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Learning Module 8
Week 13

Agricultural
Statistics
(Stat 1e)

CAMIGUIN POLYTECHNIC STATE

COLLEGE
CATARMAN CAMPUS
Institute of Agriculture
Tangaro, Catarman, Camiguin

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Table of Contents

Learning Module 8
LATIN SQUARE DESIGN

..... ii

Description .................... 1

Overview
……….……..2

Randomization of Latin Square

Design .............. 2

Sample Computation
................... 4

FINAL TERM
Submit your outputs on time.
Submission is on the schedule of
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Learning Outcomes:
At the end of the unit, the students are expected to:
1. Define the terms concerning the basic principles of
experimental design.
2. Discuss the phases of an experimental procedure.
3. Discuss the components of an experimental design.
4. Discern, calculate and layout some experimental designs.

Description

 Can be simultaneously handle 2 known sources of variation among

experimental units.
 Considers the sources as two independent blocking criteria, rather than
as only one such that in RCBD
 Two-directional blocking:
 Row-blocking and
 Column – blocking
 Two-directional blocking:
 Every treatment appears only once in both directions
 # of treatments = # of replications
 Restriction: impractical to use in experiments w/ large # of treatments
 For experiments w/ # of treatments not less than 4 and not more than 8
 Makes it possible to assess variation among row-blocks and among
column-blocks and to reduce the experimental error.

Decisions:

 If the Fcomp value ≥ than the Ftab at the 1% level of significance, the
effect of the treatments is highly significant, designated by **
 If the Fcomp value ≥than the Ftab at the 5% level of significance but
<Ftab at 1% level, the effect of the treatments is significant, designated
by *
 If the Fcomp value < than the Ftab at the 5% level of significance, the
effect of the treatments is non-significant, designated by ns

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 Overview

The Latin Square Design make use of a two-way blocking of experimental

units, that is by columns and rows. The number of treatments must be the same
as the number of columns and the number of rows (p=r=c).A treatment should
appear only once in a row and once in a column.

Linear Model Latin Square Design

𝑌𝑖𝑗𝑘 = 𝜇 + 𝜌𝑖 + 𝜑𝑗 + 𝜏𝑘 + 𝜀𝑖𝑗𝑘
Where:
𝑌𝑖𝑗𝑘 = observation taken in the ith row, jth column and kth treatment
𝜇 = average of all possible observation
𝜌𝑖 = effect of the ith row
𝜑𝑗 = effect of the ith column
𝜏𝑘 = effect of the kth treatment
𝜀𝑖𝑗𝑘 = experimental error associated with the ith row, and jth column and kth
treatment
i= 1 to r j = 1 to c k = 1 to p and p=r=c

Assumptions:
1. 𝜀𝑖𝑗𝑘~NID (0,𝜎𝜀 2 ) – normally distributed with mean zero and variance
𝜎𝜀 2 .
2. Model 1 – The treatments, columns and row are fixed and have no
interaction with each other.
∑ 𝜑𝑖 = ∑ 𝜌𝑗 = ∑ 𝜏𝑘 = 0
3. Model 2- The treatments, rows and columns are random and have no
interaction with each other.
𝜑𝑖 ~NID (0,𝜎𝜑2 ) 𝜌𝑗~NID (0,𝜎𝜌2 ) 𝜏𝑘 ~NID (0,𝜎𝜏 2 )

Randomization of Latin Square Design

Randomization of Latin Square Design
1. Start with a basic Latin Square Design using letters for treatments.
2. Randomize the rows.
3. Randomize the columns.

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 4. Randomize the treatments.

