STA408
Design and Analysis of Experiments
Lesson 3
JAVED IQBAL
Department of Statistics
Virtual University of Pakistan
Randomized Complete Block
Design
Testing the effects of two independent
variables or factors on one dependent variable.
Hypothesis Testing – General
Procedure
Main Points
• Experimental material is divided into groups
or blocks
– Why
• Each block contains a complete set of
treatments
• Treatments are assigned randomly
– Randomizations is restricted within block
Main Points
• A new randomization is done for each
block/group
• It has different layouts: 2X3, 3X5, etc
• The RCBD is perhaps the most frequently used
experimental design
CRD vs RCBD
May look like
The Agricultural Research Field
The Agricultural Research Field
Experimental Layout
Allot treatment randomly
BLOCK - I
BLOCK - II
BLOCK - III
Statistical Model
• The liner model for this design is:
Yij Bi T j eij
• Where:
= True mean effect
T = Effect of treatment
B = Effect of block/groups
e = effect of random error
Data Computations Table
Block Treatments Block
(Groups)
1 2 … j … k Total Mean
1 Y11 Y12 B1• 𝑌1•
2
r
Total B•1
Means 𝑌•1
Partitioning of Variations
Total Variation due to Variation due to Variation due to
Variation Treatment Block/Grouping Unknown factors
Total SS = Treatment SS + Block SS + Error SS
(Yij Y ) 2 r (Y j Y ) 2 k (Yi Y ) 2 (Yij Y j Yi Y )
i j j i i j
AVOVA Table for RCB
Source of
Degree of
Variation MS = Mean
Freedom SS = Sum of Squares F -test
s.o.v Squares
d.f
(Variations SSTr St2 MST
due to) K-1 SSTr r (Y j Y ) 2
S
2
Ft 2
k 1
j t
Treatment Se MSE
(Variations SSB Sb2 MSB
due to) r- 1 SSB k (Yi Y ) 2
S
2
Fb 2
r 1
i b Se MSE
Blocks
(Variations
SSTr
due to) (k-1) (r-1) SSE = By Subtraction S
2
(r 1)(k 1)
e
Error
Total rk-1 SST (Yij Y ) 2 ---
i j
Example
• Four verities of wheat ere tested in randomized complete clock design in
four replications/blocks. Yield in kg/plot is shown in the table below.
• Test the hypothesis that there is no difference in the means of four
varieties.
Blocks Varieties of wheat
V1 V2 V3 V4
I 2 5 4 1
II 2 3 3 1
III 4 6 6 2
IV 1 4 2 3
Testing Hypothesis – Example
Step 1
Stat the hypothesis.
𝐻0 ∶ 𝑇𝑗 = 0 (There is NO difference in the means of four varieties of wheat)
𝐻1 ∶ 𝑇𝑗 ≠ 0 (There is difference in the means of four varieties)
( Not ALL the four means are equal)
𝐻0 ∶ 𝐵𝑖 = 0 (There is NO difference in the means of four varieties of wheat)
𝐻1 ∶ 𝐵𝑖 ≠ 0 (There is difference in the means of four varieties)
( Not ALL the four means are equal)
Step 2
We set the level of significance (alpha) 𝜶 = 𝟎. 𝟎𝟓 (5%)
Testing Hypothesis – Example
Step 3
The test-statistic (formula) to be used is:
St2 MS (Treatment )
F 2
Se MS ( Error )
It follows F-distribution with degree of freedom
df = v1 = from treatment = k-1
df = v2 = from error = (k-1)(r-1)
Computations
Blocks Varieties (Treatment)
2
V1 V2 V3 V4 Block Block Sum of Square
Total of each value
Total
I 2 5 4 1 2+5+4+1 = 12 12*12 = 144 22 52 42 12
46
II 2 3 3 1 9 81 23
III 4 6 6 2 18 324 92
IV 1 4 2 3 10 100 30
Varities
Total
2
Varities
Total
Sum of Square
of each value
Computations
T
SST Sum of Squares of Total Y 2
n
2 5 4 ... 3
2
= 2 5 4 ... 3
2 2 2 2
16
(49) 2
191 40.94
16
T
SSTr Sum of Squares of Treatment(varities) Y 2
n
2 5 4 ... 3
2
= 2 5 4 ... 3
2 2 2 2
16
(49) 2
191 40.94
16
Computations
T2 T
SSTr Sum of Squares of Treatment(varieties)
r n
679 (49) 2
169.75 150.06 19.69
4 16
