102.4 25.600
Venus 49 27.3 27.0 24.0 28.3 106.6 26.650
MARDI Composite-I 23.1 21.4 20.1 23.3 87.9 21.975
Replication Total (Rj) 159.0 166.8 154.5 163.8 644.1 26.8375
GT GM

The different components of variations in RCBD are computed as follows.

1. The linear additive model (L.A.M.): Xij = µ + Ti + Bj + Eij where, Xij = the ith
treatment effect in jth block, µ = the overall mean, Ti = ith treatment effect (µi-µ), Bj
is jth block effect (µj-µ) and Eij = the effect of ith treatment in jth block. j=1…r,
i=1…t.
Assumptions
Ti ~ NI (0, σ2t)

56
 Biometry

Bj ~ NI (0, σ2B)
Eij ~ NI (0, σ2)
(GT ) 2
2. Correction factor (C.F.) = where, r= number of replication and
rt
(644.1) 2
= t = number of treatment
4 x6
=17286.03375
3. Total sum of squares (Total SS)
Total SS = ∑∑Xij 2 –C.F.
=(27.0)2 + (30.4)2 + … + (23.3)2 – 17286.03375
=225.856
4. Sum of squares due to treatment (SSt)
2

SSt =
∑ Ti
– C.F.
r
[(114.8) 2 + (118.1) 2 + ... + (87.9) 2 ]
= - 17286.03375
4
=155.68375
5. Sum of squares due to block (SSB)
2

SSB =
∑ Bj
– C.F.
t
[(159) 2 + (166.8) 2 + ... + (163.8) 2 ]
= - 17286.03375
6
= 14.62125
6. Sum of squares due to error (SSE) = Total SS-SSt-SSB
= 225.856-155.68375-14.62125
= 55.55129
7. Degrees of freedom for treatment (d.f. for t)
d.f. for t = number of treatment (t)-1
= 6 -1
=5
8. Degrees of freedom for block (d.f. for b)
d.f. for b =b-1
=4-1
=3
9. Degrees of freedom for error (d.f. for e)
d.f. for e = (t-1)(r-1)
= (6-1)(4-1)
= 15
10. Mean square due to treatment (MSt)
MSt = SSt/d.f. for t
= 155.68375/5
= 31.13675
11. Mean square due to block (MSB)
MSB = SSB/d.f. for b

57
 Biometry

= 14.62125/3
= 4.876
12. Mean square due to error (MSE)
MSE = SSE/d.f. for e
= 55.55129/15
= 3.7034
13. Calculate F-values
a. Replication
F-cal = MSB/MSE
= 4.87375/3.7034
= 1.316
b. Treatment
F-cal = MSt/MSE
= 31.13675/3.7034
= 8.4076

Then the calculated F-vales are compared with the tabulated F-values at 0.05 and
0.01 level of probability: Replication with d.f. 3 and 15 = 3.29 and 5.42 and for
treatment with d.f. 5 and 15 = 2.90 and 4.56, respectively. For replication, the
calculated F-value is less than the tabulated F-value while for treatment; the
calculated F-value is greater than the tabulated F-values (Table 4.24). The tabular F-
values are obtained using the degrees of freedom for replication or treatment as V1
value and error degrees of freedom as V2 value. For example, the tabulated value V1
= 5 (degrees of freedom for treatment) and V2 = 15 (degrees of freedom for error) are
2.90 at 0.05 and 4.56 at 0.01 level of significance.

Table 4.24. Analysis of variance for RCBD

Source of F-tabulated
variation d.f. SS MS F-calc. 0.05 0.01
Block 3 14.62125 4.876 1.32 3.29 5.42
Population 5 155.68375 31.13675 8.41** 2.90 4.56
Error 15 70.17254 3.7034

** = Significant at 0.01 level of probability

c.v.% = 7.2%.

14. Coefficient of variation (c.v.%)

MS E
c.v. = 100( )
GM
3.7034
= 100 x
26.8375
= 7.2%

58
 Biometry

Since the treatment effect was significant, it is necessary to proceed to calculate

standard error of the mean, standard error of difference and critical (least significant)
difference to identify the best treatment.

15. Standard error of the mean (SE (m))

MS E
SE (m) =±
r
3.7034
=±
4
= ± 0.962
2MS E
SE (d) =±
r
2x3.7034
=±
4
= ± 1.361

CD (LSD) = SE (d) x t at 0.05, at error d.f. = 1.361 x 2.131 = 2.90

Once the CD value is obtained, paired population means can be compared after
computing the number of possible pairs of means. Number of possible combination =
n (n-1)/2 = 6(6-1)/2, where n is the number of treatments (Table 4.25). The pair wise
comparison below can be summarized as
T2a T2
T1a T1
T3a or T3
T5ab T5
T4 b T4
T6 c T6

Treatments connected with the same bar or having the same letter show non-
significant difference among them.

Table 4.25. Paired population mean comparison

Paired populations Mean difference CD Status

Suwan-I vs. Synthetic GxA 28.700-29.525 = -0.825 2.90 ns
Suwan-I vs. Hercules 28.700-28.575 = 0.125 ns
Suwan-I vs. Red Iron 45 28.700-25.600 = 3.000 *
Suwan-I vs. Venus 49 28.700-26.650 = 2.050 ns
Suwan-I vs. MARDI Comp.I 28.700-21.975 = 6.725 *
Synthetic GxA vs. Hercules 29.525-28.575 = 0.950 ns
Synthetic GxA vs. Red Iron 45 29.525-25.600 = 3.925 *
Synthetic GxA vs. Venus 49 29.525-26.650 = 2.875 ns

59
 Biometry

Synthetic GxA vs. MARDI Com. -I29.525-21.975 = 7.550 *

Hercules vs. Red Iron 45 28.575-25.600 = 2.975 *
Hercules vs. Venus 49 28.575-26.650 = 1.925 ns
Hercules vs. MARDI Comp. -I 28.575-21.975 = 6.600 *
Red Iron 45 vs. Venus 49 25.600-26.650 = -1.050 ns
Red Iron 45 vs. MARDI Comp. -I 25.600-21.975 = 3.625 *
Venus 49 vs. MARDI Comp. -I 26.650-21.975 = 4.675 *

ns= non-significant, * significant at 0.05 level of probability

Relative efficiency (R.E.) of RCBD over CRD

[(r − 1) MSB + r (t − 1) MSE ]
R.E. = , where r is rep, t is treatment
(rt − 1) MSE
[(4 − 1)4.87 + 4(6 − 1)3.70]
=
(4 x6 − 1)3.70
= 1.0412

When the degrees of freedom for error is less than 20 Gomez and Gomez (1984)
suggested that the R.E. value should be multiplied by a correction factor (K) defined
as
[(r − 1)(t − 1) + 1][t (r − 1) + 3]
K=
[(r − 1)(t − 1) + 3][t (r − 1) + 1]
[(4 − 1)(6 − 1) + 1][6(4 − 1) + 3]
=
[(4 − 1)(6 − 1) + 3][6(4 − 1) + 1]
= 0.9824

Therefore, the adjusted R.E. will be

R.E. x K = 1.0412 x 0.9824

= 1.0228.

The value 1 is subtracted and then the efficiency is expressed as percentage, i.e., it
will be 1-1.0228 = 0.0228. When 100 multiply it, it will be 2.3% indicating that the use
of RCBD instead of CRD increased experimental precision by 2.3% which is quite
low.

Table 4.26a. Mean 100-grain weight (g) of six populations

evaluated in Serdang, Selangor (LSD)
______________________________________________
Population Mean
______________________________________________
Suwan-I 29.53
Synthetic GxA 28.70
Hercules 28.58

60
 Biometry

Red Iron 45 26.65

Venus 49 25.60
MARDI Composite-I 21.98
LSD 2.90
______________________________________________

Table 4.26b. Mean 100-grain weight (g) of six populations

evaluated in Serdang, Selangor (DNMRT)
______________________________________________
Population Mean
______________________________________________
Suwan-I 29.53a
Synthetic GxA 28.70a
Hercules 28.58a
Red Iron 45 26.65ab
Venus 49 25.60 b
MARDI Composite-I 21.98 c
______________________________________________

Interpretation
The analysis of variance for RCBD (Table 4.24) showed that the effect due to the
populations of maize was highly significant (p=0.01). Synthetic GxA had the highest
mean 100-grain weight (though Synthetic GxA, Suwan-I, Hercules and Venus 49
were not statistically different) followed by Suwan-I and Hercules (Tables 4.25a and
4.25b). Synthetic GxA, Suwan-I and Hercules were all better than Red Iron 45 and
MARDI composite-I. Therefore, Synthetic GxA, Suwan-I and Hercules are potential
populations for further evaluation.

Blocking effect was not significant indicating that blocking was not necessary in
controlling the gradient considered during planning or the gradient was less
heterogeneous. The relative efficiency of RCBD was very small (2.3%) showing that
RCBD and CRD had almost similar precision.

4.4. Latin Square Design

Introduction
Latin square is a type of experimental design, which is more restrictive than
randomized complete block design. In Latin square design, the number of
experimental units is the square of the number of treatments. The major feature of
this design is its capacity to simultaneously handle two known sources of variation
among experimental units.

For example, for t number of treatments, there will be t2 number of experimental

units, which are first grouped into t experimental units each on the basis of one
blocking factor and referred to as row grouping. The experimental units are then

61
 Biometry

grouped into t-groups of t experimental units each based on a different blocking

factor termed as column grouping. The treatments are then assigned to the plots in
such a way that each treatment appears once, and only once, in each row and in
each column. This procedure makes it possible to estimate variation among row-
blocks and column blocks in order to remove the variations from experimental error.

The relationship between treatments and experimental units imposes restriction on

the use of the design. When the number of treatments is large the design becomes
impractical due to the large number of replications required. On the other hand, when
the number of treatments is small the error degree of freedom becomes small for the
error to be reliably estimated. Therefore, this design is applicable in experiments with
four to eight numbers of treatments. Due to this limitation, it is not widely used in field
experiments.

4.4.2. Layout
The basic pattern of Latin square design can be shown using a 4 x 4 Latin square. In
this case the number of treatments, t equals 4, denoted as A, B, C and D, as shown
in Figure 4.4. From the Figure it’s clear that each row and column form a complete
block as in a randomized complete block design. The two modes of blocking help to
reduce sources of variations and thereby improve precision. Randomization can be
done in such a way that the first row is arranged in alphabetical order followed by
shifting subsequent letters one position to the left as ABCD, BCDA, CDAB and
DABC. The order of the rows can also be arranged as 2, 1, 3 and 4, followed by
randomization of the order of columns as 4, 2, 3 and 1 to get the following
randomization. Finally, treatments are assigned to plots.
Column
Row 1 2 3 4
1 A C D B
2 D B C A
3 B D A C
4 C A B D
Figure 4.4. Basic layout of a 4 x 4 Latin square

4.4.3. Analysis
The analysis of data from Latin square follows much the same pattern as for RCBD
except that one source of variation is added. For example, for t treatments that will
occupy t2 experimental units. Then the data collected can be organized as in the
Tables 4.41 and 4.42.

Table 4.41. Data structure for Latin square for column C and row R

Row Column Rj
1 2 . . . k
1 x11 x12 . . . x1b R1
2 x21 x22 . . . x2b R2
3 x31 x32 . . . x3b R3

62
 Biometry

. . . . . . . .
. . . . . . . .
. . . . . . . .
j x1j x2j . . . xkj Rt
Column total C1 C2 . . . Ck GT

Table 4.42 Data structure of Latin square for treatment totals (Ti)

Treatment 1 2 . . . t
Sum T1 T2 . . . Ti

Ti is obtained by adding all values in each column/row where that particular t appeared

The stepwise analysis of variance for LSD will be:

1. The linear additive model (L.A.M.): Xijk = µ + Rj, Ck, T(i) + Ejk(i) where, Xjk(i) = the
jth row of kth column, µ = the overall mean, Ti = ith treatment effect (µi-µ) and
Ejk(i) = the error associated with jkth observation of ith treatment.
Assumptions
Ti ~ NI (0, σ2t)
Rj ~ NI (0, σ2R)
Ck ~ NI (0, σ2C)
Eijk ~ NIR (0, σ2)

2. Correction factor (C.F.) = (GT)2/t2 where, t = number of treatments

3. Total sum of squares (Total SS) = ∑∑Xijk 2 – C.F.
4. Sum of squares due to Row (SSR) = (∑Rj 2)/t – C.F.
5. Sum of squares due to Column (SSC) = (∑Ck 2)/t – C.F.
6. Sum of squares due treatment (SSt) = (∑Ti 2)/t – C.F.
7. Sum of squares due to error (SSE) = Total SS – SSt – SSR – SSC
8. Degrees of freedom:
for row (d.f. for R) = (r-1)
for column (d.f. for C) = (c-1)
for treatment (d.f. for t) = t-1
for error (d.f. for e) = (t-1)(t-2)
9. Mean squares
MSR = SSR/ d.f. for R
MSC = SSC/ d.f. for C
MSt = SSt/ d.f. for t
MSE = SSE/ d.f. for e
10. Calculate F-values
a. Row = MSR/MSE
b. Column = MSC/MSE

63
 Biometry

c. Treatment = MSt/MSE

Then the calculated F-vales are compared with F-tabulated at 0.05 and 0.01 level of
probability. All components of the ANOVA and expected mean squares are shown in
Table 4.43, 4.44 and 4.45.

Table 4.43. ANOVA table for Latin square design

Source of
variation d.f. SS MS F-cal
Rows r-1 SSR MSR MSR/MSE
Columns c-1 SSC MSC MSC/MSE
Treatments t-1 SSt MSt MSt/MSE
Error (t-1)(t-2) SSE MSE

For all the three components the degrees of freedom is the same in Latin square and can be given by
simply t-1 for each of them, i.e., row, column and treatment have the same degrees of freedom.

Table 4.44. Expected mean squares for Latin square design

Source of
variation d.f. E(MS) -Random E(MS) -Fixed
Rows r-1 σ2 + tσR2 σ2 + t((∑Rj 2)/(t-1)
Columns c-1 σ2 + tσC2 σ2 + t(∑Ck 2)/(t-1)
Treatments t-1 σ2 + tσT2 σ2 + t(∑Ti 2)/(t-1)
Error (t-1)(t-2) σ2 σ2

Table 4.45. Expected mean squares for Latin square design with squares/sets

Source of
variation d.f. E(MS) Random
Squares/sets s-1 σ2+ tσS2
Rows/sets (r-1)s σ2 + tσ2R/S
Columns/sets (c-1)s σ2 + tσ2C?S
Treatments t-1 σ2 + tσT2
Error s(t-1)(t-2) + (s-1)(t-1) σ2

64
 Biometry

MS E
11. Coefficient of variation (c.v.%) = 100( ).
GM

12. Standard error of the mean (SE (m))

MS E
SE (m) = ±
t
2 MS E
SE (d) = ±
t
CD(LSD) = SE(d) x t0.05 at error d.f.

13. Relative efficiency of LSD over:

[ MS R + MSC + MS E (t − 1)]
Over CRD, RE(CRD) =
[(t − 1) MS E

[ MS R + MS E (t − 1)]
Over RCBD, RE(RCBD, row) =
t ( MS E )

[ MSC + MS E (t − 1)]
Over RCBD, RE(RCBD, column) =
t ( MS E )

4.4.4. Numerical examples

Example-I
There are experimental animals that do differ in body weight as well as in age (two
gradients). A researcher, planning to compare 5 newly formulated rations and a
check, designed his experiment in Latin square design. Body weight was labeled as
row and animal age as column. After a certain period of time, the weight gain was
recorded as in the following Table 4.46.

Table 4.46. Body weight of animals fed on different rations

Column
Row 1 2 3 4 5 6 Rj Mean
1 F(2.19) E(2.50) D(2.27) C(1.62) B(1.82) A(0.91) 11.31 1.89
2 E(2.27) C(1.41) A(0.91) D(1.91) F(2.13) B(1.95) 10.58 1.76
3 B(2.04) A(0.91) F(2.25) E(2.29) D(1.91) C(2.07) 12.06 2.01
4 A(0.77) (B2.04) E(2.40) F(1.99) C(1.82) D(2.50) 11.52 1.92
5 D(2.50) F(2.31) C(2.09) B(2.04) A(0.91) E(2.27) 12.12 2.02
6 C(1.52) D(1.86) B(1.91) A(0.77) E(2.30) F(1.98) 10.34 1.72
Ck 11.29 11.03 00.83 10.62 11.48 11.68 67.93 1.89

The different components of ANOVA can be done as follows.

65
 Biometry

(GT ) 2
1. Correction factor (C.F.) = , where, t = number of treatments
t2
(67.93) 2
=
62

=128.18
2. Total sum of squares (Total SS)
Total SS = ∑∑Xijk 2 –C.F.
= (2.19)2 + (2.27)2 + … + (1.98)2 – 128.18
= 9.93
3. Sum of squares due to Row (SSR)
2

SSR =
∑ Rj
– C.F.
t
[(11.31) 2 + (10.58) 2 + ... + (10.34) 2 ]
= – 128.18
6
= 0.46
4. Sum of squares due to Column (SSC)
2

SSC =
∑ Ck
– C.F.
t
[(11.29) 2 + (11.03) 2 + ... + (11.68) 2 ]
= – 128.18
6
= 0.17
5. Sum of squares due treatment (SSt)
2

SSt =
∑ Tk
– C.F.
t
[(5.18) 2 + (11.80) 2 + ... + (12.85) 2 ]
= – 128.18
6
= 8.86

The treatment total is obtained by adding all observations for each treatment as
shown below:
• Total of treatment A = 0.77+0.91+0.91+0.77+0.91+0.91 = 5.18
• Total of treatment B = 2.04+2.04+1.91+2.04+1.82+1.95 = 11.80
• Total of treatment C = 1.52+1.41+2.09+1.62+1.82+2.07 = 10.53
• Total of treatment D = 2.50+1.86+2.27+1.91+2.50+2.50 = 13.54
• Total of treatment E = 2.27+2.50+2.40+2.29+2.30+2.27 = 14.03
• Total of treatment F = 2.19+2.31+2.25+1.99+2.13+1.98 = 12.85

7. Sum of squares due to error (SSE) = Total SS - SSt - SSR - SSC

= 9.93 - 8.86 - 0.46 - 0.17
= 0.44
8. Degrees of freedom for row (d.f. for R)

66
 Biometry

d.f. for R = (r-1) = (6-1) = 5

9. Degrees of freedom for column (d.f. for C)
d.f. for C = (c-1) = (6-1) = 5
10. Degrees of freedom for treatment (d.f. for t) = t-1 = 6-1 = 5,
11. Degrees of freedom for error (d.f. for e) = (t-1)(t-2) = (6-1)(6-2) = 20
12. Mean squares
SS R 0.46
MSR = = = 0.092
d . f . for R 5
SSC 0.17
MSC = = = 0.034
d . f . forC 5
SSt 8.86
MSt = = = 1.772
d . f . for t 5
SS E 0.44
MSE = = = 0.022
d . f . for e 20
13. Calculate F-values
MS R 0.092
a. Row = = = 4.18
MS E 0.022
MSC 0.034
b. Column = = = 1.55
MS E 0.022
MSt 1.772
c. Treatment = = = 80.55
MS E 0.022

Then the calculated F-vales are compared with tabulated F-value at 0.05 and 0.01
level of probability: For row and treatment the calculated F-value is greater than the
tabulated F-value (2.71 at 5%) while for column the calculated F-value is less than
the tabulated F-value (Table 4.47).

Table 4.47. ANOVA table for 6 x 6 Latin square design

Source of F-tabulated
variation d.f. SS MS F-cal 0.05 0.01
Rows 5 0.46 0.092 4.18** 2.71 4.10
Columns 5 0.017 0.034 1.55 2.71 4.10
Treatments 5 8.86 1.772 80.55* * 2.71 4.10
Error 20 0.44 0.022

** Significant at 0.01 level of probability

MS E 0.022
14. Coefficient of variation (c.v.%) = 100 ( ) = 100 x (
) = 7.8%.
GM 1.89
Row and treatment effects are significant and hence, we proceed to calculate
standard error of the mean, standard error of difference and critical (least
significant) difference to identify the best ration.

67
 Biometry

15. Standard error of the mean (SE (m))

MS E 0.022
SE (m) = ± =± = ±0.06
t 6
MS E 2x0.022
SE (d) = ± =± = ± 0.09
t 6
CD(LSD) = SE(d) x t0.05 at error d.f. = 0.09 x 2.086 = 0.19

16. Relative efficiency of LSD over:

[ MS R + MSC + MS E (t − 1)]
Over CRD, RE(CRD) =
[(t − 1) MS E
[0.092 + 0.034 + 0.022(6 − 1)]
=
[(6 − 1)0.022
= 1.53 or 53%

[ MS R + MS E (t − 1)]
Over RCBD, RE(RCBD, row) =
t ( MS E )
[0.092 + 0.022(6 − 1)]
=
6(0.022)
= 1.53 or 53%
[ MSC + MS E (t − 1)]
Over RCBD, RE(RCBD, column) =
t ( MS E )
[0.034 + 0.022(6 − 1)]
= 6(0.022)

= 1.09 or 9%

The use of LSD increased precision by 53% over RCBD, but the addition of column
blocking in the LSD did not increase precision. Hence, RCBD with rows as blocks
would have been as efficient as LSD. Since there was a significant difference among
treatments, it is necessary to do mean separation to identify the best treatment
(Table 4.48) and then present the data as in Table 4.49.

Paired treatment mean comparison

There are t(t-1)/2 number of possible comparisons (Table 4.48).

Table 4.48. Paired population mean comparison

Paired populations Mean difference CD Status
A vs. B 0.86-1.97=1.11 0.19 *
A vs. C 0.86-1.76=0.90 0.19 *
A vs. D 0.86-2.26=1.40 0.19 *

68
 Biometry

A vs. E 0.86-2.34=1.48 0.19 *

A vs. F 0.86-2.14=1.28 0.19 *
B vs. C 1.97-1.76=0.21 0.19 *
B vs. D 1.97-2.26=0.29 0.19 *
B vs. E 1.97-2.34=0.37 0.19 *
B vs. F 1.97-2.14=0.17 0.19 ns
C vs. D 1.76-2.26=0.50 0.19 *
C vs. E 1.76-2.34=0.58 0.19 *
C vs. F 1.76-2.14=0.38 0.19 *
D vs. E 2.26-2.34=0.08 0.19 ns
D vs. F 2.26-2.14=0.12 0.19 ns
E vs. F 2.34-2.14=0.20 0.19 *
ns = non-significant, * significant at 0.05 level of probability
Data presentation

Table 4.49. Mean body weight gain (kg) due to different rations
Ration Mean
A 0.86
B 1.76
C 1.97
D 2.26
E 2.34
F 2.14
5% CD 0.19
CV 7.8%

Interpretation
The treatment effect was significant indicating that there is a significant difference
among the treatment means (Table 4.47). Those treatments that had their mean
difference greater than the critical value are statistically different where as those
having their difference less than the critical values were not statistically different.

All the new formulations were superior to the standard ration (treatment A).
Treatment E gave the highest (although treatment E and D are not statistically
different) mean body weight followed by treatment D and F. Treatment E had 172%
advantage in increasing the body weight of animals compared to the standard ration,
treatment A. Hence, treatment E is a potential ration for use (Table 4.49).
The effect due to rows was significant implying that blocking in the row direction
significantly reduced the variation considered (initial body weight). However, the
effect due to column was not significant indicating that considering age as a source
of variation was not important in this particular experiment. Therefore, randomized
complete block design with blocking in the row direction (blocking the animals by
initial body weight) could have been satisfactory.

69
 Biometry

4.5. Augmented Design

4.5.1. Introduction
In plant breeding programme there are cases where treatments include new
progenies generated through breeding. The progenies are supposed to be compared
against certain checks. Any of the experimental design can be used depending on
the number of treatments and stage of the breeding programme.

In the early stages of selection process there could be insufficient seed of the new
progenies (genotypes) to undertake replicated experiments or the number of
genotypes could be very large to manage in terms of resources. In such cases, some
plant breeders use single row plots to evaluate the newly developed progenies and at
a certain regular intervals check varieties are planted. The performances of the new
genotypes are then compared with the performance of the nearest check,
subjectively. The standard statistical analysis is not possible as the new genotypes
are not replicated but an objective comparison can be made. The disadvantage to
the subjective judgment is that the checks are systematically placed and no provision
is made to adjust a given measurement for environmental variation from one part of
the experiment to another.

A better method, when there are many progenies at the early stage of the breeding
programme, is to use an experimental design called augmented design that was
developed by Federer and well illustrated by Federer and Ragavarao (1975). This
design is of particular interest in an extensive plant breeding programme.

4.5.2. Layout
The first step is to divide the experimental area into blocks and then assigning plots
within the blocks in order to have a single row representing a treatment. Two or more
checks are assigned to each block at random and the remaining rows are assigned
to the new genotypes. The design is more efficient if blocks of the same size are
used but it does not necessarily be the same size. The resulting design is one in
which the checks are replicated but the new genotypes are not. By doing so the
design helps to estimate experimental error and provides a means to adjust the yield
of the new genotypes for variations from block to block.

The number of checks and blocks is determined by the statistically acceptable error
degrees freedom, which are at least 12. This restriction determines the minimum
number of blocks and checks to be used. The maximum number of blocks depends
on the number of new genotypes and the experimental area. If for example, there are
c number of checks and b number of blocks, then the degrees of freedom for error is
(c-1) (b-1). Hence, the number of blocks can be equated as follows: b ≥ 12/(c-1) + 1.
The minimum number of blocks for 3 checks will be 7, for example.

4.5.3.Analysis
In the data analysis, the experimental error is estimated considering the checks
placed in randomized complete block design, which has been dealt in detail in
Section 4.3. After conducting the analysis on the checks, the error mean square from

70
 Biometry

this analysis is used to compute the standard errors for different comparisons. The
difference between the mean of the checks in a given block and the mean of the
checks over the entire experiment.

The first step in the analysis is to conduct ANOVA on the data measured for checks.
Then, a table is constructed (Table 4.61) to organize the different entities (check,
block, sum for the checks & blocks, mean for the checks & blocks and the
adjustment). Let Xij be the yield of the ith check in jth block. Then, adjustment aj =
mean of j – the grand mean and the sum of aj = 0. The variable of interest (example
yield) of the new genotypes is adjusted for the effect of the block in which it was
grown as shown in Table 4.62.

Table 4.61. Two way table of totals for checks x block

Check Block Sum (Ci) Mean
variety 1 2 3 ... b Check
1 X11 X12 X13 … X1b C1 Mean 1
2 X21 X22 X23 … X2b C2 Mean 2
3 X31 X32 X33 … X3b C3 Mean 3
. . . . . . .
. . . . . . .
. . . . . . .
c Xc1 Xc2 Xc3 … Xcb Cc Mean c
Sum (Bj) B1 B2 B3 … Bb Grand total
Mean Mean1 Mean2 Mean3…Mean b Grand
(block) mean
Adjustment a1 a2 a3 ….. ab
Where, Ci = the sum of the ith check, Bj = the sum of the checks in the jth block, mean of a
check = total of that check/number of blocks, mean of a block = total of that block/number of
checks and grand mean = grand total/bc.

Table 4.62. Adjusted yield of new genotypes

Genotype Yield
Observed Adjusted
1 y1j ŷ1j
2 y2j ŷ2j
3 y3j ŷ3j
. . .
. . .
. . .
g ygj ŷgj
Where yij = yield of ith genotype grown in jth block, ŷij = yij – aj, adjusted yield of ith genotype is
the difference between observed yield minus block adjustment.

71
 Biometry

Experimental error is obtained by running ANOVA for randomized complete block

design using the check yield data. The step wise computation is given below.
Correction factor is computed as square of the grand total divided by number of
observations (bc), C.F. = GT2/bc, Total sum of squares (Total SS) is obtained by
squaring each observation, sum them all and subtract the correction factor.

Total SS = ∑Xij2 – C.F.,

SSR =
∑ R j 2 – C.F.,
c
2

SSC =
∑ Ci
– C.F.,
b
SSE = Total SS – SSR – SSC and
SS E
Mean square of error (MSE) = .
(c − 1)(b − 1)

The analysis of variance for the checks can be given as in Table 4.63

Table 4.63. ANOVA for the checks in augmented design

Source d.f. SS MS
of variation
Blocks b-1 SSB
Checks c-1 SSC
Error (b-1)(c-1) SSE MSE

Standard errors
There are four types of standard error of differences to compare the means in
augmented design. These standard error of differences are computed as follows. To
compare two check means standard error of difference:
2 MS E
SE(d) = .
b

For comparing two genotypes in the same block:

SE(d) = 2MS E .
For comparing two genotypes in different blocks:
2(c + 1) MS E
SE(d) = and for comparing a genotype and a check:
c
2(c − 1)(b − 1) MS E
SE(d) = .
c

72
 Biometry

These standard errors of differences can be used to compute the least significant
difference (LSD) values.

4.5.4.Numerical example

A rice breeder would like to evaluate 16 progenies generated by an International rice

research institute (IRRI). He planned to evaluate the progenies in augmented design
(because the seed for each progeny is not enough for replicated trials, there is a
limited resource for the research or observation/preliminary information is enough to
narrow down the number of progenies for further test). There are 4 checks to be used
as a reference to evaluate the new progenies. The 20 treatments (16 new progenies
and + 4 checks) were put in four block each containing 8 treatments (4 new + 4
checks) as shown in Figure 4.5.

P1 A P2 B P3 C P4 D Total
120 83 100 77 90 70 85 65 690

P5 B P6 C P7 A P8 D Total
88 76 130 71 105 84 110 64 728

P9 D P10 A P11 B P12 C Total

102 63 140 86 135 78 138 69 811

P13 A P14 D P15 C P16 B Total

84 82 90 63 95 68 103 75 660
Figure 4.5. Layout of augmented design for progeny evaluation GT = 2889

At this point the checks and the check effects are estimated followed by progeny
effect as shown in Tables 4.64 and 4.65.

1. Compute check effects: Let C = check total, c = check effect and cC = the product
of the check total and check effect for each checks.

Table 4.64. Check total, mean and check effects

Check B1 B2 B3 B4 check check check ciCi

total mean effect
A 83 84 86 82 335 87.75 -16.68 -5587.80
B 77 76 78 75 306 76.50 -23.93 -7322.00
C 70 71 69 68 278 69.50 -30.93 -8598.54
D 65 64 63 63 255 63.75 -36.68 -9353.40
Total 1174 293.5 -30862.32

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 Biometry

2. Compute block effects

1
bi = [total of bi – total of all checks means– total of progenies in bi],
c
where c = number of checks
1
b1 = [total of b1 – total of all checks means– total of progenies in b1]
c
= ¼[690-293.5-395]
= 0.375
1
b2 = [total of b2 – total of all checks means– total of progenies in b2]
c
= ¼[728-293.5-433]
= 0.375
1
b3 = [total of b3 – total of all checks means– total of progenies in b3]
c
= ¼[811-293.5-515]
= 0.625
1
b4 = [total of b4 – total of all checks means– total of progenies in b4]
c
= ¼[660-293.5-372]
= -1.375
3. Sum up all bi values and summarize the block effect in tabular form
∑ bi = 0.375 + 0.375 + 0.625 + (-1.375) = 0

Table 4.65. Progeny total, block total and block effects

Check ni Block Progeny No. of

total total progenies bi Pbi
B1 8 690 395 4 0.375 1.5
B2 8 728 433 4 0.375 1.5
B3 8 811 515 4 0.625 2.5
B4 8 660 372 4 -1.375 -5.5

ni = number of treatments (checks + progenies) in each block, Pbi = bi x no. of progenies

4. Compute adjusted grand mean (AGM)

1
AGM = [GT – (b-1)(total of all checks means) - ∑Pbi]
c+ p

1
= [2889 -(4-1)(293.5) – 0.5] c = number of checks
4 + 16
p = number of progenies
= 100.4 b = number of blocks

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 Biometry

5. Compute check effective

Gi = check mean – adjusted grand mean (AGM)
G1 = 83.75 – 100.4 = -16.68, that is for check A
G2 = 76.50 – 100.4 = -23.93, for check B
G3 = 69.50 – 100.4 = -30.93, for C
G4 = 63.75 – 100.4 = -36.68, for D

6. Compute adjusted progeny means

Adjusted progeny mean = observed progeny mean – effect of the block
For example, adjusted progeny mean for P1 = 120 – 0.375 = 119.83 and for the
remaining progenies it is given in Table 4.66, ANOVA in Table 4.67 and data
presentation in Table 4.68.
7. Compute the different components
(GT ) 2 (2889) 2
Correction factor (C.F.) = = = 260822.53, where
N 32
N=total N0. of plots
2
Total SS =∑Xijk – CF
= [(120)2 + (83)2 + … + (75)2] - 260822.53
= 15798.47
2

8. Crude block SS =
∑ Bi
ni
[(690) 2 + (728) 2 + (811) 2 + (660) 2
=
8
=262425.63
9. True block SS (TBSS)
2

a. Ignoring entries =
∑ Bi
– C.F.
ni
= 262425.63 – 260822.53
= 1603.1
b. Eliminating = TBSS – [SSeib– SSeeb] where eib= entries ignoring blocks
eeb = entries eliminating block
= 1603.10 – [15776.97 – 14192.7]
= 18.83
Table 4.66. Adjusted progeny mean

Observed Adjusted Progeny

Progeny Mean bi mean effect piPi biBi
P1 120 0.375 119.83 19.2 2304 258.75
P2 100 0.375 99.63 -0.8 -80 258.75
P3 90 0.375 89.63 -10.8 -972 258.75
P4 85 0.375 84.63 -15.8 -1343 258.75
P5 88 0.375 87.63 12.8 -1126.4 273.00
P6 130 0.375 129.63 29.2 3796 273.00

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 Biometry

P7 105 0.375 104.63 4.2 441 273.00

P8 110 0.375 109.63 0.95 1012 273.00
P9 102 0.625 101.38 0.95 96.9 506.875
P10 140 0.625 139.38 38.95 5453 506.875
P11 135 0.625 134.38 33.95 4583.25 506.875
P12 138 0.625 137.38 36.95 5099.1 506.875
P13 84 -1.375 85.38 -15.05 -1264.2 –907.5
P14 90 -1.375 91.38 -9.05 -814.5 –907.5
P15 95 -1.375 96.38 -4.05 -384.75 –907.5
P16 103 -1.375 104.38 3.95 406.85 –907.5
Total 1715 17207.25 131.125

biBi = block effect x block total of each block

10. SSeeb = (AGM x GT) + (∑biBi + ∑ciCi + ∑piPi) – Crude BSS
= (100.47 x 2889) + (131.125 + -30862.32 + 17207.25)
= 14192.7
2

11. SSeib =
∑ Ci + Crude SS progenies - (C.F.)
b
[(120) 2 + (100) 2 + ... + (103) 2 ]
= – 260822.53
4
= 15776.97
2 2

12. SS checks =
∑ Ci – CF’ (of the checks), where C.F.’ = ∑ Ci = (1174) 2 = 86142.25
b cb 4 x4
[(335) 2 + (306) 2 + (278) 2 + (255) 2 ]
= – 86142.25
4
= 900.25
2
2
13. SS progenies = ∑Pj – CF’’ (of the progenies), C.F.’’ =
∑ Pi
=
(1715) 2
= 183826.56
P 16
= [(120)2 + (100)2 + … + (103)2] – 183826.56
= 5730.44
14. SS check x progenies = SSeib – SS checks – SS progenies
= 15776.97 – 900.25 – 5730.44
= 9146.28
15. SS error = Total SS – TBSS ignoring - SSeeb
= 15798.47 – 1603.10 – 14192.70
= 2.67

Table 4.67. ANOVA for Augmented design

Source of d.f. SS MS F-cal F-tabulated

Variation 0.05
Block 3
a. Ignoring entries 1603.10 534.37 1781.23 3.86
b. Eliminating entries 18.83 6.28 20.93 3.86

76
 Biometry

Entries 19
a. eliminating blocks 14192.70 746.98 2489.93 2.96
b. ignoring blocks 15776.97 830.37 2767.90 2.96

Checks 3 900.25 300.08 1000.27 3.86

Progenies 15 5730.44 382.03 1273.43 3.00
Progenies vs. checks 1 9146.28 9146.28 30487.6 5.12
Error 9 2.67 0.3

*, significant at 0.05 level of probability

16. Compute standard error of mean difference and critical difference

a. To compare two checks
MS E
SE(m) =
b
0.3
=
4
= 0.27, where b = number of blocks
2 MS E
SE(d) =
b
2x0.3
=
4
= 0.39
CD = SE(d) x t0.05 at error d.f.
= 0.39 x 2.262
= 0.88
b. To compare two progenies in the same block
SE(m) = MS E
= 0.3
= 0.55
SE(d) = 2 MS E
= 2 x0.3
= 0.77
CD = SE(d) x t0.05 at error d.f.
= 0.77 x 2.262
= 1.74
c. To compare two progenies in different blocks
1
SE(d) = 2MS E (1 + )
c

77
 Biometry

1
= 2 x0.3(1 + )
4
= 0.87 where c = no. of checks
CD = SE(d) x t0.05 at error D.F.
= 0.87 x 2.262
= 1.97
d. To compare two progenies vs. checks
1 1 1
SE (d) = MS E (1 + + + )
b c bc
1 1 1
= 0.3(1 + + + )
4 4 4 x4
= 0.68

CD = SE (d) x t0.05 at error d.f.

= 0.68 x 2.262
= 1.54
17. Mean comparison is shown in Table 4.68.

Table 4.68. Mean comparison

Adjusted Progenies Progenies Progenies

Treatment Mean different blocks Same block vs. checks
1. Progeny CD = 1.97 CD = 1.74 CD = 1.54
P10 139.38 P10 P1 P10
P12 137.38 P12 P2 P12
P11 134.38 P11 P3 P10
P6 129.63 P6 P4 P11
P1 119.63 P1 P1
P8 109.63 P8 P5 P8
P7 104.63 P7 P6 P7
P16 104.38 P16 P7 P16
P9 101.38 P9 P8 P9
P2 99.63 P2 P2
P15 96.38 P15 P9 P15
P14 91.38 P14 P10 P14
P3 89.63 P3 P11 P3
P5 87.63 P5 P12 P5
P13 85.38 P13 P13
P4 84.63 P4 P13 P4
P14 A
2. Check P15 B
A 83.75 P16 C
B 76.50 D
C 69.50

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 Biometry

D 63.75

Treatments with the same line are not statistically different

Interpretation
There was a significant difference among progenies, checks and progenies vs.
checks. Among the progenies, P10 had the highest yield (120 g/plot) followed by P12
and P11 while the least was obtained from P4 (but P4and P13 are statistically not
different). Among the checks, the highest yield (83.75 g/plot) was obtained from A
followed by B while the least was obtained from D. When progenies and checks were
compared, the highest yield was obtained from P10 followed by P12. However,
progeny 4 and check A were not statistically different. Therefore, all progenies except
P4 can be promoted for further test based on the yield per plot. P4 can only be
considered if it has special feature like disease or insect resistance.

4.6. Incomplete Block Design

4.6.1. Introduction
When the number of treatments to be tested becomes very large, it is so difficult to
get uniform blocks large enough to accommodate a complete replication of all the
treatments. That is, as the number of treatment increases, the complete block
designs become less efficient due to the increase block size that entails an increase
in the variability within-blocks (experimental error). In such cases, incomplete block
designs, with their reduced block size are recommended to provide a higher degree
of precision than complete block designs.

Incomplete block designs are types of designs where the experimental units are
grouped into blocks that cannot contain a full replication of all treatments. One of the
basic principles of modern experimental design is to group the experimental units into
blocks of homogeneous units to increase precision and to make comparisons under
uniform conditions. In general, precision increases as the block size decreases and
hence, smaller blocks are preferred to larger ones. The disadvantages of incomplete
block designs are: restricted number of treatments and replications or both, unequal
degree of precision in comparing treatment means and complex data analysis.
The incomplete block designs are categorized into two: balanced designs and partial
balanced designs. In balanced designs each treatment occurs together in the same
block with every other treatment equal number of times, usually once. In a balanced
incomplete block design all pairs of treatments are compared with the same precision
even if the differences between blocks may be large.

The balanced lattice designs are restricted in such a way that the number of
treatments must be a perfect square, the block size (k) is equal to the square root of
the number of treatments and the number of replications is one more than the block
size (k+1). In such cases, the minimum number of replication is fixed to balance and
that makes the balanced design less practical. As a result of this, a class designs
called partially balanced incomplete block designs were developed.