B2 T
SSB Sum of Squares of Block
r n
649 (49) 2
162.25 150.06 12.19
4 16
SSE Sum of Squares of Error SST SSTr SSB
40.96 19.69 12.19 9.06
ANOVA Table
Source of
Degree of
Variation SS = Sum of Computed
Freedom MS = Mean Squares
s.o.v Squares F -test
d.f
(Variations
6.56
due to) 4-1 = 3 19.69 19.69 / 3 = 6.56 Ft 6.50
Treatment 1.01
(Variations
4.06
due to) 4–1=3 12.19 12.19 / 3 = 4.06 Fb 4.02
Blocks 1.01
(Variations
due to) 9.06 9.06 / 9 = 1.01
Error
(4*4) – 1
Total 40.96 ---
=15
F – Table Value
Comparison and Conclusion
Step 5
Calculated Value of F (treatment) = 6.56
Critical/Table Value of F = 3.86
6.56 3.86
Ft F (Table / CriticalValue)
Conclusion and Summarizing:
Since the computed value of F (treatment) is GREATER THAN the
F Table value, So it falls in the critical region. We reject our null
hypothesis and may conclude that the means of four varieties of
wheat are significantly different.
Comparison and Conclusion
Step 5
Calculated Value of F (Block/Group) = 4.06
Critical/Table Value of F = 3.86
4.06 3.86
Ft F (Table / CriticalValue)
Conclusion and Summarizing:
Since the computed value of F (Block/group) is GREATER THAN
the F Table value, So it falls in the critical region. We reject our
null hypothesis and may conclude that the BLOCKING or
GROUPING was effective.
Another example
• The following is the plan of a filed layout testing four varieties A, B, C and
D of wheat in each of % blocks. The plot yields in Kgs are also indicated.
Block I Block II Block III Block IV Block V
D 29.3 B 33.0 D 29.8 B 36.8 D 28.8
B 33.3 A 34.0 A 34.3 A 35.00 C 35.8
C 30.8 C 34.3 B 36.3 D 28.0 B 34.5
A 32.3 D 26.0 C 35.3 C 32.3 A 36.5
Computations
Blocks Varieties (Treatment)
2
V1 V2 V3 V4 Block Block Sum of Square
Total of each value
Total
I
II
III
IV
Varities
Total
2
Varities
Total
Sum of Square
of each value
ANOVA Table
Source of
Degree of
Variation SS = Sum of Computed
Freedom MS = Mean Squares
s.o.v Squares F -test
d.f
(Variations
due to)
Treatment
(Variations
due to)
Blocks
(Variations
due to)
Error
Total
Advantages
• The source of extraneous variation is controlled by grouping
the experimental material and hence the estimate of the
experimental error is decreased.
• The design is flexible. Any number of replications may be run
and any number of treatments may be tested.
• The experiment can be set up easily
• The statistical analysis is simple and straightforward.
• It is easy to adjust the missing observations
Disadvantages
• It control the variability only in one direction
• It is not suitable design when the number of treatments is
very large or when the blocks are not homogeneous.
• Extraneous Variables are undesirable variables that influence the
relationship between the variables that an experimenter is examining.
• Another way to think of this, is that these are variables that influence the
outcome of an experiment, though they are not the variables that are
actually of interest. These variables are undesirable because they add
error to an experiment. A major goal in research design is to decrease or
control the influence of extraneous variables as much as possible.
Summary
• Experimental material is divided into groups or blocks
• Each block contains a complete set of treatments
• Treatments are assigned randomly for each block
• Data is organized in ANOVA table
• F-test is used for testing
• Rest of the testing procedure is same.
The END