79
 Biometry

Partially balanced designs are difficult to analyze statistically and comparisons

among some pair of treatments are made with greater precision than for other pairs.
Cochran and Cox (1957) illustrated the details of these designs. In this manual
emphasis is given to the most commonly used incomplete block designs.

Among the most commonly used incomplete block designs, lattice designs are called
fore. In lattice designs the number of treatment must be a perfect square. The
number of blocks and the number of plots in each block is the square root of the
number of treatments. In lattice design the incomplete blocks are combined into
groups that form separate, complete replications. When a lattice design consists of
two or more complete replications it has been shown that the design can be analyzed
as if it were an ordinary randomized complete block design (Cochran and Cox, 1957).
Hence the blocking restriction can be ignored without destroying the validity of the
analysis even thought the precision may be reduced.

In all the lattice designs the number of treatment must be a perfect square, and the
number plots per incomplete block is the square root of the number of treatments.
The designs differ from each other primarily by the number of replications. Field plan
for selected lattice designs are given in Appendix 8.

The blocks in lattice designs should be composed of plots that are as homogeneous
as possible. Blocks in the same replication should also be as uniform as possible to
maximize the variation among replications. This can maximize precision if the
experiment is analyzed as a randomized complete block design. Lattice designs can
be categorized into two classes: balanced lattice and partially balanced lattice.

4.6.2. Balanced lattice

In balanced lattice, the number of treatments must be a perfect square and the size
of block is the square root of the number of treatments. The special feature of the
balanced lattice, as distinguished from other lattices, is that every pair of treatments
occurs once in the same incomplete block. Consequently, all pairs of treatments are
compared with the same degree of precision (Cochran and Cox, 1957).

In balanced lattice each treatment occurs together with every other treatment in the
same block equal number of times. However, if there are b number of blocks, there
must be b + 1 number of replication to achieve balance. This restriction in the number
of replication and treatments makes the design less practical in terms of resources.
Balanced lattice design cannot be constructed for 36, 100 and 144 number of
treatments.

4.6.2.1. Layout

For example, in an experimental field where there is a high variation that runs in more
than one direction, RCBD cannot be used because the high variability in the field can

80
 Biometry

introduce within block variation. It is also not possible to use Latin square design as it
requires larger experimental area that definitely minimizes our precision. The next
option is to seek for a suitable experimental design. Lattice designs are suitable in
such cases. Suppose there are nine treatments that need bigger plot size in a site
where the experimental area has got fertility variation which can not be controlled
through blocking in one direction (as in RCBD) or in two directions (Latin SD as it
makes the experimental area very large, 9 x 9 LSD). Hence, the experiment is
planned as a 3 x 3 balanced lattice design so that the fertility difference can not
contribute to intra-block error. The treatments were placed into three blocks each
containing three treatments as shown in Figure 4.7.

Block Rep-I Block Rep-II

1, 1 2 3 4, 1 4 7
2, 4 5 6 5, 2 5 8
3, 7 8 9 6, 3 6 9

Block Rep-III Block Rep-IV

7, 1 5 9 10, 1 6 8
8, 2 6 7 11, 2 4 9
9, 3 4 8 12, 3 5 7

Figure 4.7. Field plan for 3 x 3 balanced lattice design

4.6.2.2. Analysis

1. Calculate the block totals, the replication totals, the grand total and the treatment
total as shown in Table 4.69.

Table 4.69. Data summary for 3 x 3 balanced lattice design

Block Rep-I Total Block Rep-II Total

1 1 2 3 B1 4, 1 4 7 B4
2 4 5 6 B2 5, 2 5 8 B5
3 7 8 9 B3 6, 3 6 9 B6
Rep total R1 Rep total R2

Block Rep-III Block Rep-IV

7, 1 5 9 B7 10, 1 6 8 B10
8, 2 6 7 B8 11, 2 4 9 B11
9, 3 4 8 B9 12, 3 5 7 B12

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 Biometry

Rep total R3 Rep total R4

Grand total GT

1. For each treatment, Bt = sum of all blocks for a treatment containing that a
particular treatment will be calculated. For example, for treatment 1,
Bt = B1 + B4 + B7 + B10
2. The quantities, W = kT - (k+1)Bt +GT, where ∑W = 0.
3. Total SS, SSt, SSR are computed as usual and the SSB within replications,

adjusted for treatment effects = 3

∑ W2
k (k + 1)
4. The adjustment factor,
E −E
A or µ = b 2 e ,
k Eb
where Eb and Ee are mean squares for blocks and error (intra-block), respectively.

The symbol for adjustment factor (A) can also have different symbols, for example
Cochran and Cox (1957) used µ but the most important thing is to understand the
meaning. In this manual as mentioned in the introductory part, we use notations that
are easy to pick, like the first letter of the term (example, Adjustment factor = A,
Grand total = GT, grand mean = GM, Correction factor = CF and so on) and
notations used by most authors like µ. If Eb < Ee, A is taken as zero and no
adjustment is applied to the treatment totals.

Table 4.70. ANOVA table for 3 x 3 balanced lattice design

Sources of variation d.f. SS MS

Replications k SSR
Treatments k2 - 1 SSt
Blocks(adj.) k2 – 1 SSB Eb
Intra-block error (k-1)(k2-1) SSE Ee
Total (k3 +k2 – 1) Total SS

5. For t-test, calculate the effective error mean square, E’e = Ee(1+kA). The variance
of the difference between two adjusted treatment totals is 2r E’e, while for the
difference between two adjusted treatment means is
2 E 'e
.
r

The treatment mean square in the above case (Table 4.70) cannot be tested against
the intra-block error mean square because the treatment mean square contains

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 Biometry

some block effects. Rather the adjusted treatment mean square is tested against the
effective error mean square, E’e

4.6.2.4. Numerical example

A breeder would like to evaluate 16 highly advanced hybrids in balanced lattice
design as the experimental area has variability in terms soil acidity with unknown
direction of the gradient. Then he conducted the experiment and obtained the
following measurements (Table 4.71). The statistical objective of this example is to
get familiarize with the analysis of variance for balanced lattice design and compare
the treatment means using the appropriate standard error of differences. The
biological objective of the experiment is to identify the best hybrid for commercial use.

83
 Biometry

Table 4.71. Grain yield data (t/ha) of 16 hybrids evaluated in 4 x 4 balanced lattice design
Rep
Trt I II III IV V
b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 Tj Bj Wj
T1 14.9 14.0 15.0 22 14.8 80.7 329.9 55.8
T2 15.2 15.6 13.2 15.9 20.5 80.4 336.2 23.1
T3 16.7 19.2 15.3 20.5 19.6 91.3 346.2 16.7
T4 15.6 16.2 19.5 17.6 22.6 91.5 361.2 -57.5
T5 12.7 16.0 18.0 16.8 17.8 81.3 343.7 -10.8
T6 15.5 14.2 16.5 17.8 21.3 85.3 333.0 58.7
T7 16.2 15.6 18.5 17.8 19.6 87.7 359.2 -62.7
T8 17.2 20.9 15.9 18.9 14.8 87.8 334.4 61.3
T9 14.3 18.0 13.9 15.6 18.0 80.2 332.3 41.8
T10 10.0 9.8 12.5 15.6 11.5 59.4 316.6 37.1
T11 19.2 19.5 17.9 18.6 20.6 95.8 356.2 -15.3
T12 17.0 21.2 16.6 20.8 19.5 95.1 362.4 -49.1
T13 15.5 15.2 18.5 20.8 17.5 87.5 358.0 -57.5
T14 19.0 19.3 18.6 20.5 22.5 99.9 360.9 -22.4
T15 19.2 18.2 17.9 18.5 17.0 90.8 339.5 48.2
T16 20 18.5 15.7 16.8 16.9 87.9 360.3 -67.4
Bi 62.4 61.6 60.9 73.7 63.2 58.9 72.5 76.8 65.1 65.7 63.7 69.0 81.1 74.2 69.7 69.5 58.1 75 77.9 83.5 1382.5 5530 0
Rk 258.6 271.4 263.5 294.5 294.5

84
The stepwise analysis is as follows:

2. Compute Bj
a. B1 = Bi1 + Bi5 + Bi9 + Bi13 + Bi17
= 62.4 + 63.2 + 65.1 + 81.1 + 58.1
=329.9
b. B2 = Bi1 + Bi6+ Bi10 + Bi14 + Bi18
= 62.4 + 58.9 + 65.7 + 74.2 + 75
=336.2
c. B3 = Bi1 + Bi7 + Bi11 + Bi15 + Bi19
= 62.4 + 72.5 + 63.7 + 69.7 + 77.9
=346.2
d. B4 = Bi1 + Bi8+ Bi12 + Bi16 + Bi20
= 62.4 + 76.8 + 69.0 + 69.5 + 83.5
=361.2
e. B5 = Bi2 + Bi5 + Bi10 + Bi15 + Bi20
= 61.6 + 63.2 + 65.7 + 69.7 + 83.5
=343.7
f. B6 = Bi2 + Bi6 + Bi9 + Bi16 + Bi19
= 61.6 + 58.9 + 65.1 + 69.5 + 77.9
=333.0
g. B7 = Bi2 + Bi7 + Bi12 + Bi13 + Bi18
= 61.6 + 72.5 + 69.0 + 81.1 + 75.0
=359.2
h. B8 = Bi2 + Bi8 + Bi11 + Bi14 + Bi17
= 61.6 + 76.8 + 63.7 + 74.2 + 58.1
=329.9
i. B9 = Bi3 + Bi5 + Bi11 + Bi14 + Bi17
= 60.9 + 63.2 + 63.7 + 74.2 + 58.1
=332.3
j. B10 = Bi3 + Bi6 + Bi12 + Bi15 + Bi17
= 60.9 + 58.9 + 69 + 69.7 + 58.1
=329.9
k. B11 = Bi3 + Bi7 + Bi9 + Bi14 + Bi20
= 60.9 + 72.5 + 65.1 + 74.2 + 83.5
=356.2
l. B12 = Bi3 + Bi8 + Bi10 + Bi13 + Bi19
= 60.9 + 76.8 + 65.7 + 81.1 + 77.9
=362.4
m. B13 = Bi4 + Bi5 + Bi12 + Bi14 + Bi19
= 73.7 + 63.2 + 69 + 74.2 + 77.9
=358
n. B14 = Bi4 + Bi6 + Bi11+ Bi13 + Bi20
= 73.74 + 58.9 + 63.7 + 81.1 + 83.5
=360.9
 Biometry

o. B15 = Bi4 + Bi7 + Bi10 + Bi16 + Bi17

= 73.7 + 72.5 + 65.7 + 69.5 + 58.1
=339.5
p. B16 = Bi4 + Bi8 + Bi9 + Bi15 + Bi18
= 73.7 + 76.8 + 65.1 + 69.7 + 75
= 360.3
3. Compute W j
a. W1 = qT1 – (q+1)B1 + GT
= (4x80.7) – (4+1) x 329.9 + 1382.5
= 55.8
b. W2 = qT2 – (q+1)B2 + GT
= (4x80.4) – (4+1) x 336.2 + 1382.5
= 23.1
c. W3 = qT3 – (q+1)B3 + GT
= (4x91.3) – (4+1) x 346.2 + 1382.5
= 16.7
d. W4 = qT4 – (q+1)B4 + GT
= (4x91.5) – (4+1) x 361.2 + 1382.5
= -57.5
e. W5 = qT5 – (q+1)B5 + GT
= (4x81.3) – (4+1) x 343.7 + 1382.5
= -10.8
f. W6 = qT6 – (q+1)B6 + GT
= (4x85.3) – (4+1) x 333 + 1382.5
= 58.7
g. W7 = qT7 – (q+1)B7 + GT
= (4x87.7) – (4+1) x 359.2 + 1382.5
= -62.7
h. W8 = qT8 – (q+1)B8 + GT
= (4x87.7) – (4+1) x 334.4 + 1382.5
= 61.3
i. W9 = qT9 – (q+1)B9 + GT
= (4x80.2) – (4+1) x 332.3 + 1382.5
= 41.8
j. W10 = qT10 – (q+1)B10 + GT
= (4x59.4) – (4+1) x 316.6 + 1382.5
= 37.1
k. W11 = qT11 – (q+1)B11 + GT
= (4x95.8) – (4+1) x 356.2 + 1382.5
= -15.3
l. W12 = qT12 – (q+1)B12 + GT
= (4x95.1) – (4+1) x 362.4 + 1382.5
= -49.1
m. W13 = qT13 – (q+1)B13 + GT
= (4x87.5) – (4+1) x 358 + 1382.5
= -57.5

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 Biometry

n. W14 = qT14 – (q+1)B14 + GT

= (4x99.9) – (4+1) x 360.9 + 1382.5
= -22.4
o. W15 = qT15 – (q+1)B15 + GT
= (4x90.8) – (4+1) x 339.5 + 1382.5
= 48.2
p. W16 = qT16 – (q+1)B16 + GT
= (4 x 87.9) – (4+1) x 360.3 + 1382.5
= -67.4

4. Compute Sum of squares for the different components. The best way is to
start with the adjusted block sum of squares because the mean square of the
block is an important component for making decision whether we continue the
analysis as lattice or as RCBD after comparing it with MS of error.
2

a. SS block (adjusted) = 3
∑ Wj
q (q + 1)

[(55.8) 2 + (23.1) 2 + ... + (−67.4) 2 ]

=
43 (4 + 1)
=109.14
SS block (adj )
b. MS block =
q2 − 1

109.14
=
42 − 1

= 7.28 = Eb
(GT ) 2 (1382.5) 2
c. Correction factor(C.F.) = = = 23891.33
rq 2 5 x 42
d. Total SS = ∑Xijk2 – CF = [(14.9)2 + (15.2)2 + … + (22.5)2] - 23891.33
= 566.4
e. SS treatment (unadj.)
2

SSt (unadj.) =
∑ Ti
− C.F .
r
(80.7) + (80.4) 2 + ... + (87.9) 2 ]
=
5
= 257.13
f. SS replication (SSR)

SSR =
∑ R j 2 − C .F .
q2

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 Biometry

[(258.6) 2 + (271.4) + ... + (294.5) 2

= -23891.33
42
= 72.71
g. SS due to error (SSE)
SSE = Total SS – SSt(unadj) – SS block(adj) – SSR
= 566.4 – 257.13 – 109.14 – 72.71
= 127.42
h. Degree of freedom for error = (q-1)(q2-1) = (4-1)(42-1) = 45
i. MSE = SSE/d.f. for error = 127.42/45 = 2.83 = Ee, Once the two statistics are
obtained, it is possible to check whether µ is positive or not. If it is positive we
will continue the analysis as lattice, if not as in RCBD.

Eb − Ee 7.28 − 2.83
j. µ = 2
= = 0.04, since µ is positive we will proceed to adjust
q Eb 4 2 x 7.28
treatment means as in Table 4.72.

Let T’j = Tj + µW j where Tj is unadjusted treatment total

Table 4.72. Computing adjusted treatment means

Treatment Tj Bj Wj T’j = Tj + µW j Adjusted
mean(T’j/r)
T1 80.7 329.9 55.8 82.93 16.59
T2 80.4 336.2 23.1 81.32 16.26
T3 91.3 346.2 16.7 91.97 18.39
T4 91.5 361.2 -57.5 89.20 17.84
T5 81.3 343.7 -10.8 80.87 16.17
T6 85.3 333.0 58.7 87.65 17.53
T7 87.7 359.2 -62.7 85.19 17.04
T8 87.7 334.4 61.3 90.15 18.03
T9 80.2 332.3 41.8 81.87 16.37
T10 59.4 316.6 37.1 60.88 12.18
T11 95.8 356.2 -15.3 95.19 19.04
T12 95.1 362.4 -49.1 93.14 18.63
T13 87.5 358.0 -57.5 85.20 17.04
T14 99.9 360.9 -22.4 99.00 19.80
T15 90.8 339.5 48.2 92.73 18.55
T16 87.9 360.3 -67.4 85.20 17.04

5. SSt (adjusted) =
∑T i
– CF
q +1
[(82.93) 2 + (81.32) 2 + ... + (85.20) 2 ]
= - 23891.33
4 +1

88
 Biometry

= 223.77
6. MSt = SSt(adj)/q2 –1= 223.77/42-1 = 14.92
7. Effective error MS = Ee (1 + qµ) = 2.83(1+4x0.04) = 3.2828
8. F-calculated = MSt/Effective error MS = 14.92/3.2828 = 4.54
9. Finally ANOVA table can be constructed as in Table 4.73

Table 4.73. ANOVA for balanced lattice design

Sources of d.f. SS MS F-cal F-tabulated

Variation 0.05
Replication 4 72.71 18.18
Block (adj) 15 109.14 7.28 2.57* 1.90
Treatment(adj) 15 223.77 14.92 4.54* 1.90
Intra-block error 45 127.13 2.83
Effective error 45 3.2828

*, significant at 0.05 level of probability,

Effective MS E 3.28
CV = ( ) x100 = = 11.56
GM 17.28

10. Compute standard errors

[ E (1 + qµ )] 2.83(1 + 4 x0.04)
SE(m) = e = = 0.81
r 5
[2 Ee (1 + qµ )] [2 x 2.83(1 + 4 x0.04)]
SE(d) = = ]=
r 5
CD/LSD = SE(d) x t0.05 at error degree of freedom
= 1.15 x 2.019
= 2.32
11. Compute relative efficiency of lattice over RCBD
SS B (adj ) + SS E
MSERCBD =
d . f . block + d . f . error

(109.14) + (127.42)
=
15 + 45
= 3.94

Effective error MS = 3.2828

RE = MSERCBD/Effective error MS = 3.94/3.2828 = 1.20

Table 4.74. Mean grain yield (t/ha) of the hybrids

_________________________________

Treatment Mean grain yield (t/ha)

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 Biometry

_________________________________
T14 19.80
T11 19.04
T12 18.63
T15 18.55
T3 18.39
T8 18.03
T4 17.84
T6 17.53
T16 17.04
T13 17.04
T7 17.04
T1 16.59
T9 16.37
T2 16.26
T5 16.17
T10 12.18
_________________________________
CD/LSD 2.54
c.v. 11.56%
_________________________________

Interpretation
Interpretation
Treatment effect was found to be significant (Table 4.73). T14 had the highest grain
yield (19.80 t/ha) followed by T11 and T12 (but statistically these three treatments did
not show differences among themselves) as shown in Table 4.74. There was also a
significant block effect implying that blocking helped in reducing experimental error.
The relative efficiency of 1.20 indicates that the use of lattice design instead of RCBD
improved precision by 20%.

4.6.3. Partially balanced lattice

4.6.3.1. Introduction
In partially balanced lattice designs, the number of replications is not restricted, but
the number of treatment must be a perfect square and the block size is equal to the
square root of the number of treatments. However, all treatments do not occur
together in the same block. This leads to differences in the precision with which some
comparisons are made relative to other comparisons. But still, the reduced replication
makes the experiment manageable for most research programmes while providing
the possibility of increased precision associated with smaller block size.

The names of the sub categories of partially balanced design follow the number of
replication. For example, the balanced lattice with two replications is called simple
lattice, with three replication triple lattice, with four replications quadruple lattice and
so on. In such arrangement, some treatment pairs never appear together in the same

90
 Biometry

incomplete block (λ = 1 or λ = 0). That is why treatment means are compared with
different level of precision, as there is more than one level of precision.

4.6.3. 2. Simple lattice

Simple lattices are types of partially balanced designs with two replications. Plans for
simple lattices can be obtained by taking the first two replications from the set of field
plans for balanced designs (Appendix 8). In simple lattice, unlike balanced lattices,
plans can be done for 36, 100 and 144 treatments as well as for those numbers of
treatments for which balanced designs are given (Cochran and Cox, 1957).

Let us assume that the experimental area has variability with unknown direction that
RCBD could not control the variability. We cannot use Latin square as it requires
larger experimental area that definitely minimize our precision. The resource we
have is not enough to use a balanced lattice. The next option is to use the simple
lattice design as it fulfills the requirements.

For 9 and 16 treatments, simple lattices are unlikely to be more accurate than RCBD
unless the variation among incomplete blocks is great compared with that within
incomplete blocks (Cochran and Cox, 1957). Besides, the degrees of freedom are
lower in simple lattices than RCBD.

Cochran and Cox, (1957) recommend to analyze the data as RCBD if missing
observations are numerous. The full analysis can also be done by estimating the
missing values by the formula, which minimizes the intra-block error sum of squares
as shown below. Hence, simple lattice design is suggested for use where the number
of treatments ≥ 16.

4.6.3.2.1. Layout
Suppose there are 16 treatments that need bigger plot size in a site where the
experimental area has got variation in soil acidity with unknown gradient which
cannot be controlled through blocking in one direction (as in RCBD) or in two
directions (Latin SD as it makes the experimental area very large, 16 x 16 LSD).
Therefore, the experiment is planned as a 4 x 4 simple lattice design so that the soil
acidity difference cannot contribute to intra-block error. The treatments are placed
into four blocks each containing four treatments as shown in Figure 4.8.

Block Rep-I Block Rep-II

1 1 2 3 4 5 1 5 9 13
2 5 6 7 8 6 2 6 10 14
3 9 10 11 12 7 3 7 11 15
4 13 14 15 16 8 4 8 12 16
Figure 4.8. Field plan for 4 x 4 balanced lattice design

91
 Biometry

4.6.3.2.2. Analysis

The stepwise analysis of variance for t number of treatments, k number of blocks and
r number of replication is as follows.

1. The totals for blocks, replications and treatments are calculated as shown in Table
4.75.

Table 4.75. Summary of data from lattice design

Rep Block Yield Sum B Sum Adj. Sum
1 1 x11(l) x12(l) ... x1k(l) B11 C11 AC11
2 x1k+1(2) x1k+2(2) ... x1k+k(2) B12 C12 AC12
. . . . . . .
. . . . . . .
. . . . . . .
k x1k2-k(k) ... x1k2(k) B1k C1k AC1k
Sum R1 C1
2 1 x21(l) x22(l) ... x2k(l) B21 C21 AC21
2 x2k+1(2) x2k+2(2) ... x2k+k(2) B22 C22 AC22
. . . . . . .
. . . . . . .
. . . . . . .
k x2k2-k(k) ... x2k2(k) B2k C2k AC2k
Sum R2 C2
. . . . . .
. . . . . .
. . . . . .
r 1 xr1(l) xr2(l) ... xrk(l) Br1 Cr1 ACr1
2 xrk+1(2) xrk+2(2) ... xrk+k(2) Br2 Cr2 ACr2
. . . . . . .
. . . . . . .
. . . . . . .
k xrk2-k(k) ... xrk2(k) Brk Crk ACrk
Sum Rr Cr
Sum GT 0 0
It is also necessary to compute the unadjusted treatment totals. These can be
tabulated with the adjusted treatment total and means as in Table 4.76.

Table 4.76. Summary of treatment total and adjusted means of simple lattice design
Total Mean
Treatment Unadjusted (Ti) Adjusted (TAi) Adjusted (AM)
1 T1 TA1 MA1
2 T2 TA2 MA2
. . . .
. . . .
. . . .
q Tq2 TAq2 MAq2

92
 Biometry

Ti = sum of the ith treatment, TA = treatment total adjusted (Ti + ACil over all blocks where the
ith treatment appear) and AM = adjusted total/number of replication.

The sum of squares for total, replication, treatment and error are computed as in any
other designs. The sum of squares due to block is a new statistic to be computed in
lattice designs.

(GT ) 2
1. Correction factor C.F. =
rq 2
2. Total SS = ∑∑X2ij(l) – C.F.
2

3. SSR =
∑ Rj
–C.F.
q2
2 2

4. SSB =
∑∑ C ij
-
∑C i

qr (r − 1) q 2 r (r − 1)
2

5. SSt =
∑T i
– C.F.
r
6. SSE = Total SS – SSR – SSB – SSt

7. Then the ANOVA table can be constructed as in Table 4.77.

Table 4.77. Analysis of variance table for simple lattice design

Sources d.f. SS MS
of variation
Replications (r-1) SSR
Treatments (t-1) SSt
Blocks (adj.) r(q-1) SSB Eb
Intra block error (q-1)(rq-q-1) SSE Ee

Total rq-1 Total SS

8. The mean square of block and error are computed as usual dividing the sum of
squares of block and error by their respective degrees of freedom:

93
 Biometry

SS B
Eb = and
r (q − 1)
SS E
Ee =
(q − 1)(rq − q − 1)

Then these two mean squares are compared either to go for adjustment factor or not.
If Eb ≤ Ee, the adjustment for block has no effect. That will lead us to ignore the
blocking restriction and analyze the data as if the design had been a randomized
block design with replications as blocks. If on the other hand, Eb> Ee, an adjustment
factor, µ, is computed for the design.
E − Ee
µ= b
qEb
This adjustment factor is used to compute ACil in Table 4.75 and adjusted treatment
means in the table of total, Table 4.76.

Finally, the effective error mean square that can be used in calculating t-test and
interval estimates is calculated as:
1 + rqµ
E’e = .
(q + 1) Ee
Adjusted treatment mean square is computed to test whether there is a significant
difference among adjusted treatment means or not. In order to do that, it is necessary
to compute first the unadjusted blocks within replications sum of squares (SSBu).
2

SSBu =
∑∑ Bil
– C.F. – SSR.
q
Then the adjusted treatment sum of squares, SSt(adj.), is computed as
2
SSt(adj.) = SSt(unadj.) – µq( )[SSBu/(r-1)(1+µq) – (SSB(adj.))].
1 + qµ
SSt
MSt =
t −1
MSt
F=
Ee
2 Ee
SE(d)1 = [1 + (r − 1) µ ] , is used for comparing treatments within a block
r
2 Ee
SE(d)2 = [1 + rµ ] , is used for comparing treatments in different blocks
r

CD1 = SE(d)1 x T-0.05 at error d.f. (same block)

CD2 = SE(d)2 x T-0.05 at error d.f. (different blocks)
SS B adj + SS E
A. MSE (in RCBD) =
block d . f . + error d . f .

94
 Biometry

Ee 1 + 2kµ
B. Effective MSE = E’e = [ ].
r (k + 1)
A
Relative efficiency (R.E.) of simple lattice over RCBD =
B
% efficiency = (R.E. – 1)100

4.6.3.2.3. Numerical example

A pathologist planned to study the reaction of 16 promising lines for rust resistance.
He inoculated all the lines uniformly except treatment-1, which was a control. He
suspected variability in terms of soil acidity in the experimental site and decided to
use simple lattice to conduct the experiment. Then, he collected data for grain yield to
determine loss assessment due to the pathogen. The data was given as in Table
4.78

Table 4.78. Data summary for 4 x 4 simple lattice design

Block Rep-I BK
1 650 (1) 670(2) 720(3) 680(4) 2720
2 685(5) 655(6) 670(7) 908(8) 2918
3 685(9) 680(10 680(11) 685(12) 2730
4 725(13) 690(14) 735(15) 685(16) 2835
CT1 2745 2695 2805 2958 11203
R1
Block Rep-II BK
5 735(1) 690(5) 840(9) 805(13) 3070
6 650(2) 685(6) 820(10) 735(14) 2890
7 670(3) 675(7) 685(11) 745(15) 2775
8 910(4) 790(8) 825(12) 855(16) 3380
CT2 2965 2840 3170 3140 12115
R2
GT = R1 + R2 = 11203 + 12115 = 23318

BK= block total, ( ) = treatment, CT1 = column total of rep-I, = column total of rep-II, xij = ith treatment
in jth block GT1 grand total of rep-I, GT2 = grand total of rep-II, R1 = total of rep-l, R2 = total of rep-II
and GT = grand total.

1. Compute treatment total

T1 = (t1 in rep-I + t1 in Rep-II) = (650 + 735) = 1385, in the same way,
T2 = 1320
T3 = 1390
T4 = 1590
T5 = 1375
T6 = 1340

95
 Biometry

T7 = 1345
T8 = 1698
T9 = 1525
T10 = 1500
T11 = 1365
T12 = 1510
T13 = 1530
T14 = 1425
T15 = 1480
T16 = 1540
2. Compute correction term for each block
B’1 = (B1 + C5) – rB1
= (2720 + 2965) – 2 x 2720
= 245 or = (T1+T2+T3+T4)-rB1
B’2 = (B2 + C6) – rB2
= (2918 + 2840) – 2 x 2918
= -78
B’3 = (B3 + C7) – rB3
= (2730 + 3170) – 2 x 2730
= 440
B’4 = (B4 + C8) – rB4
= (2835 + 3140) – 2 x 2835
= 305
B’5 = (B5 + C1) – rB5
= (3070 + 2745) – 2 x 3070
= -325
B’6 = (B6 + C2) – rB6
= (2890 + 2695) – 2 x 2890
= -195
B’7 = (B7 + C3) – rB7
= (2775 + 2805) – 2 x 2775
= 30
B’8 = (B8 + C4) – rB8
= (3380 + 2958) – 2 x 3380
= -422

∑B’K = B’1 + B’2 + B’3 + B’4 + B’5 + B’6 + B’7 + B’8

= 245 + (-78) + 440 + 305 + (-325) + (-195) + 30 + (-422)
=0
R’1 = B’1 + B’2 + B’3 + B’4
= 245 + (-78) + 440 + 305
= 912
R’2 = B’5 + B’6 + B’7 + B’8
= (-325) + (-195) + 30 + (-422)
= -912
∑R’j = R’1 + R’2

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 Biometry

= 912 + (-912)
=0

3. The different components are computed as follows

(GT ) 2 (23318) 2
a. Correction factor C.F. = = = 16991535.13
rq 2 2 x 42
b. Total SS = ∑∑X2ijk – C.F.
=(650)2 + (685)2 + … + (855)2 – 16991535.13
=173628.87

c. Sum of squares due to replication (SSR)

2

SSR =
∑ R j – C.F.
q2
(11203) 2 + (12115) 2
= – 16991535.13
42

=25991.995
d. Sum of squares due to block(unadjusted (SSB(unadj)
2

SSB (unadj) =
∑ Bk
– C.F.
q
(2720) 2 + (2918) 2 + ... + (3380) 2
= – 16991535.13
4
=84783.37
e. Sum of squares due to treatment (SSt)
2

SSt =
∑ Ti
– C.F.
r
(1385) 2 + (1320) 2 + ... + (1540) 2
= – 16991535.13
2
=84341.87
4. The SSB (adj), d.f. for block , MSB and Eb

2 2

SSB(adj) =
∑ B' k
–
∑ R' j

kr (r − 1) k 2 r (r − 1)
= [(B’12 + B’22 + . . . + B’82)/4x2(2-1)] - [(R’12 + R’22)/42x2(2-1)]
(245) 2 + (−75) 2 + ... + (−422) 2 (912) 2 + (−912) 2
= -
4 x 2(2 − 1) 42 x 2(2 − 1)
= 32437
5. Degree of freedom for block = 2(k-1) = 2(4-1) = 6

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 Biometry

SS B adj 32437
MSB(adj.) = = = 5406.17
block d . f . 6
6. The error sum of squares, degrees of freedom and mean square
SSE = Total SS – SSR – SSt(unadj.) – SSB(adj.)
= 173628.87 – 25991.995 – 84341.87 – 32437
= 30858.01
7. d.f. for error = (q-1)(2q-q-1) = (4-1)(2x4 – 4 – 1) = 9
SS E
MSE =
error d . f .
30858.01
=
9
= 3428.67

8. Compute µ
Eb − Ee
µ=
qEb
5406.17 − 3428.67
=
4(5406.17)
= 0.09

Since µ is positive we will proceed to analyze the data as lattice.

9. Compute sum squares of treatment adjusted (SSt(adj.)).

2
SSt(adj.) = SSt(unadj.) - qµ[( qµ)(SSB-unadj.)- SSB-adj.]
1 + qµ
2
= 84341.87 – 4x0.09[( ) [(2/(1+4x0.09))(84783.37) – 32437]
1 + 4 x0.09
= 51133.88

10. The ANOVA can be constructed as in Table 4.79

Table 4.79. Analysis of variance table for a lattice design

Source d.f. SS MS F-cal F-tabulated

of variation 0.05
Replications 1 25991.995 25991.995 7.58 5.12
Blocks /rep (adj.) 6 32437 5406.17 1.58 3.37
Treatments (adj.) 15 51133.88 3408.93 0.99ns3.00
Intra block error 9 30858.01 3428.67

ns = non-significant
c.v. = 8.04

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 Biometry

In the practical situations, the analysis is terminated at this point because the
treatment effect was not significant. As the manual is intended to show the statistical
methods related to this particular design, the standard error of differences and
adjusted treatment means will be computed to make the statistical procedure
complete. The remaining component is to compare the treatments following the
adjusted treatment means.

11. The adjusted treatment means

Adjusted treatment mean is computed as:
T + µ ( Bi1 '+ Bi 2 ' )
Ti = i ,
r
where T i = adjusted treatment mean, Ti = treatment total for that particular treatment
and B’ij = block correction terms in which that particular treatment is involved. Hence,

T’1 = [T1 + µ(B’1 + B’5)]/r

= [1385 + 0.09(245 + (-325))]/2
= 688.9
T’2 = [T2 + µ(B’1 + B’6)]/r
= [1320 + 0.09(245 + (-195))]/2
= 662.25
T’3 = [T3 + µ(B’1 + B’7)]/r
= [1390 + 0.09(245 + 30)]/2
= 707.38
T’4 = [T4 + µ(B’1 + B’8)]/r
= [1590 + 0.09(245 + (-422))]/2
= 787.04
T’5 = [T5 + µ(B’2 + B’5)]/r
= [1375 + 0.09(-78 + (-325))]/2
= 669.37
T’6 = [T6 + µ(B’2 + B’6)]/r
= [1340 + 0.09(-78 + (-195)]/2
= 657.72
T’7 = [T7 + µ(B’2 + B’7)]/r
= [1345 + 0.09(-78 + 30)]/2
= 670.34
T’8 = [T8 + µ(B’2 + B’8)]/r
= [1698 + 0.09(-78 + (-422))]/2
= 826.50
T’9 = [T9 + µ(B’3 + B’5)]/r
= [1525 + 0.09(440 + (-325))]/2
= 767.68
T’10 = [T10 + µ(B’3 + B’6)]/r
= [1500 + 0.09(440 + (-195))]/2
= 761.03
T’11 = [T11 + µ(B’3 + B’7)]/r

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 Biometry

= [1365 + 0.09(440 + 30)]/2

= 703.65
T’12 = [T12 + µ(B’3 + B’8)]/r
= [1510 + 0.09(440 + (-422))]/2
= 755.81
T’13 = [T13 + µ(B’4 + B’5)]/r
= [1530 + 0.09(305 + (-325))]/2
= 764.10
T’14 = [T14 + µ(B’4 + B’6)]/r
= [1425 + 0.09(305 + (-195))]/2
= 717.45
T’15 = [T15 + µ(B’4 + B’7)]/r
= [1480 + 0.09(305 + 30)]/2
= 755.08
T’16 = [T16 + µ(B’4 + B’8)]/r
= [1540 + 0.09(305 + (-422))]/2
= 764.74
12. Compute standard error of differences

2 Ee
SE(d)1 = [1 + (r − 1) µ ] , (for comparing treatments within a block)
r
2 x3428.67
= [1 + (2 − 1)0.09]
2
= 61.13

2 Ee
SE(d)2 = [1 + rµ ] , (for comparing treatments in different blocks)
r
2 x3428.67
= [1 + 2 x0.09]
2
= 63.61

CD = SE(d)1 x t0.05 at error d.f. (same block)

= 61.13 x 2.262
= 138.28
CD = SE(d)2 x t0.05 at error d.f. (different blocks)
= 63.61 x 2.262
= 143.89
13. Effective error mean square = Ee/r[1 + 2kµ/(k + 1)]
= (3428.67/2)[1 + (2 x 4 x 0.09]/(4+1)
= 1961.20
14. Error MS in RCBD = [SSB(adj.) + SSE]/(d.f.block + d.f.error)
= [32437 + 30858.01]/(6 + 9)
= 4219.67
15. Relative efficiency of simple lattice over RCBD (R.E.)

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 Biometry

R.E. = Effective error mean square/ Error MS in RCBD

= 4219.67/1961.20
= 2.15
% efficiency = (R.E. –1) x 100
= (2.15 – 1) x 100
= 115%

Interpretation
The effect due to block was significant indicating that blocking in lattice manner was
effective in controlling the variability of the soil acidity (helped in reducing
experimental error). The relative efficiency was 115% implying that the use of lattice
design increased precision by 115% compared to RCBD. The treatment effect was
not significant.

5. ANALYSIS OF VARIANCE – TWO OR MORE

FACTORS
FACTORS
5.5. Split-plot
5.2.1. Introduction
Split-plot design is an experimental design in which the levels of one factor could be
applied to larger plots while the levels of another factor are applied to smaller plots.
The levels of a specified factor are assigned to larger plots (main- or whole-plot) at
random that are grouped into blocks. The larger plots are then divided into smaller
plots within the larger plots. The larger plots are called main-plots or whole plots,
while the smaller units are called sub-plots.

In split-plot design with two factors, randomization is done at two levels

independently, one for the levels of the factor to be assigned in the main-plots and
the other one for the sub-plots. The levels of the main-plot factor are randomly
assigned to the main plots with in the blocks using separate randomization for each
block. The levels of the split-plot factor are randomly assigned to the sub-plots within
the main-plots using a separate randomization in each main plot.

The main-plot factor effects are estimated from the large units, while the sub-plot
factor effects and the interaction of the sub-plot factor with the main-plot are
estimated from the sub-plots. There are two experimental errors, one coming from
the main-plot and the other one from the sub-plot, as they differ in size of the plots.
Generally the main-plot error is larger than the sub-plot error and as a result the
main-plot factor is estimated with less precision than the factor in the sub-plot and the
interaction. Hence, it is important to consider the difference in precision at the time of
assigning factors into either of the plots. Gomez and Gomez (1984) suggested the
following guidelines in assigning factors into either main- or sub-plot: degree of
precision, relative size of the main effect and management practices.

101
 Biometry

The advantages of this design is that it permits the efficient use of some factors that
require different plot size, permits introduction of new treatments into an on-going
experiment and also provides increased precision in the estimation of some of the
factorial effects.

The disadvantages of the design are the less precision associated with the main-plot
(which requires large differences for significance) and that the statistical analysis is
more complex compared to designs for single factor experiments.

5.2.2. Lay-out
Two independent randomizations are necessary in this design. The first
randomization is done for the main-plot and the second for the sub-plot. In both
cases, the use of random number is important. Let the number of the main plot
treatment is denoted by a, the number of sub-plot treatments by b and the number of
replication by r. Then, there will be abr number of plots. Other components remain as
in RCBD.
5.2.3. Analysis
The analysis of variance for split-plot design involves two steps. The first step is main
plot analysis and the second step is the sub plot analysis. For example, if there are
two factors: factor A at a levels as a main plot and factor B at b levels replicated r
times. The computation will be done as follows:

(GT ) 2
1. Correction factor (C.F.) = , where a is the number of levels of A
abr
b is the number of levels of B
r is the number of replication
2. Total sum of squares (Total SS)
Total SS =∑∑∑Xijk 2 –C.F.

3. Sum of squares due to replication (SSR)

2

SSR =
∑ Rk
– C.F.
ab

4. Sum of squares due Main-plot (SSA). Here we need to have a two-way table
between main-plot and replication (Table 5.1).

Table 5.1. Factor A x Replication table of totals

A\rep I II . . . r Ai
a1 X11 X12 . . . X1r Ai
a2 X21 X22 . . . X2r A2
. . . . . . . .
. . . . . . . .
. . . . . . . .
an Xi1 Xi2 . . . Xir Ai
Rk

102
 Biometry

Then, SSA =
∑A i
– C.F.
br

5. Sum of squares due to Error (a)

Total SS ( AxR)
SSE(a) = – C.F. – SSA - SSR
b
6. Sum of squares due sub-plot (SSB)

We need to do a two-way table between A and B (Table 5.2)

2

SSB =
∑ Bj
– C.F.
ar

Table 5.2. Data structure for split plot design

S\rep b1 b2 . . . bj Ai
a1 X11 X12 . . . X1j Ai
a2 X21 X22 . . . X2j A2
. . . . . . . .
. . . . . . . .
. . . . . . . .
an Xi1 Xi2 . . . Xij Ai
Bj B1 B2 . . . Bj

7. Sum of squares due A x B (SSAxB)

Total SS ( AxB)
SSAxB = – C.F. – SSA - SSB
r
8. Sum of squares due to error (b)
SSE(b) = Total SS-SSR-SSA-SSE(a) – SSB - SSAxB
9. Degrees of freedom for replication (d.f. for R)
d.f. for R = number of replication (r)-1
10. Degrees of freedom for sowing time (d.f. for A)
(d.f. for A) = (a-1)
11. Degrees of freedom for error(a)
d.f. for Ea = (r-1)(a-1)
12. Degrees of freedom for plant density (d.f. for B) = b-1
13. Degrees of freedom for sowing time x Density (d.f. for AxB)
d.f. for AxB =(a-1)(b-1)
14. Degrees of freedom for error (b) (d.f. for Eb) = a(b-1)(r-1)
15. Mean squares
SS of each component divided by the corresponding degree of freedom
as in the ANOVA Table.
16. Calculate F-values

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 Biometry

F-calculated is obtained by dividing the MS of a component to the proper MS

error. In here we have to test Error(a) for significance by dividing its MS by MSE(b)
as in Table 5.3 and the expected mean square is given in Table 5.4.

Table 5.3. ANOVA for split plot design

Source of d.f. SS MS F-cal F-tabulated

variation 0.05 0.01
Replications r-1 SSR MSR MSR/MSE(a)
Factor A a-1 SSA MSA MSA/MSE(a)
Error(a) (r-1)(a-1) SSE(a) MSE(a) MSE(a)/MSE(b)
Factor B b-1 SSB MSB MSB/MSE(b)
AxB (a-1)(b-1) SSAxB MSAxB MSAxB/MSE(b)
Error (b) a(r-1)(b-1) SSE(b) MSE(b)

Table 5.4. Expected mean squares for split plot design

Source of d.f. E(MS)

variation
Replications r-1 δ2 + σ2 + abσR2
Factor A a-1 δ2 + σ2 + rσ2AxB + rbσA2
Error(a) (r-1)(a-1) δ2 + bσ2
Factor B b-1 δ2 + rσ2AxB + raσB2
AxB (a-1)(b-1) δ2 + rσ2AxB
Error (b) a(r-1)(b-1) δ2

5.2.4. Numerical examples

Example-I: Two factors
An experiment was conducted to study the effects of 3 population densities and 3
sowing time on grain yield of a wheat variety. The design was a split-plot with 6
replications. The main-plot factor was planting time at three levels (S1 = onset of the
rainfall, S2 = two weeks after the first rain, and S3 = 30 days after the first rain). The
sub-plot factor was population density at three levels ( D1, D2 and D3).

The statistical objectives of this exercise are to get familiarized with the analysis of
variance for split-plot, to identify the important factor, optimum level of each factor
and combination of factors, and to investigate whether there is interaction effect
between the two factors or not.

The biological objective is to investigate the optimum sowing time and plant density, if
factors are interdependent to identify best combination of the factors that can give
highest yield.

104
 Biometry

The hypothesis of the experiment is that the different levels of each factors act
independently and the alternative hypothesis at least two levels of the factors
interact. The data collected from the experiment was provided in Table 5.5 for
analysis.

Table 5.5. Grain yield (t/ha) of a wheat variety at different density and sowing time
Replication
Sowing date Total
I II III IV V VI Sowing
Density S1 Date(Si)
D1 5.59 5.50 5.25 5.31 4.63 4.69
D2 5.69 6.66 6.22 5.84 5.97 6.03
D3 6.90 5.28 6.66 6.19 6.40 6.38
Sum 18.18 17.44 18.13 17.34 17.0 17.1 105.19
S2
D1 5.84 5.63 6.14 5.94 6.13 6.08
D2 6.90 6.86 7.03 6.94 6.22 6.88
D3 7.09 6.72 6.25 6.36 6.34 6.81
19.83 19.21 19.42 19.24 18.69 17.77 116.16
S3
D1 5.23 5.41 5.25 5.71 5.06 5.75
D2 6.60 6.55 7.00 6.28 6.75 7.03
D3 6.03 6.52 6.12 5.72 6.13 6.88
Sum 17.86 18.48 18.37 17.71 17.94 19.66 110.02

Rk 55.87 55.13 55.92 54.29 53.63 56.53 331.37

Let the main plot is denoted by A, the sub plot by B and the number of replication by
r. Then, the stepwise analysis of variance for Split-plot with 6 replications will be done
as follows.

1. The linear additive model (L.A.M.): Xijk = µ + Ai, Bj, (AB)ij + Rk + Eijk where, Xijk =
ijkth observation, µ = the overall mean, Ai = the ith level of factor A, Bj = the jth
level of factor B, (AB)ij = ijth level of combination(interaction of factor A and B, Rk =
kth replication effect and Eijk = the error associated with ijkth observations.
Assumptions
Both factors (Density and sowing date) ~ NI (0, σ2t)
Rj ~ NI (0, σ2R)
Eijk ~ NIR (0, σ2)
(GT ) 2
2. Correction factor (C.F.) = , where a is the no. of levels of A
abr
(331.37) 2
= b is the no. of levels of B
3 x3 x 6
r is the no. of replication

105
 Biometry

=2033.45
3. Total sum of squares (Total SS)
Total SS = ∑∑∑Xijk 2 – C.F.
= [(5.59)2 + (5.50)2 + … + (6.88)2] – 2033.45
=21.00
4. Sum of squares due to replication (SSR)
2

SSR =
∑ Rk
– C.F.
ab
(55.87) 2 + (55.13) 2 + ... + (56.53) 2
= – 2033.45
3 x3
= 0.67
5. Sum of squares due sowing date (SSS)

Here we need to have a two-way table between main-plot and replication(Table 5.6).

Table 5.6. Grain yield (t/ha) of a wheat variety at different density and sowing time
S\rep I II III IV V VI Si
S1 18.18 17.44 18.13 17.34 17.0 17.1 105.19
S2 19.83 19.21 19.42 19.24 18.69 17.77 116.16
S3 17.86 18.48 18.37 17.71 17.94 19.66 110.02
Rk 55.87 55.13 55.92 54.29 53.63 56.53 331.37

Then, SSS =
∑S i
– C.F.
br
(105.19) 2 + (116.16)2 + (110.02) 2
= – 2033.45
3 x6
= 3.35
5. Sum of squares due to Error (a)
Total SS ( SxR)
SSE(a) = – C.F. – SSS - SSR
b
(18.18) 2 + (17.44) 2 + ... + (19.66) 2
= – 2033.45 - 0.67 - 3.35
3
= 0.91
6. Sum of squares due Density (SSD)

It is necessary to make a two-way table between sowing time and density as Table
5.7.

Table 5.7. Grain yield (t/ha) of a wheat variety at different density and sowing time
S\rep D1 D2 D3 Si
S1 30.97 36.41 37.81 105.19
S2 35.76 40.83 39.57 116.16
S3 32.41 40.21 37.40 110.02
Dj 99.14 117.45 114.78 331.37

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 Biometry

SSD =
∑D j
– C.F.
ar
(99.14) 2 + (117.45) 2 + ... + (114.76) 2
= – 2033.45
3x6
= 10.87
7. Sum of squares due sowing time x Densities (SSSxD)
Total SS ( SxD)
SS SxD = – C.F.
r
(30.97) 2 + (36.41) 2 + ... + (37.40) 2
= – 2033.45
6
= 1.01
8. Sum of squares due to error (b)
SSE (b) = Total SS – SSR – SSS - SSE(a) – SSD - SS SxD
= 21.00 – 0.67 – 3.35 – 0.91 – 10.87 – 1.01
= 4.19
9. Degrees of freedom for replication (d.f. for R)
d.f. for R = number of replication (r)-1
= 6-1
=5
10. Degrees of freedom for sowing time (d.f. for S)
d.f. for S = (a-1)
= (3-1)
=2
11. Degrees of freedom for error(a)
d.f. for Ea = (r-1)(a-1)
= (6-1)(3-1)
= 10
12. Degrees of freedom for plant density (d.f. for D)
d.f. for D = b-1
= 3-1
=2
13. Degrees of freedom for sowing time x Density (d.f. for IxD)
d.f. for IxD = (a-1)(b-1)
= (3-1)(3-1)
=4
14. Degrees of freedom for error (b) (d.f. for Eb)
d.f. for Eb = a(b-1)(r-1)
= 3(3-1)(6-1)
= 30
15. Mean squares
SS of each component divided by the corresponding degree of freedom
as in the ANOVA Table 5.8.
16. Calculate F-values

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 Biometry

F-calculated is obtained by dividing the MS of a component to the proper MS error. In

here we have to test Error(a) for significance by dividing its MS by MSE(b).

F- value for Error (a) = MSE(a)/MSE(b)

= 0.091/0.1397
= 0.65.

Then the calculated F-vales are compared with F-tabulated at 0.05 and 0.01 level of
probability: (10, 30) = 2.16 and 2.98. Error (a) has F-calculated less than F-tabulated,
i.e. not significant, and hence can not be used for testing the effect of replication and
sowing time. Because non-significant F-calculated indicates that the mean square
error (a) is zero and dividing any number by zero is not mathematically logical.

The interpretation when mean square error (a) is non significant is that our
consideration during planning of assigning the sowing time as main-plot was not
effective in controlling the error associated with the main plot. You may find some
literature using error (a) without testing but at this level it is important to be
considerate enough with this kind of apparent confusion.

Table 5.8. ANOVA table for sowing time x plant density study in split-plot design

Source of d.f. SS MS F-cal F-tabulated

variation 0.05 0.01
Replications 5 0.67 0.134 0.96 2.53 5.53
Sowing, S 2 3.35 1.675 18.41** 3.32 8.77
Error(a) 10 0.91 0.091 0.65 2.16 4.24
Densities, D 2 10.87 5.435 38.90** 3.32 8.77
SxD 4 1.01 0.2525 1.81 2.69 6.12
Error (b) 30 4.19 0.1397

**, significant at 0.01 level of probability

The analysis of variance showed that the effect of sowing time and plant density were
significant but the interaction effect was not-significant implying that the two factors
acted independently in this particular experiment. Hence the best level of each factor
can be identified by computing the critical value for each factor.

17. Standard error of the mean (SE (m))

MS E (b )
SE(m) for all components = ±
ar

108
 Biometry

0.1397
=±
3 x6
= ±0.09
2 MS E ( b )
SE(d) for all components =±
ar
2 x0.1397
=±
3x6
= ±0.13
CD(LSD) = SE(d) x t0.05 at error d.f.
= 0.1397 x 2.042
= 0.27

SE (m) to compare sub-plots within the same main-plot,

MS E (b )
=±
r
0.1397)
=±
6
= ±0.15
SE (d) to compare sub-plots within the same main-plot,
2MS E (b )
=±
r
2x0.1397
=±
6
= ±0.22
CD(LSD) to compare sub-plots within the same main-plot,
= SE(d) x t0.05 at error d.f.
= 0.22 x 2.042
= 0.45

Table 5.9. Mean yield (t/ha) of different sowing date and density on wheat

Treatment Mean yield CD c.v.(%)

1. Sowing date
S-1 5.84 0.22 4.91
S-2 6.45
S-3 6.11

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 Biometry

2. Density
D-1 5.51 0.27 6.10
D-2 6.53
D-3 6.38

Interpretation
The analysis of variance for split-plot design (Table 5.8) showed that the effects due
sowing dates and spacing were significant while the interaction was not indicating
that the two factors acted independently. The MS due to error (a) was not significant
indicating that assigning sowing date as the main plot was not correct.

The highest yield was obtained at sowing date (S-2) while the least was due to check
(S-1). Likewise, the highest yield was obtained at density (D-2). Hence, S-2 and D-2
can be recommended for the area where the experiment was conducted (Table 5.9).

5.3. Split-block (Strip-plot)

5.3.1. Introduction
The split-block (also called strip plot) design is a particular useful variation from split-
plot design in such a way that Latin square arrangements are used either on the
main-plots or sub-plots. This design is meant for an experiment with two factors each
at different levels. In this design the interaction effect between the two factors is
measured with higher precision than either of the factors. Accordingly, there are three
plot sizes: vertical strip plot, horizontal strip plot and the intersection plot.

Since there are three plot sizes, there will be three experimental errors, one for each
plot size. The interaction is measured with greater precision than the main effects.
The principal advantage of the split-block design is that it permits the efficient
application of factors that would be difficult to apply on small plots. It is also useful to
introduce new factor into an ongoing experiment. The disadvantages of this design
are similar to split-plot in that it has unequal precision (in estimating the interaction
and the main effects) and complicated statistical analysis.

5.2.1.Lay-out
In a split-block design, the levels of one factor are assigned to strips of plots running
through the block in one direction with a separate randomization in each block. The
levels of the second factor are then applied to strips of plots that are oriented
perpendicular to the strips for the first factor. The field plan for any number of
replication of two factors can be done as in RCBD, separate randomization for each
factor. Details can be obtained from (Gomez and Gomez, 1984).

5.2.2.Analysis
The analysis of data from split-block design follows the same general pattern as any
other design. For example, if there are two factors, i.e., factor A at a levels assigned
at random to a strip of b plots each and factor B at b levels to b strips of a plots each

110
 Biometry

crossing the strips used for the A factor. If the treatments are replicated r times and
Xijk represents an observation of the ith level of factor A, jth level of factor B in kth
replication, then the data can be summarized as in Tables 5.18. As there are three
plots, there will be three summary tables (Tables 5.18 to 5.20).

Table 5.18. Factor A x Replication table of totals

A\rep I II . . . r Ai
a1 X11 X12 . . . X1r Ai
a2 X21 X22 . . . X2r A2
. . . . . . . .
. . . . . . . .
. . . . . . . .
an Xi1 Xi2 . . . Xir Ai
Rk

Table 5.19. Factor B x Replication table of totals

B\rep I II . . . r Bj
b1 X11 X12 . . . X1r B1
b2 X21 X22 . . . X2r B2
. . . . . . . .
. . . . . . . .
. . . . . . . .
bn Xi1 Xi2 . . . Xir Bj
Rk

Table 5.20. Factor A x B table of totals

A\B 1 2 . . . b Ai
1 X 11 X 12 . . . X1b A1
2 X21 X22 . . . X2b A2
. . . . . . . .
. . . . . . . .
. . . . . . . .
a Xa1 Xa2 . . . Xab Ai
Bj B1 B2 . . . Bb GT

The different sources of variations are computed as follows:

(GT ) 2
1. Correction factor (C.F.) = , where a is the no. of levels of A
abr
b is the no. of levels of B
r is the no. of Replication
2. Total sum of squares (Total SS)
Total SS = ∑∑∑Xijk 2 –C.F.

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 Biometry

3. Sum of squares due to replication (SSR)

2

SSR =
∑ Rk – C.F.
ab
4. Sum of squares due Main-plot (SSA). Here we need to have a two-way table
between main-plot and replication (Table 5.18).

SSA =
∑A i
– C.F.
br
5. Sum of squares due to Error (a)
Total SS ( AxR)
SSE(a) = – C.F. – SSA - SSR
b

6. Sum of squares due to factor B (SSB)

We need to do a two-way table between factor B and replication (Table 5.19).

SSB =
∑ B j 2 – C.F.
ar

7. Sum of squares due to error (b)

Total SS ( BxR )
SSE (b) = – C.F. - SSR – SSB
a

8. Sum of squares due A x B (SSAxB)

Total SS ( AxB)
SS AxB = – C.F. – SSA - SSB
r
9. Sum of squares due to error (c)
Total SS – SSR – SSA – SSE(a) - SSB – SSE(b) – SSAxB

10. Degrees of freedom for replication (d.f. for R)

d.f. for R = number of replication (r)-1
11. Degrees of freedom for factor A (d.f. for A)
d.f. for A = (a-1)
12. Degrees of freedom for error(a)
d.f. for E(a) = (r-1)(a-1)
13. Degrees of freedom for factor B (d.f. for B) = b-1
14. Degrees of freedom for AxB (d.f. for A x B)
d.f. for A x B) =(a-1)(b-1)
15. Degrees of freedom for error (b) (d.f. for Eb) = (b-1)(r-1)
16. Degrees of freedom for error (c) (d.f. for Ec) = (a-1) (b-1)(r-1)
17. Mean squares
SS of each component divided by the corresponding degree of freedom
as in the ANOVA Table 5.21.
18. Calculate F-values

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 Biometry

F-calculated is obtained by dividing the MS of a component to the

proper MS error. In here we have to test MSA by MSE(a) and MSB by
MSE(b) and the interaction MS AxB by MSE(c).

Table 5.21. ANOVA for split block design

Source of d.f. SS MS F-cal F-tabulated

variation 0.05 0.01
Replication r-1 SSR MSR MSR/MSE(a)
Factor A a-1 SSA MSA MSA/MSE(a)
Error(a) (r-1)(a-1) SSE(a) MSE(a)
Factor B b-1 SSB MSB MSB/MSE(b)
Error (b) (r-1)(b-1) SSE(b) MSE(b)
AxB (a-1)(b-1) SSAxB MSAxB MSAxB/MSE(c)
Error (c) (a-1)(r-1)(b-1) SSE(b) MSE(b)

5.2.4. Numerical example

An agronomist planned to determine the interaction between green manure and date
of transplanting on tomato yield. Green manure was assigned to be horizontal factor
and has four levels (g1, g2, g3 and g4). Date of transplanting (taking seedlings from
seedbed to permanent field) was assigned to be vertical factor and has four levels
(d1, d2, d3 and d4). A tomato variety was used to study the different transplanting
dates. The experiment was designed in split-block (strip-plot) with four replications.
The data was collected on yield of tomato and formatted as in Table 5.22. Analyze
the data and make valid recommendation.

Table 5.22. Yield of tomato as affected by green manure and date of transplanting

Replication
I II III IV Gi
g1 d1 23.7 25.5 28.0 27.2
d2 40.8 42.3 48.8 49.2
d3 44.5 41.6 45.8 43.1
d4 25.7 27.2 28.2 26.6
Sub total 134.7 136.6 150.8 146.1 568.2
g2 d1 30.1 29.2 33.5 31.5
d2 45.6 48.3 50.2 46.6
d3 48.0 52.1 55.5 50.2
d4 29.2 31.5 30.2 53.5
Sub total 152.9 161.1 169.4 181.8 665.2
g1 d1 35.2 38.4 39.5 40.2
d2 50.5 52.6 58.5 54.5
d3 54.1 53.6 59.5 56.2
d4 32.0 31.8 34.0 32.8

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 Biometry

Sub total 171.8 176.4 191.5 183.7 723.4

g1 d1 38.3 40.2 42.5 41.5
d2 60.2 63.1 64.0 61.5
d3 65.3 68.1 69.2 67.6
d4 38.0 40.2 42.0 39.2
Sub total 201.8 211.6 217.7 209.8 840.9
Rj 661.2 685.7 729.4 721.4
Grand total 2797.7

Here, it is necessary to introduce notations that will be used in the analysis.

G = green manure and gi levels of G
D = date of transplanting, dj levels of D
g = number of levels of G
d = number of levels of D
GR = two way table between green manure and replication
DR = two way table between date of transplanting and replication
GD = two way table between green manure and date of transplanting

1. The linear additive model (L.A.M.): Xijk = µ + Ai, Bj, (AB)ij + Rk, + Eijk
where, Xijk = ijkth observation, µ = the overall mean, Ai = the ith level of factor A, Bj
= the jth level of factor B, (AB)ij = ijth level of combination(interaction of factor A
and B, Rk = kth replication effect and Eijk = the error associated with ijkth
observations.
Assumptions
Factors A and B ~ NI (0, σ2t)
Rj ~ NI (0, σ2R)
Eijk ~ NIR (0, σ2)
(GT ) 2
3. Correction factor (C.F.) = , where g= number of levels of G and
gdr
(2797.7) 2
= d = number of levels of D
4 x4 x4
=122298.83
4. Total sum of squares (Total SS)
Total SS = ∑∑Xijk 2 –C.F.
=(23.7)2 + (40.8)2 + … + (39.2)2 – 122298.83
=9558.08

A two way table between G and R is required to proceed (Table 5.23).

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 Biometry

Table 5.23. A two way table between G and R

Replication
Green manure I II III IV Gi
g1 134.7 136.6 150.8 146.1 568.2
g2 152.9 161.1 169.4 181.8 665.2
g3 171.8 176.4 191.5 183.7 723.4
g4 201.8 211.6 217.7 209.8 840.9
Rk 661.2 685.7 729.4 721.4 2797.7

5. Total sum of squares (Total SS for GR)

Total SSGR =
∑∑ X i.k
– C.F.
d
(134.7) 2 + (136.6) 2 + ... + (209.8) 2
= – 122298.83
4
= 2682.67

6. Sum of squares due to replication (SSR)

2

SSR =
∑ Rk
– C.F.
gd

(661.2) 2 + (685.7) 2 + ... + (721.4) 2

= – 122298.83
4 x4
= 189.44
7. Sum of squares due green manure (SSG)

SSG =
∑ Gi 2 – C.F.
dr
(568.2) 2 + (665.2) 2 + ... + (840.9) 2
= – 122298.83
4 x4
= 2436.34
8. Sum of squares due to Error green manure (g)
Total SSGxR
SSE(a) = – SSG - SSR
d
= 2682.67 – 2436.34 – 189.44
= 56.89

A two way table between D and R is required to proceed (Table 5.24).

Table 5.24. A two way table between D and R

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 Biometry

Replication
Green manure I II III IV Dj
d1 127.3 133.3 143.5 140.4 544.5
d2 197.1 206.3 221.5 211.8 836.7
d3 211.9 215.4 230.0 217.1 874.4
d4 124.9 130.7 134.4 152.1 542.1
Rk 661.2 685.7 729.4 721.4 2797.7

9. Total sum of squares (Total SS for DR)

Total SS DxR
Total SSDR = – C.F.
g
(127.3) 2 + (133.3) 2 + ... + (152.1) 2
= – 122298.83
4
= 6406.08

10. Sum of squares due to replication (SSR) the same as above = 189.44

11. Sum of squares due to transplanting date (SSD)

2

SSD =
∑ Bj
– C.F.
gr
(544.5) 2 + (836.7) 2 + ... + (542.1) 2
= – 122298.83
4 x4
= 6138.35

12. Sum of squares due to Error date of transplanting(b)

SSE(b) = Total SSDR – SSD – SSR
= 6406.08 – 6138.35 – 189.44
= 78.29

A two way table between G and D is required to proceed (Table 5.25).

Table 5.25. A two way table between G and D

Date of transplanting
Green manure d1 d2 d3 d4 Gi
g1 104.4 181.1 175.0 107.7 568.2
g2 124.3 190.7 205.8 144.4 665.2
g3 153.3 216.1 223.4 130.6 723.4
g4 162.5 248.8 270.2 159.4 840.9
Dj 544.5 836.7 874.4 542.1 2797.7

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 Biometry

6. Total sum of squares (Total SS GD)

Total SSGxD
Total SSGD = – C.F.
r
(104.4) 2 + (181.1) 2 + ... + (159.4) 2
= – 122298.83
4
= 8908.91

14. Sum of squares due to interaction between G and D

SSGxD = Total SS (DxR) – SSG - SSD
= 8908.91 – 2436.34 – 6138.35
= 334.22

15. Sum of squares due to error (c)

SSE(c) = Total SS - SSG-SSR - SSE(a)-SSD – SSE(b) - SSF-SSGxD
= 9558.08 – 2436.34 – 189.44 – 56.89 – 6138.35 – 78.29 – 334.22
= 324.55

Table 5.26. ANOVA Table for split-block design

Source of d.f. SS MS F-cal F-tabulated

variation 0.05 0.01
Replication 3 189.44 63.15
Green manure, G 3 2436.34 812.11 128.50** 3.86 6.99
Error (g) 9 56.89 6.32
Date of transp., D 3 6138.35 2046.12 235.19** 3.86 6.99
Error (d) 9 78.29 8.70
GxD 9 334.22 37.14 3.09* 2.25 3.15
Error (c) 27 324.55 12.02

*, **, significant at 0.05 and 0.01 level of probabilities

MS E ( a )
c.v. for green manure = ( ) 100
GM
6.32
=( ) 100
43.71
= 5.8%
MS E ( b )
c.v. for transplanting = ( ) 100
GM
8.70
=( ) 100
43.71

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 Biometry

= 6.7%
MS E ( c )
c.v. for interaction =( ) 100
GM
12.02
=( ) 100
43.71
= 7.9%
16. Standard error of the mean, difference and critical difference
a. Green manure
MS E ( a )
SE(m) = ±
dr
6.32
=
4 x4
= ±0.63

2 MS E ( a )
SE (d) = ±
dr
2 x6.32
=
4 x4
= ±0.89
CD(LSD) = SE(d) x t0.05 at error d.f.
= 0.89 x 2.262
= 2.01
b. Date of transplanting
MS E ( b )
SE(m) = ±
gr
8.7
=
4 x4
= ±0.74
2 MS E ( b )
SE (d) = ±
gr
2 x8.7
=
4 x4
= ±1.04
CD(LSD) = SE(d) x t0.05 at error d.f.
= 1.04 x 2.262
= 2.35
c. Interaction (any two means)

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 Biometry

MS E ( c )
SE(m) = ±
r
12.02
=
4
= ±1.73
2 MS E ( c )
SE (d) = ±
r
2x6.32
=
4
= ±2.45
CD(LSD) = SE(d) x t0.05 at error d.f.
= 2.45 x 2.052
= 5.03

Then, the mean values can be summarized as in Table 5.27.

Table 5.27. Mean values of green manure, date of transplanting and combinations

Treatment Mean
1. Green manure, G
g1 35.51
g2 41.58
g3 45.21
g4 52.56

SE(m) = 0.63
c.v. = 5.8%

2. Date of transplanting, D
d1 33.88
d2 34.03
d3 52.29
d4 54.65

SE(m) = 0.74
c.v. = 6.7%

3. Interaction, G x D
g1d1 26.10
g1d4 26.93
g2d1 31.08
g3d4 32.65
g2d4 36.10

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 Biometry

g3d1 38.33
g4d4 39.85
g4d1 40.63
g1d3 43.75
g1d2 45.28
g2d2 47.68
g2d3 51.45
g3d2 54.03
g3d3 55.85
g4d2 62.20
g4d3 67.55

SE(m) = 1.73
c.v = 7.9%

Mean separation

1. Green manure
g1
g2
g3
g4
2. Date of transplanting, D
d4
d1
d2
d3
3. Interaction, G x D
g1d1
g1d4
g2d1
g3d4
g2d4
g3d1
g4d4
g4d1
g1d3
g1d2
g2d2
g2d3
g3d2
g3d3

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 Biometry

g4d2
g4d3

Interpretation
The analysis of variance for split-block design (Table 5.26) showed that the effects
due green manure, date of transplanting and their interaction (green manure x date
of transplanting) were significant.

Among the green manure levels, g4 (52 units/plot) had the highest yield per plot,
followed by g3 while g1 had the lowest yield. Among the date of transplanting, d3 (54
units/plot) had the highest yield per plot, followed by d2 while d4 (which was not
statistically different from d1) had the lowest yield. The least yield might be due to late
planting that could not allow sufficient time for decomposition or the crop used as
green manure was inefficient to undergo decomposition and release the necessary
nutrients for the main crop (tomato) to be grown.

Of the over all combinations, g4d3 (67.55 units/plot) revealed the highest performance
may be due to the legume used could easily undergo decomposition to supply the
necessary nutrients and the date of transplanting could be the optimum that
maximized the utilization of growth resources.

5. FACTORIAL EXPERIMENTS
6.1. Introduction
There are many cases where experiments involve the study of the effects of two or
more factors. If interaction exist and are common, experiments are planned in such a
way that they can be measured and tested. This cannot be complete if we vary a
factor at a time. It is necessary to include combination of two or more factors at
different levels in the same experiment to deal with the interaction.

In factorial experiment, all possible combinations of the levels of the factors are
investigated. A factor is a kind of treatment, and in factorial experiments, any factor
will supply several treatments. The term levels refer to the several treatments within
any factor. Thus, a factorial experiment is one in which the set of treatments consists
of all possible combinations of the levels of several factors. For example, if there are
a levels of factor A and b levels of factor B, each replication contains all ab treatment
combinations. If the ab treatment combinations are replicated r times then, there will
be abr experimental units.

The term factorial refers to the nature of the treatments to be included in the
experiment but not to the experimental design. In short Factorial experiment is the
arrangement of treatments, it is not a type of design. Factorial experiments are used
in so many fields of research. They are of great value in exploratory research where
little is known concerning the optimum levels of the factors.

121
 Biometry

Factorial experiments have got several advantages. When the factors are
independent, the factorial approach can result in a saving of time and material as the
main effects can describe the results and serve as hidden replication, one factor
becomes an addition replicate of the other. When factors are not independent, the
factorial approach provides a systematic set of treatment combinations for estimating
and testing interactions.

The primary disadvantages of factorial experiments are that as the number of factors
increases, the total number of treatments becomes very large and large factorials can
be difficult to interpret, especially when interactions are all significant.

6.2. Two Factor Experiment

6.2.1.Introduction
There are several important inputs to maximize production from a sector. For
example, chemical fertilizer is one of the most important inputs for crop production of
which nitrogen and phosphorus are the principal elements required by plants in large
quantities. Here, Nitrogen and phosphorus are considered as two factors each with
different levels. Hence, determining the optimum levels of the two factors that can
maximize production is important. The levels of these two factors may act
independently or may have interaction. When the levels of the two factors are
independent, the focus will be more on the main effect and the best level is identified
from each factor. However, when factors are not independent, the focal point will be
the interaction effect and recommendation is based on the best combination.

6.2.2. Layout
Let us assume that there are four levels of nitrogen (n1, n2, n3 and n4) and 4 levels of
phosphorus (p1, p2, p3 and p4). Then, there be 4 x 4 = 16 combinations (treatments).
If these treatments are replicated three times then there will be 16 x 3 = 48
experimental units (plots). Let us further assume that a rice variety will be used and
data are collected on grain yield to identify the best combination or levels. As the
experiment is conducted in the field, RCBD is used. In conducting this experiment, all
possible combinations should be done (n1p1, n1p2, n1p3, n1p4, n2p1, n2p2, n2p3, n2p4,
n3p1, n3p2, n3p3, n3p4, n4p1, n4p2, n4p3 and n4p4). Secondly, these 16 treatments will be
randomized in each block independently and finally, the layout will be prepared as in
RCBD.

6.2.3. Analysis
The analysis of variance is also done as in RCBD taking combinations as treatments.
If the treatment effect is significant, then the treatment effect will be partitioned into
due to nitrogen, phosphorus and the interaction of nitrogen by phosphorus. If the
interaction is significant then, the best combination will be chosen, and if it is not
significant, recommendation will be based on the best level of each factor. The detail
analysis of variance for two factors are given below.

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 Biometry

6.2.4. Numerical example

An agronomist conducted an experiment to determine the effect of different levels of
nitrogen fertilizer (N0, N1, N2, N3 and N4) on three varieties (V1, V2 and V3). A 3 x 5
factorial experiment in RCBD with three replication was used, keeping all other
cultural practices as recommended for the area. The grain yield data were given in
Table 6.1 for analysis and appropriate interpretation.

Analysis of variance for factorial experiment is done twice. In the first part, only three
source of variations (block, treatment and error) are considered as in RCBD. If the
treatment effect is found to be significant, then the second part of analysis will be
done. Here, treatment refers to the different combinations of the levels of the two
factors.

Table 6.1. Yield data of three varieties tested at different levels of nitrogen fertilizer.

Treatment Block Treatment Total

I II II (Ti)
V1 N0 3.85 2.61 3.14 9.60
N1 4.79 4.94 4.56 14.29
N2 4.58 4.45 4.88 13.91
N3 6.03 5.27 5.91 17.21
N4 5.87 5.92 5.98 17.77

V2 N0 2.84 3.79 4.11 10.74

N1 4.96 5.13 4.15 14.24
N2 5.93 5.70 5.81 17.44
N3 5.66 5.36 6.46 17.48
N4 5.46 5.55 5.79 16.80

V3 N0 4.19 3.75 3.74 11.68

N1 5.25 4.58 4.90 14.73
N2 5.82 4.84 5.68 16.34
N3 5.89 5.82 6.04 17.75
N4 5.86 6.26 6.06 18.18

Rk 76.98 73.97 77.21 228.16

Part-I
1. The linear additive model (L.A.M.): Xikj = µ + Ai + Bj + (AB)ij + Rk + Eijk, where, Xijk =
the ith level of factor A, jth level of factor B, in kth block, µ = the overall mean, Ai =
effect of factor A, Bj = effect of factor B, (AB)ij, interaction effect, Rk , effect of block
and Eijk, = the error term associated with the ith level of factor A, jth level of factor
B, and (AB)ij in kth block.
Assumptions:

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 Biometry

All factors ~ NI (0, σ2t)

Bk ~ NI (0, σ2B)
Eijk ~ NI (0, σ2)
(GT ) 2
2. Correction factor (C.F.) = where, r= number of replication and
rt
(228.16) 2
= t = number of treatment, vn
3 x15
= 1156.82
3. Total sum of squares (Total SS)
Total SS = ∑∑Xij 2 –C.F.
= (3.85)2 + (2.61)2 + … + (6.06)2 – 1156.82
= 40.26
4. Sum of squares due to treatment (SSt)
2

SSt =
∑ Ti
– C.F.
r
(9.6) 2 + (14.29) 2 + ... + (18.18) 2
= – 1156.82
4
= 35.79
5. Sum of squares due to block (SSB)
2

SSB =
∑ Bj
– C.F.
t
(76.98) 2 + (73.97) 2 + ... + (77.21) 2
= – 1156.82
15
= 0.44
6. Sum of squares due to error (SSE) = Total SS – SSt - SSB
= 40.26 – 35.79 – 0.44
= 4.03

At this point we need to construct ANOVA and test the treatment effect as in Table 6.2

Table 6.2. ANOVA for 3 x 5 factorial experiment in RCBD

Source of d.f. SS MS F-cal F-tabulated

variation 0.05 0.01
Blocks 2 0.44 0.22 1.53 3.34 5.45
Treatments 14 35.79 2.556 17.75* * 2.06 2.80
Error 28 4.03 0.144

**, significant at 0.01 level of probability

Since the treatment effect was significant, it is necessary to proceed to compute the
three components of the treatment sum of squares (variety, nitrogen and Vx N) as
follows.

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 Biometry

Part-II: Make two way table between factor 1 and 2 (variety and nitrogen) (Table 6.3)

Table 6.3. Variety x Nitrogen

Nitrogen Variety Nitrogen Total

V1 V2 V3 (Nj)
N0 9.60 10.74 11.68 32.02
N1 14.29 14.24 14.34 43.26
N2 13.91 17.44 16.34 47.69
N3 17.21 17.48 17.75 52.44
N4 17.77 16.80 18.18 52.75
Variety Total (Vi) 72.78 76.70 78.68 228.16

7. Sum of squares of variety (SSV) where, n = levels of nitrogen

2

SSV =
∑ Vi
– C.F.
nr
(72.78) 2 + (76.70) 2 + (78.68) 2
= – 1156.82
5 x3
= 1.20

8. Sum of squares due to nitrogen (SSN)

2

SSN =
∑ Nj
– C.F.
vr
(32.02) 2 + (43.26) 2 + ... + (52.75) 2
= – 1156.82
3 x3
= 32.46

8. Sum of squares due to variety x nitrogen (SSVxN)

SSVxN = SSt – SSV – SSN
= 35.79 – 1.20 – 32.46
= 2.13

We partitioned the treatments sum of squares into three (SSV, SSN and SSVxN). We
can import the remaining two components (SSB and SSE) from Part-I ANOVA table
and then summarize the ANOVA as in Table 6.4.

Table 6.4. ANOVA for 3x5 factorial experiment in RCBD

Source of d.f. SS MS F-cal F-tabulated

variation 0.05 0.01
Blocks 2 0.44 0.22 1.53 3.34 5.45
Treatments 14 35.79 2.556 17.75* * 2.06 2.80
Varieties, V 2 1.20 0.60 4.17* 3.34 5.45
Nitrogen, N 4 32.46 8.115 56.35** 2.71 4.07

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 Biometry

VxN 8 2.13 0.266 1.85 2.29 3.23

Error 28 4.03 0.144

**,*, Significant at 0.01 and 0.05 level of probabilities, respectively

c.v. = 7.5%

9. Standard error of the mean (SE (m))

MS E
Variety: SE (m) =±
rn
0.144
=
3 x5
= ±0.10
MS E
Nitrogen: SE (m) =±
rv
0.144
=
3 x3
= ±0.13
10. Standard error of difference
2MS E
Variety: SE(d) =±
rn
2 x0.144
=
3 x5
= ± 0.14
2MS E
Nitrogen: SE(d) =±
rv
2 x0.144
=
3 x3
= ± 0.18

Once we have the CD values, we can compare paired means as follows.

Table6.5. Mean grain yield (t/ha) of three varieties evaluated under 5 levels of
nitrogen fertilizer.

Treatment Mean
1. Variety
V1 4.85
V2 5.11
V3 5.25

SE(m) = 0.10

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 Biometry

CD = 0.29

2. Nitrogen
N0 3.56
N1 4.81
N2 5.30
N3 5.83
N4 5.86

SE(m) = 0.13
CD = 0.37

c.v. = 7.5%

Mean comparison (DNMRT)

V1 N0
V2 N1
V3 N2
N3
N4

Interpretation
The effects due to variety and nitrogen were significant while their interaction was not
implying that these three varieties were not responsive to the different levels of
nitrogen in different ways, may be due to their similar genetic background. Hence,
instead of selecting the best combination, the two factors can be recommended
independently, that is, the best variety and the most economical nitrogen dose (Table
6.5).

Of the three varieties, V3 had the highest grain yield but V3 and V2 were not
statistically different. There was a significant difference between V1 and V3 and hence
it is better to use V3.

The highest yield was obtained at N4 level of nitrogen fertilizer but N4 and N3 were
not statistically different. Hence, N3 is expected to be more economical.

6.4. Fractional factorial experiment

As the number of factors increases the number of treatment combinations rapidly
increases which in turn increases within a block. Such variation within a block
reduces the precision of the experiment. Besides, the main effects can be estimated
if the interactions of all the factors are known to be negligible. In such a situation, a
fraction of the treatment combinations are selected and compared.

127
 Biometry

The difficulty of making formal significant tests using data from fractional factorial
experiments lies in the determination of proper error terms. Unless there are data
available from prior experiments, the error must come from a pooling of contrasts
representing effects that are presumed to be negligible.

7. ANALYSIS OF EXPERIMENTS OVER TIME

AND SPACE
7.1.
7.1. Introduction
Repetitions of the same experiment over space and/or time are quite common in a
research programme aiming at releasing technologies where they are targeted fore.
This is true because recommendations can seldom be effective if they are based on
the analysis of data from single isolated experiment. In line with this concept,
Cochran and Cox (1957), concluded that there will not be enough information from a
single experiment on which we can base recommendation. For example, a plant
breeding programme is often interested in generating new genotypes/progenies that
can give higher performance over several environments. The programme then is
interested in the response of selected new genotypes over a wide ranges of
environments for several seasons (years) within the domain that the genotypes are
developed fore. Maximum and valid research information can be obtained on such
events by running combined analysis of experiments.

The appropriate statistical analysis for data from series of experiments can vary with
the objective of the research (Cochran and Cox, 1957). The purpose of the analysis
of combined experiments is to generate valid recommendations for users to apply the
research results within a defined target area. According to Crossa (1990), data from
multi location experiments have three major objectives: to accurately estimate and
predict yield based on limited experimental data, to determine yield stability and
pattern of response of treatments (genotypes, agronomic practices, cropping
systems, feeds, etc.) across environments and to provide reliable guidance for
selecting the best treatments (genotypes, agronomic practices, cropping systems,
feeds, etc.) for use in future years and new environments.

7.2. The principle behind analysis of several experiments

Performance of a treatment depends on the treatment, the environment in which the
treatments are placed and the interaction between the treatment and the
environment. Treatments can be controlled by the researcher but most environmental
factors (climatic variables and some soil properties) are generally difficult to control
or modify for a given site and season. Thus a researcher with a one-time experiment
at a single site can evaluate only the controllable factors but not the environmental
factors that are beyond his control (Gomez and Gomez, 1984).

The effect of uncontrollable environmental factors on treatment performance is

important and quantifying their effects are essential. As the uncontrollable factors are
expected to change with season and site, and these changes are measurable, their

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 Biometry

effects on treatment performance can be evaluated. The most common way of

evaluating the effects of uncontrollable environmental factors on treatment response
is to repeat the experiment across environments and years (seasons).

A combined analysis of experiments can provide results on how treatment effects

and differences among treatment effects change in response to differences in soil
and climatic factors in the test environments (Crossa, 1990). When treatments are
evaluated in different environments, they interact with the environment. This
interaction is part of the behavior of the treatment and confounds its observed mean
performance with its true value. Assessing any treatment without including its
interaction with the environment is incomplete and thus limits the accuracy of yield
estimates. Therefore, a significant portion of the resource of crop improvement
programmes is devoted to determining this interaction through multiplication trials
(Crossa, 1990).

The function of the experimental design and statistical analysis of multilocation trials
is to eliminate as much as possible of the uncontrollable contained in the data.
Gauch (1988) mentions that statistical analysis of multilocation trials may have two
different objectives. With the first, a statistical model is constructed for a data set, and
success is measured in terms of model’s ability to fit this data set. With the second,
the data from a yield trial are portioned into modeling data and validation data.

7.3. Analysis
Analysis of combined experiments is done after doing some preliminary analysis.
Separate analysis of variance should be done for each experiment followed by
testing experimental errors for homogeneity and a preliminary estimate of the
interaction of treatment with the environment/season.

When considering the combed analysis of data from several experiments, the first
requirement is to assess the homogeneity of the error variance at from each
experiment. If the errors are homogeneous, the analysis can proceed. If the errors
are heterogeneous, then the data can be transformed to produce homogeneous
variance or the locations may be separated into groups within which the variance is
homogeneous.

In the process of preliminary combined analysis, the pooled error mean square could
be used to test effects of treatment, locations and the interaction (T x E). If the T x E
effect is found to be significant, then we will have to candidate mean squares to test
the effect of treatment and location, the T x E MS or the pooled MS from error SS
and T x E SS as denominator.

7.3.1. The same experiment across environments

In the analysis of combined experiment of data from several environments, the first
requirement is to assess the homogeneity of the error variance at the various
environments. If the errors are homogeneous, the analysis can proceed. If the error
variances are heterogeneous, the data will be transformed (data transformations are

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 Biometry

presented in Section 11) to produce homogeneous variance or the locations may be

separated into groups within which the variance is homogeneous.

The data presented in Table 7.1 are the mean performance of five genotypes tested
at six environments for two years using randomized block designs with six
replications.

Table 7.1. Yield (kg/ha) performance of five genotypes tested at six environments

Genotype Environments Genotype Total

___________________________________________
E1 E2 E3 E4 E5 E6 Gi
G1 702 1498 583 2140 1454 2262 8639
G2 770 1266 614 2071 1194 2187 8102
G3 833 1429 723 2115 1276 2137 8513
G4 527 1557 718 2056 1098 2133 8089
G5 905 1482 626 2074 1454 2399 8940
Ej 3737 7232 3264 10456 6476 11118 42283

Individual location analysis of variance is the first step towards combined analysis of
variance. One way of combing such data is to calculate the error associated with
each mean (Me’). This can be done by importing error mean square, degree of
freedom for error and number of replications from individual location analysis of
variance as follows.

E1 E2 E3 E4 E5 E6

MSE 5776 4028 4516 9526 7056 5535

d.f. 20 20 20 20 20 20
Replication 6 6 6 6 6 6

Me’ 962.67 671.33 752.67 1587.67 1176 922.5

MSE = mean square error at each environment, r = number of replicate at each environment,
E = environment, Gi = total of each genotype, and Ej = total of each environment
MSE
Me’ = = the error associated with each mean,
r

In order to combine the data the error with which each mean is measured should be
tested for homogeneity which is one of the basic assumptions of analysis of variance.
A quick test of homogeneity of variance is provided by the ratio of the largest mean
square error to the smallest mean square error in the set (Gomez and Gomez, 1984).
The result is then compared with tabulated F-value at error degrees of freedom for
individual location. In our case,

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 Biometry

L arg est MS E 9526

= = 2.36.
Smallest MS E 4028

F-value with f1 = f2 = 20 is 2.94. Since 2.36 is less than 2.94, it can be concluded that
the error variances are not heterogeneous.

An alternative procedure, which is more sensitive than the ratio test, is Bartlett’s test
of homogeneity of variance (Snedecor and Cochran, 1980). This test is based on the
natural logarithm of sample variances. The procedure is as follows: let kj is error
degrees of freedom for individual location, Mej error mean square for individual
location, log is the natural logarithm to the base ten, the Bartlett’s χ2 can be
calculated as follows (Table 7.2).

Table 7.2. Statistical components for Bartlett’s test

Environment Kj Mej LogMej KjlogMej 1/Kj
E1 20 5776 3.76 75.2 0.05
E2 20 4028 3.61 72.2 0.05
E3 20 4516 3.65 73.0 0.05
E4 20 9526 3.98 79.6 0.05
E5 20 7056 3.85 77.0 0.05
E6 20 5535 3.74 74.8 0.05
120 36437 451.8 0.30

R j log MS E − ∑ K j log MS E
χ2 = 2.3026 [ ] , n = number of environments
1 1 1
1 + (n − 1)(∑ −
3 kj ∑kj
453.6 − 451.8
= 2.3026 [ ]
1 1
1 + (6 − 1)(0.30 −
3 120
= 4.06

Since the calculated χ2 value (4.06) is less than the tabulated χ2-value (11.07), then
χ2 test is non-significant. Hence the data can be combined because non-significant χ2
test implies the errors associated with each mean are homogenous.

(GT ) 2 (42283) 2
1. Correction Factor C.F= = =59595069.63; g= genotype & e= environment
ge 5 x6
2. Total SS = ∑x.j2 – C.F
= [(702) 2+ (770) 2+…+ (2399) 2] – 59595069.63
= 11067063.37
3. Sum of squares due to Genotype (SSG)

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 Biometry

SSG =
∑G i
– C.F., where Gi is total of ith genotype
e
(8639) 2 + (8102) 2 + ... + (8940) 2
= - 59595069.63
6
= 88499.54
1. Sum squares due to environment (SS)
2

SSE =
∑ Ej
- C.F., where Ej is total of jth environment
g
(3737) 2 + (7232) 2 + ... + (11118) 2
= – 59595069.63
5
= 10764355.37
2. Sum squares due to G x E interaction (SSGxE)
SSGxE = TSS- SSG- SSE; where G x E is the interaction between G and E
= 11067063.37 - 88499.54 - 10764355.37
= 214208.46
3. Sum squares due to pooled error (SSE)

SSE (pooled error) =

∑ kl ME j
r
(20 x5776) + (20 x 4028) + ... + (20 x5535) 2
=
6
= 1214556.67
After computing all the components we can construct ANOVA for the preliminary
analysis as in Table 7.3.

Table 7.3. Preliminary analysis of variance

Source of d.f. SS MS E(MS)

variation
Environments, E 5 107643355.37 2152871.07** σ2e + rσ2GxE+ rgσ2E
Blocks/E 30 1574832.72 52494.42** σ2e + gσ2B/E
Genotypes, G 4 88499.54 22124.89** σ2e + rσ2GxE+ reσ2G
GxE 20 214208.46 10710.42** σ2e+ rσ2GxE
Pooled Error 120 121456.80 1012.12 σ2e

**, significant at 0.01 level of probability

The genotypes in the above ANOVA table are tested using pooled error mean square
and that will tell us whether the effect of genotypes is significant or not. On the other
hand, the mean square of the genotypes can also be tested using the G x E mean
square and that will tell us whether there is consistency in the genotype performance
across the environment or not. For the genotypic effect it will be

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 Biometry

MSG 22124.89
F= = = 3.07
MSGxE 10710.42

The calculated F(3.07) is greater than the tabulated F(2.87) indicating that the effect
due to genotype was significant. For the environmental effect
MS E 2152871.07
F= = = 201.01.
MSGxE 1071042

The calculated F(201.01) is greater than the tabulated F(2.71) indicating there was
significant effect of the environment.

Since effects of genotypes, environment and genotype by environment interaction

were significant it is necessary to calculate SE(m), SE(d) and the critical difference as
well as partition the G x E interaction sum of squares into a set of orthogonal
contrasts to get more information on the nature of interaction (why the relative
performance of genotypes differ over environments). For this example, we compute
SE(m), SE(d) and the critical difference as there is another example on partitioning of
treatment x location interaction SS into a set of orthogonal contrasts in Section 7.2.2.

Me' 1012.14'
SE(m)= ± =± = ±12.99
e 6
2Me' 2x1012.14
SE(d) = ± =± =± 18.37
6 6
CD = SE(d) x t0.5 at error D.F. = 18.37 x 1.98 = 36.37

Then the mean performance of the genotypes across environments is summarized in

Table 7.4. Since the table does not show performance of each genotype across
environments, this pattern will be illustrated using Figure 7.1.

Table 7.4. Mean yield (kg/ha) of five genotypes tested at six environments

Genotype Environment
___________________________________________________
E1 E2 E3 E4 E5 E6
G1 702 1498 583 2140 1454 2262
G2 770 1266 614 2071 1194 2187
G3 833 1429 723 2115 1276 2137
G4 527 1557 718 2056 1098 2133
G5 905 1482 626 2074 1454 2399

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 Biometry

INTERPRETATION
The combined analysis showed the effect of genotypes, the environment and their
interaction were significant at 0.01 level of probability. Genotype 5 had the best mean
performance across locations (Figure 7.1). It out yielded all others at E1, E5 and E6
(Table 7.4) but G1 yielded the same as G5 at E5. G1 is the second best though G1 and
G3 were not statistically different. It performed highest at E2, E4 and as equal as G5 at
E5.

Of all the testing sites E6 is the best followed by E4 while E3 is the poorest. Significant
Genotype X Environment indicates that genotypes were not consistent in
performance across the environments (Figure 7.1.)

Figure 7.1. Genotype x environment interaction

3000

2500

2000
Mean yeild (kg/ha)

G1
G2
1500 G3
G4
1000 G5

500

0
E1 E2 E3 E4 E5 E6
Environment

7.4. Summary
Data from several experiments help researchers estimate treatment performance
more accurately, select better production alternatives, and understand the interaction

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 Biometry

of these technologies with environments. Several methodologies have been

presented for efficient statistical analysis of such data to generate relevant
information from experiments conducted over space and time. The results are often
difficult to interpret if treatment by environment interaction is significant. The addition
of certain detailed analysis is required to get the real effect of the treatment especially
when the error terms are heterogeneous.

There are several criticisms towards the analysis of combined experiments especially
when data violates the basic assumptions of analysis of variance. The first criticism is
that some components of the T x E SS may be much larger than others, the
interaction variance is not constant which happens when treatment response varies
greatly from place to place while others vary little or not at all. A method of coping
with this difficulty is to divide the treatment SS into a set of orthogonal components
that will supply all the necessary information. The interaction SS is partitioned in the
same way so as to isolate the interaction of each component with places. If the
interaction variance is not still constant , it is valid to test any component with its own
interaction with places. The drawback of this method is that the degree of freedom for
the denominator of F- are reduced.

A second criticism related to the assumption that the experimental error variances
are the same in all experiments. This assumption hold if all experiments have been
conducted in the same way, with the same control over environmental variables and
experimental material of the same variability. In practice this is seldom achieved
because of the natural variability of soil, climate and crop species. Therefore, the
experimental error variances varies from place to place and when we feel that the
variation is high (> 3X the smaller) it is necessary to run Bartlett’s test of
homogeneity.

Hence, it is better to use additional statistics and available soft wares and more
attention has to be devoted to the collection, analysis, and interpretation of
biophysical variables. This can help to characterize particular genotypes and
geographical regions and therefore, better explain certain aspects of the interaction.

8. ANALYSIS OF COVARIANCE
8.1. Introduction
The analysis of covariance is concerned with two or more measured variables where
any measurable independent variable is not at the predetermined levels as in
factorial experiment (Steel and Torrie, 1980). The covariance between two random
variables is a measure of the nature of the association between the two. It makes use
of the concept of both analysis of variance and of regression, i.e., it detects the
variances and covariances of specific variables to estimate treatment effect more
accurately than the use of ANOVA alone.

The application of covariance analysis can be extended to any number of covariates

and to any functional relationship between variables. However, this manual is limited

135
 Biometry

to a single covariate that has linear relationship with the trait of primary interest. The
importance of covariance analysis in improving precision has been emphasized by
several authors (Cochran and Cox, 1957; Steel and Torrie, 1980; Gomez and
Gomez, 1984; and Walpole et al., 2002). The core remarks of the authors can be
summarized as follows. The most important uses of covariance analysis are to:
control error (increase precision), adjust treatment means; assist in the interpretation
of data, partition the total covariance into component parts and estimate missing
data.

The use of covariance to control error is by means of regression in such a way that a
certain recognized effects that can not be controlled effectively by experimental
design. The main task here is to control the variation due to the physical conduct of
the experiment but not usually on inherent variation. Examples of variation that can
be controlled by covariance analysis are: when crop varieties with different number of
plants per plot are evaluated together for yield (number of plants per unit area and
yield are positively associated in most cases) we can use number of plants per plot
as a covariate and there by estimate the true treatment effect. Estimation of the true
treatment effect will enable us to adjust treatment means. The adjusted treatment
means in turn help in proper interpretation of data.

By measuring additional variable (covariate, X) that is known to be associated with

the primary variable (Y) linearly, the source of variation associated with the covariate
can be deducted from experimental error. After that the primary variable can be
adjusted linearly upward or downward, depending on the relative size of its
respective covariate. By doing so, the treatment mean is adjusted and the
experimental error is reduced and the precision for comparing treatment means is
increased. In this case the covariate must not be affected by treatments being tested.

7.2. Covariance Analysis and Blocking

Covariance analysis should be considered in experiments in which blocking can not
adequately reduce experimental error. This is true because blocking is done before
the start of the experiment, it can be used only to cope with sources of variation that
are known or predictable. Analysis of covariance can take care of unexpected
sources of variation that occurs during the process of experimentation. Hence,
covariance analysis is a supplementary procedure to take care of sources of variation
that can not be accounted for by blocking.

Covariance analysis is essentially an extension of the analysis of variance and

hence, all the assumptions for a valid analysis of variance which was mentioned in
Section 1.2 are also important here. Besides, covariance analysis requires that the
covariates to be fixed (measured without error and independent of treatment), the
regression of the dependent variable (Y) on the covariate (X) after removal of block
and treatment differences is linear and the errors are normally and independently
distributed with mean zero and a common variance.

136
 Biometry

The identification of covariate is an important task in the application of covariance

analysis. Assigning the covariate is highly determined by the purpose for which the
covariance technique is applied. When there is irregular stand establishment in field
experiments, the number of plants per plot becomes an important source of variation.
Here, covariance analysis can be used with stand count as a covariate. Another
example in animal science could be determination of rate of body weight gain of
different animals feeding on certain ration (the experimental animal do vary in age,
age and body weight gain are positively associated with age) then the initial age of
the animals can be used as covariate.

7.3. Numerical example

Example-
Example-I
An experiment was conducted to identify the best variety in RCBD with five
replications. In the process of execution of the experiment, there was water stress
that resulted in variation in the number of plants per plot within a block. Such
variation can not be controlled by experimental design since there was no gradient
for the stress at the planning time. To minimize the effect of variation caused by the
stress, the researcher decided to use covariance analysis using plant stand as a
covariate and yield as dependent character as shown in Table 8.1.

Table 8.1. Grain yield and number of plants (stand count) of eight varieties evaluated
in field experiment
Replication
Variety

Ti Si TiSi
I II III IV V
S T S T S T S T S T
1 20 30.50 16 27.25 15 17.50 12 13.25 14 18.50 107.00 77 8239
2 20 36.25 15 21.75 15 18.50 16 16.50 15 19.75 112.75 81 9132.75
3 15 24.25 12 18.75 13 20.25 11 21.75 14 18.75 103.75 65 6743.75
4 14 18.25 20 25.25 19 26.25 14 18.25 14 16.50 104.50 81 8464.50
5 20 30.50 20 36.50 15 20.50 16 23.50 15 19.25 130.25 86 11201.5
6 20 40.50 10 22.25 12 20.75 10 21.75 8 12.25 117.50 60 7050.00
7 15 20.75 16 25.75 10 21.25 12 20.25 9 10.50 98.50 62 6107.00
8 20 48.75 15 21.75 16 20.75 11 20.50 12 22.50 134.25 74 9934.50
Bj 144 249.75 124 199.25 115 165.75 102 155.75 101 138 908.5 586 66873
BjBj 35964 24707 19032.5 15886.5 13938
Si = number of plants per plot and Ti = variety total

The different components for yield are computed as follows.

(GT ) 2
1. Correction factor (C.F.) = where, r= number of replication and
rt

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 Biometry

(908.5) 2
= t = number of treatment
5 x8
= 20634.31
3. Total sum of squares (Total SS)
Total SS = ∑∑Xij 2 –C.F.
=(30.5)2 + (27.25)2 + … + (22.5)2 – 20634.31
= 2160.32
4. Sum of squares due to treatment (SSt)
2

SSt =
∑ Ti
– C.F.
r
(107) 2 + (112.75) 2 + ... + (134.25) 2
= – 20634.31
5
= 234.19
5. Sum of squares due to block (SSB)
2

SSB =
∑ Bj
– C.F.
t
(249.75) 2 + (199.25) 2 + ... + (138) 2
= – 20634.31
8
= 972.03
6. Sum of squares due to error (SSE)
SSE = Total SS – SSt – SSB
= 2160.32 – 234.19 – 972.03
= 954.10

Table 8.2. Analysis of variance for grain yield data alone

Sources of F-tabulated
variation d.f. SS MS F-cal 0.05 0.01
Blocks 4 972.03 243.01 7.13* 2.71 4.07
Varieties 7 234.19 33.46 0.98 2.36 3.36
Error 28 954.10 34.08

*, significant at 0.05 level of probability

The different components for number of plants per plot (stand) are computed as
follows.

(GT ) 2
1. Correction factor (C.F.) = where, r= number of replication and
rt
(586) 2
= t = number of treatment
5 x8
= 8584.90

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 Biometry

2. Total sum of squares (Total SS)

Total SS = ∑∑Xij 2 –C.F.
=(20)2 + (16)2 + … + (12)2 – 8584.90
= 437.10
3. Sum of squares due to treatment (SSt)

SSt =
∑ Ti 2 – C.F.
r
(77) 2 + (81) 2 + ... + (74) 2
= – 8584.90
5
= 133.5
4. Sum of squares due to block (SSB)
2

SSB =
∑ Bj
– C.F.
t
(144) 2 + (124) 2 + ... + (101) 2
= – 20634.31
8
= 972.03
5. Sum of squares due to error (SSE)
SSE = Total SS – SSt – SSB
= 437.10 – 133.5 – 157.85
= 145.75

Table 8.3. Analysis of variance for number of plants per plot (stand)

Sources of F-tabulated
variation d.f. SS MS F-cal 0.05 0.01
Block 4 157.85 39.46 7.57* 2.71 4.07
Treatment 7 133.50 19.07 3.66* 2.36 3.36
Error 28 145.75 5.21

*, significant at 0.05 level of probability

The covariance analysis will be computed as follows.

(GTS xGTT )
1. Correction factor (C.F.) =
rt
(586 x908.5)
=
5 x8
= 13309.53
2. Total sum of products (Total SP)
Total SP = ∑Sij2 x Xij – C.F.
=(30.5x20) + (27.25 16) + … + (22.5x12) – 13309.53
= 712.97

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 Biometry

3. Sum of product due to treatment (SPt)

2

SPt =
∑ Ti
– C.F.
r
(107 x77) + (112.75 x81) + ... + (134.25 x74)
= – 13309.53
5
= 65.07
4. Sum of product due to block (SPB)
2

SPB =
∑ Bj
– C.F.
t
(249.75 x144) + (199.25 x124) + ... + (138 x101)
= – 13309.53
8
= 381.47
5. Sum of product due to error (SPE)
SPE = Total SP – SPt – SPB
= 712.97 – 65.07 – 381.47
= 266.43
SPE XY
6. b(regression of y on x) =
SSE X
266.43
=
145.75
= 1.83
( SPE XY ) 2
7. SSy adjusted for x error = SSy error –
SSE X
(266.43) 2
= 954.10 –
145.75
= 467.07
8. Adjusted d.f. = (r-1)(t-1) – 1
= (5-1)(8-1) – 1
= 27
SSy − adjusted for x
9. MSE adjusted =
Error d . f . − Adjusted
467.07
=
27
= 17.30
10. Treatment + Error
d.f.(t+e) = d.f. trt + d.f. error = 7 + 28 =35
SSx(t+e) = SSt of x + SSE of x = 133.50 + 145.75 = 279.25
SPxy(t+e) = SPxy trt + SPxy error = 65.07 + 266.43 = 331.50
SSy(t+e) = SSt of y + SSE of y = 234.19 + 954.10 = 1188.29
Adjusted d.f. (trt + error) = [(t-1)(r-1)] + (t-1) – 1 = [(8-1)(5-1)] + (8-1) – 1 = 34

140
 Biometry

(trt + eSPXY ) 2
Adjusted SS (trt + e) = trt + e SSy –
trt + eSS X
(331.5) 2
= 1188.29 –
279.25
= 794.76
Adjusted SS (trt + e)
Adjusted MS (trt+ e) =
Adjusted d . f .(trt + e)
794.76
=
34
= 23.38
Treatment Adjusted d.f. = Adjusted d.f. (trt + e) - Adjusted d.f. error 34 – 27 =7
Treatment adjusted SS = Adjusted SS (trt + e) – Adjusted SS error
= 794.76 – 467.07
= 327.69
Treatment adjusted MS = Treatment Adjusted SS/ Treatment adjusted d.f.
= 327.69/7
= 46.81
Treatment Adjusted MS 46.81
F-cal = = = 2.71
Adjusted MS E 17.30

Table 8.4. Analysis of covariance for grain yield data with plant stand as covariate

Source of d.f. SSx SPxy SSy Reg d.f. SS MS F-cal F-tabulated

Variation 0.05 0.01
Blocks 4 157.85 381.47 972.03
Treatments 7 133.5 65.07 234.19
Error 28 145.75 266.43 954.1 1.83 27 467.07 17.30
Error +trt 35 279.25 331.5 1188.29 34 794.76 23.38
Trt adjusted 7 327.69 46.81 2.71* 2.37 3.25

*, significant at 0.05 level of probability

c.v = 18.3%

In order to adjust mean of the grain yield, the following components are necessary
which will be illustrated in tabular form for simplifying the computation (Table 8.5).

Table 8.5. Adjusted mean

Trt Mean mean

Yield number Mean yield
(y) of plants (xi – x) b( x i – x ) y-adjusted
T1 21.40 15.4 0.75 1.3725 20.03
T2 22.55 16.2 1.55 2.8365 19.71
T3 20.75 13.0 -1.65 -3.0195 23.77

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 Biometry

T4 20.90 16.2 1.55 2.8365 18.06

T5 26.05 17.2 2.55 4.6665 21.38
T6 23.50 12.0 -2.25 -4.8495 28.35
T7 19.70 12.4 -2.25 -4.1175 23.82
T8 26.85 14.8 0.15 0.2745 26.58
Sum = 0.0 Mean = 22.71

x = grand mean = 14.65

Mean yield adjusted = mean yield - b( x i – x ), for example, for T1:

Mean yield adjusted = 21.4 – 1.83(15.4 –14.65) = 20.03

It can also easily be done as:

Mean yield adjusted = Column II – Column V for example, for T2
Mean yield adjusted = 22.55 – 2.8365 = 19.71

In the same way, it can be done for:

T3: 20.75 – (-3.0195) = 23.77
T4: 20.90 –2.8365 = 18.06
T5: 26.05 – 4.6665 = 21.38
T6: 23.50 – (-4.8495) = 28.35
T7: 19.70 – (-4.1175) = 23.82
T8: 26.85 – 0.2745 = 26.58

MSt X
Effective MSE = MSE adjusted [1+ ]
SSE X
19.07
= 17.30 [1+ ]
145.75
= 19.56

% gain in efficiency = [(MSE unadjusted of y/Effective MSE) – 1] x 100

= [(34.08/19.56) – 1] x 100
= 174%

Interpretation
The analysis of variance for grain yield alone (Table 8.2) showed that treatment
effects were nonsignificant. However, after covariance analysis, taking number of
plants per plot as covariate, the treatment effects were found to be significant
indicating that covariance analysis helped to reduce experimental error, that is the
true treatment effects were revealed after covariance analysis (Table 8.4). The
ANOVA table for number of plants per plot showed that there was a significant effects
of stand count (Table 8.3).

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Covariance analysis increased the efficiency of the experiment by 74% implying that
covariance analysis is important in improving precision. Covariance analysis is also
an important method to adjust treatment means in such a way that treatments
affected by the plant stand (number of plants per plot during harvesting) was
adjusted and showed differences in their rank.

9. CORRELATION
9.1. Introduction
Correlation is used to study the relationship between two types of measurements
made on the same individuals. Correlation should not be used without first examining
the data using a scatter diagram. The scatter diagram provides a visual impression of
the nature of relation between two variables (x and y) in a bivariate data set. In many
cases, the points appear to band around a straight line. The visual impression of the
closeness of the scatter to a linear relation can be quantified by calculating a
numerical measure, called correlation coefficient.

The correlation coefficient, denoted by r, is a measure of strength of the linear

relation between the x and y variables. Before introducing its formula, some important
features of the correlation coefficient are outlined and the manner in which it serves
to measure the strength of a linear relation are discussed.

The value of r is always between –1 and +1. The magnitude of r indicates the
strength of a linear relation, whereas its sign indicates the direction. More specifically,
r >0 if the pattern of x and y values is a band that runs from the lower left to upper
right, r < 0 if the pattern of x and y values is a band that runs from the upper left to
lower right, r = +1 if all x and y values lie exactly on a strait line with positive slope
(perfect positive relation) and r = -1 if all x and y values lie exactly on a strait line with
negative slope (perfect negative relation). A high numerical value of r, closer to either
+1 or -1 represents a strong relationship. A value of r closer to zero means that the
linear association is very weak or nil.

The correlation coefficient is closer to zero when there is no visible pattern of relation;
that is, the y value does not change in any direction as the x values change. A value
of zero could also happen because the points band around the curve that is far from
linear (may be curvilinear).

9.2. Computing r Values

The sample correlation coefficient quantifies the association between two numerically
valued characteristics. It is calculated from n pairs of observations on two characters
(x1,y1), (x2,y2), . . . , (xn, yn). The correlation coefficient is best interpreted in terms of
the standardized observations:
Observation − means xi − x ∑ (x i − x )2
= , where = sx = .
s tan dard deviation sx n −1

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The sample correlation coefficient is the sum of the products of the standardized x
observation times the standardized y observations, divided by n-1.
1 x −x y −y
r= ( i )( i )
n −1 ss sy

When the pair ((xi,yi) has both components above their sample means or both below
their sample means, the product of the standardized observations will be positive;
otherwise it will be negative. Consequently, if most pairs have both components
simultaneously above, or simultaneously below, their means, r will be positive. An
alternative formula for r is used for calculation. It is obtained by canceling the
common term n-1. Then
s xy
r= where sxy = (xi – mean)(yi – mean) = covariance of x and y
s xx s yy

It is sometimes convenient, when using hand-held calculators, to check r-values

using the alternative formulas for sxx, syy and sxy. The quantity sxx and syy are the
variance of the x observations and the y observations, respectively. sxy is the sum of
the cross products of the x deviations with the y deviations (Table 9.1).

Table 9.1. Measurement on two variables (x and y) for calculating r

x y x- x y- y (x- x )2 (y- y )2 (x- x )(y- y )

2 5 0 1 0 1 0
1 3 -1 -1 1 1 1
5 6 3 2 9 4 6
0 2 -2 -2 4 4 4

Total 8 16 0 0 14 10 11
x =2 x =4 sxx syy sxy

s xy
r =
s xx s yy
11
=
14 x10
= 0.93

A third option to compute r is shown below for the data in Table 9.2.

Table 9.2. Measurement on two variables (x and y)

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for calculating correlation, r

_______________________________
x y x2 y2 xy
_______________________________
2 5 4 25 10
1 3 1 9 3
5 6 25 36 30
0 2 0 4 0
________________________________
Total 8 16 30 74 43
∑x ∑y ∑x2 ∑y2 ∑xy

(∑ x) 2 (∑ y ) 2
∑ xy − ∑ n ∑
( x)( y )
sxx = ∑ x2 − n
, syy = ∑ y2 − n
and sxy =

∑ xy − ∑ n ∑
( x)( y )

r = ___________________________

(∑ x −
∑ ( x) )(∑ y − ∑ ( y)
2
2
2
2

)
n n

8 x16
43 −
= 4
82 162
(30 − )(74 − )
4 4

= 0.93

9.3. Correlation and Causation

Data analysis often makes unjustified conclusions by making mistakes on an

observed correlation for a cause-and-effect relationship. A high sample correlation
coefficient does not necessarily signify a causal relation between two variables. The
observation that two variables tend to simultaneously vary in a certain direction does
not imply the presence of a direct relationship between them. If the number of people
smoking are recorded x and level of education y for several cities of widely varying in
size, the data may probably indicate a high positive correlation. It is the fluctuation of
the third variable (namely, the city population) that cause x and y to vary in the same
direction, despite the fact that x and y may be unrelated or even negatively related.
The third variable, which is causing the observed correlation between smokers and
level of education, is referred to as a lurcking variable. The false correlation that it
produces is called spurious correlation (Trumbo, 2002).

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 Biometry

It is a matter of common sense than of statistical reasoning to determine if an

observed correlation has a practical interpretation or is spurious. Therefore, when
using the correlation coefficient as a measure of relationship, it is important to be very
careful to avoid the possibility that a lurking variable is affecting any of the variables
under consideration.

10. REGRESSION
10.1. Introduction
Regression analysis is a statistical technique for investigating and modeling the
relationship between variables (Montgomery et al., 2001). Regression is similar to
correlation as it is used for testing a linear relationship between two types of
measurements made on the same individuals. However, regression goes further in
that it is also possible to produce an equation describing the line of best fit through
the points on the graph. When using regression analysis, unlike in correlation, the
two variables have different roles. Regression is used when the value of one of the
variables is considered to be dependent on the other, or at least reliably predicted
from the other. The dependent variable is denoted by y and the independent variable
by x. Hence, the consideration is very important because regression of y on x is not
the same as regression of x on y.

Regression analysis concerns the study of the relationships between variables with
the objective of identifying, estimating and validating the relationship. The objective of
regression analysis is the development of a statistical model that can predict the
values of a variable based upon the values of another variable. In this Section the
subject will be presented with specific reference to the straight-line model. Then, on
the basis of the model, it is possible to test whether one variable actually influences
the other or not.

Regression analysis is one of the most widely used techniques for analyzing
multifactor data. Its usefulness results from the logical process of using an equation
to express the relationship between a variable of interest and a set of related
predictor variables. For example, an experimental study of the relation between two
variables is often motivated by a need to predict one variable from the other. The
director of a job-training programme may wish to study the relation between the
duration of training and the score of the trainee on subsequent skill test. A forester
may wish to estimate the timber volume of a tree from the measurement of the trunk
diameter a few meters above the ground (breast height). An agronomist may be
interested in predicting the grain yield of maize at different levels of nitrogen fertilizer.
A medical technologist may be interested in predicting the blood alcohol
measurement from the read-out of newly devised breath analyzer.

In such context as these, the predictor or input variable is denoted by x, and the
response or output variable is labeled y. The objective is to find the nature of
relation between x and y from experimental data and use the relation to predict the
response variable y from the input variable x. The first step in such a study is to plot
and examine the scatter diagram. If a linear relation emerges, the calculation of the

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numerical value of r will confirm the strength of the linear relation. Its value indicates
how effectively y can be predicted from x by fitting a straight line to the data.

A regression problem involving a single predictor (also called simple regression)

arises when there is a need to study the relation between two variables x and y and
use to predict y from x. The variable x acts as an independent variable whose values
are controlled by the experimenter. The variable y depends on x and also subjected
to unaccountable variations or errors (Trumbo, 2002).

For any experiment, n is used to denote the sample size or number of runs of the
experiment. Each run gives a pair of observations (x, y) in which x is the fixed setting
of the independent variable and y denotes the corresponding response as shown in
Table 10.1.

Table 10.1. Data structure for a simple regression

Setting of the independent variable Response
x1 y1
x2 y2
x3 y3
. .
. .
. .
xn yn

It is advisable to begin the analysis by plotting the data because the eye can easily
detect patterns along a line curve. The scatter diagram for the observations reveals
that the relationship is approximately linear in nature, that is, the points seem to
cluster around a straight line. Because a linear relation is the simplest relation to
handle mathematically, details of the statistical regression analysis will be presented.

10.2. Determining a Linear Regression Line with a Single Predictor

A line is determined by two constants: its height above the origin (intercept) and the
amount that y increases whenever x is increased by one unit (slope). The equation of
the line fitted by method of least squares can be shown as follows.

If the relation between x and y is exactly a straight line, then the variables are
connected by the formula y = β0 + β1x, where β0 indicates the intercept of the line
with the y-axis and β1 represents the slope of the line, or the changes in y per unit
change in x.

Statistical ideas must be introduced into the study of relation when the points in a
scatter diagram do not lie perfectly on a line. The data on an underlined linear
relation that is being masked by experimental error due to in part differences in
severity of the disease, application of treatment, the plot to plot variation and so on.

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 Biometry

With these points in mind, the following regression model can be presented as a
tentative representation of the mode of relationship between y and x. Assuming that
the response y is a random variable that is related to the input variable x by

yi = β0 + β1xi + ei, i = 1, …, n

where yi denotes the response corresponding to the ith experimental run in which the
input variable x is set at the value xi, e1, …, en are the unknown error components
that are super imposed on the true linear relation. These are unobservable random
variables, which we assume are independently and normally distributed with mean
zero and unknown standard deviation σ, and the parameters β0 and β1, which
together locate the straight line, are unknown.

As in this model, the observation yi corresponding to level xi of the controlled variable

is one observation from the normal distribution with mean = β0 + β1xi and standard
deviation = σ. One interpretation of this is that as we attempt to observe the true
value on the line, nature adds the random error e to this quantity. All these
distributions have the same standard deviation and their means lie on the unknown
true straight-line β0 + β1x. As σ is unknown, the line on which the means of these
normal distributions are located is also unknown (Draper and Smith, 1981). An
important objective of the statistical analysis is to estimate this line.

10.3
10.3. Numerical Example

A chemist wishes to study the relation between the drying time of a paint and a
concentration of a base solvent that facilitates a smooth application. The data of
concentration setting x and the observed drying times y are recorded in the first two
columns (Table 10.2). Plot the data, calculate r and determine the fitted line.

Table 10.2. Data on concentration (x) and drying time (y) in minutes
Concentration, x Drying time, y x2 y2 xy
0 1 0 1 0
1 5 1 25 5
2 3 4 9 6
3 9 9 81 27
4 7 16 49 28
Total 10 25 30 165 66

To calculate r and determine the equation of the fitted line, the basic quantities mean
of x, mean of y, variance of x variance of y and covariance of xy should be calculated
first using the total in Table 10.2.
10
Mean of x =
5
=2

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 Biometry

25
Mean of y =
5
=5

102
Sxx = 30 -
5
= 10
252
Syy = 165 –
5
= 40
25
Sxy = 66 – 10 x
5
= 16,

16
Then, r =
40 x10
= 0.8
16
β1 =
10
= 1.6
β0 = 5 - (1.6)2
= 1.8, then the equation of the fitted line is

ŷ = 1.8 + 1.6x .

The simple correlation r was introduced as a measure of association between two

variables. When r is near 1 or –1, points in the scatter diagram are closely clustered
about a straight line and the association is high. In these circumstances, the value of
one variable can be accurately predicted from the value of the other. In other words,
when the value of r2 (coefficient of determination) is near 1, we can predict the value
of y from its corresponding x value can be predicted. In all cases, the slope of the
least squares line β1 and the sample correlation r are related since
S yy
β1 = r .
S xx

If the sample correlation is positive, then the slope of the lease squares line is
positive. Otherwise both are negative or both zero. The main idea of regression in the
context of specific experiment can explained using the following example of a new
fungicide against purple blotch (fungal disease of onion).

An experiment was conducted to study the effect of different doses of the fungicide
treatment on the disease (purple blotch). A replicated experiment was conducted
having the different doses. Seven different dosages were used in the experiment and
some of these were repeated for more than one plot which was done purposely to

149
 Biometry

show that regression and correlation can be done either on plot basis, on treatment
mean basis or a combination of the two. Each plot received a specific dosage of the
fungicide and yield data per plot were recorded as in Table 10.3.

Table 10.3. Dosage x (mg) and bulb yield per plot y (kg) of onion
Dosage, x Bulb yield, y
3 9
3 5
4 12
5 9
6 14
6 16
7 22
8 18
8 24
9 22

A glance at the table and Fig. 10.1 shows that y generally increases with x, but it is
difficult to say much more about the form of the relation simply by looking at this
tabular data that will lead us to the method of least squares.

Figure10. 1. Effect of fungicide on the control of onion

disease

30
25
Bulb yield

20
15 Series1
10
5
0
0 5 10
Fungicide dosage

11. DATA TRANSFORMATION

When data violet the assumption of the analysis of variance (normality,
independence, randomness, additivity and homogeneity), there may be two
possibilities: either transforming the data to fulfill the assumptions or to use non-

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 Biometry

parametric statistics. The disadvantage of using a non-parametric statistics is that it is

less powerful. However, for most biological science experiments where there are two-
way and three-way analysis of variance and regression, there is no straightforward
equivalent to non-parametric statistics (Townend, 2002). Therefore, transforming the
data is the only option although interpretation of the results of transformed data is
more difficult.

To transform the data, it is important to start with the measured values (x) and
perform transformation to get new set of values (x’). The analysis of variance or
regression is then carried out on this new sets of values. Some of the most
commonly used transformations are described below.

Log transformation is the most commonly used method. It is usually used for data
that violet normality (positively skewed distribution). The new value,

x’ = log10(x+1).

If the measurement has zero value, it is necessary to add one to all values before
applying the log transformation for the reason that natural logarithm of zero is
undefined. Adding one to each value does not affect the outcome (variance) but it is
necessary to deduct the same value at the end of the analysis from the mean.

The square root transformation is given by x’ = √x. This kind of transformation is used
when there are data from Poisson distribution (data from counts).

Angular or arcsine transformation is often used when each value in the data set is a
proportion or percentage of something. It only applies where the minimum and
maximum values are 0 and 1 (or 0 and 100%). If the original data is expressed as
percentages, then x must be converted to values between 0 and 1 before
transforming (eg. 60% = 0.6). Then,

x’ = sin-1√x or x’ = arcsin√x.

Box-Cox transformation is given by x’ = (xλ-1)/λ (if λ≠0) or x’ = ln x (if λ=0). It looks

unfriendly but it has the advantage that it is possible to calculate the value of λ which
best achieves a normal distribution. This is not a straightforward to do by hand, but a
function is available on some computer packages to calculate the best value of λ.
Once the value of λ is obtained, it is fairly easier to put it in the above formula to get
the transformed values.

After the transformed values are generated, it is necessary to check the new values
meet the assumptions by graphing for normality, for example. If the transformed
values do not meet the assumptions, it is better to go back to the original data and try
another type of transformation. If none of the above transformations do not work, it is
better to try another simple formula like cube root transformation x’ = 3√x, 1/x or 1/x2.
It is not always possible to achieve a normal distribution using simple transformation,

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especially when the data have more than one peak (multi-modal distribution). This
probably indicates that the population is actually made up of two or more populations
which would be better treated as separate populations in the analysis.

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12.
12. REFERENCES
Cochran, W.G and G.M. Cox. 1957. Experimental designs. 2nd edition. Wiley,
New York.
Gomez, A. and A. Gomez. 1984. Statistical procedure for agricultural research. John
Wiley & Sons, Inc., New York.
Little, T.M. and F.J. Hills. 1978. Agricultural experimentation: design and analysis.
John Wiley & Sons, Inc., New York.
Mandefro Nigussie. 2005. Statistical procedures for designed experiments. Ethiopian
Agricultural Research Organization (EARO), Addis Ababa, Ethiopia. pp 241.
Steel, R.G.D. and J.H. Torrie.1980. Principles and procedures of statistics, 2nd
edition. McGraw-Hill Manual Company, New York.
Wardlaw, A.C. 2000. Practical statistics for experimental biologist. John Wiley and
Sons, LTD, New York.

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