Example: Randomization of 3x3 Latin Square

1. Start with a basic Latin Square Design Treatments: A, B, C (T1, T2, T3)
1 2 3
1 A B C
2 B C A
3 C A B

2. Randomize the rows. Example random result: 3,1, 2

1 2 3
3 C A B
1 A B C
2 B C A

3. Randomize the columns, Example random result: 2, 1,3

2 1 3
3 A C B
1 B A C
2 C B A

4. Finally randomize the treatments (T1, T2, T3) to the letters (A,B,C)
Example random result 2, 1,3
Assign:
T2 for A A – T2
T1 for B B – T1
T3 for C C – T3

2 1 3
3 T2 T3 T1
1 T1 T2 T3
2 T3 T1 T2

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ANOVA table for Latin Square Design
SV df SS MS Fc Ftab Ftab1%
5%
Row p-1 RSS MSR MSR/MSE F[dftreat,dferror]

Column p-1 CSS MSC MSC/MSE

Treatment p-1 TrSS MSTr MSTr/MSE

Error (p-1) (p-2) ESS MSE

Total p2-1 TSS

Formulas for CF, SS, MS and Fc (Latin Square Design)

CF = (y….)2/p2 Correction Factor
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TSS = 𝛴𝛴𝑦𝑖𝑗𝑘 − 𝐶𝐹 using all observations
RSS = 1/p (𝛴𝛴𝑦𝑖 2 ) −𝐶𝐹 using the rows totals
2
CSS = 1/p (𝛴𝛴𝑦𝑗 ) −𝐶𝐹 using the column totals
2
TrSS = 1/p (𝛴𝛴𝑦𝑘 ) −𝐶𝐹 using the treatment totals
ESS = TSS – RSS – CSS - TrSS by subtraction
MSR = RSS/dftreat MS Row
MSC = CSS/dftreat MS Column
MSTr = TrSS/dftreat MS Treatment
MSE = ESS/dferror MS Error
Fc- row= MSR/MSE F – computed for Row
Fc - column = MSC/MSE F – computed for Column
Fc-treat = MSTr/MSE F – computed for Treatment

Sample Computation

Example, Computation Procedure for Latin Square Design

Data below are yield of rice in kgm/plot fertilized using four different
brands of fertilizers arranged in latin square based in soil fertility gradient and

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sunlight exposure. The random assignments of the different brands of fertilizers
are shown by letters A, B, C, and D.
Rows Columns
1 2 3 4
1 40 (C) 40 (D) 65 (A) 50 (B)
2 40 (D) 95 (A) 42 (B) 38 (C)
3 58 (B) 35 (C) 40 (D) 75 (A)
4 85 (A) 46 (B) 48 (C) 39 (D)
Number
Treatment: ______________________ ____________
Row: ___________________________ ____________
Column: ________________________ ____________
Experimental units: ________________ ____________
Response Variable: ________________ ____________

The Null Hypothesis (Ho)

1. Row
Ho: The effects of rows or soil fertility gradient to the yield are the same.
2. Column
Ho: The effects of columns or sunlight exposure to the yield are the
same.
3. Treatment, Brand of Fertilizer
Ho: The effects of different brands of fertilizer to the yield are the same.

Row and Column Totals

Rows Columns Row
1 2 3 4 Totals
1 40 (C) 40 (D) 65 (A) 50 (B) 195
2 40 (D) 95 (A) 42 (B) 38 (C) 215
3 58 (B) 35 (C) 40 (D) 75 (A) 208
4 85 (A) 46 (B) 48 (C) 39 (D) 218
Column 223 216 195 202 836
Totals

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The Treatments Totals

Treatment
Totals
A 85 95 65 75 320
B 58 46 42 50 196
C 40 35 48 38 161
D 40 40 40 39 159

Computing the Degrees of Freedom

SV df
Row df row = (p-1) = 4-1 = 3
Column df column = (p-1) = 4-1 = 3
Treatment df treatment= (p-1) = 4-1 = 3
Error df Error = (p-1)(p-2) = (4-1)(4-2) = 6
Total df Total = p2 – 1 = 42-1 = 15

Check: 15= 3+3+3+6

Critical values of F
From F-table, using df Numerator = df row/column/treat = 3
and df Denominator =dferror = 6

SV df Ftab 5% Ftab1%
Row 3 F[3,6] = 4.76 F[3,6] = 9.78
Column 3 F[3,6] = 4.76 F[3,6] = 9.78
Treatment 3 F[3,6] = 4.76 F[3,6] = 9.78
Error 6
Total 15

Decision Rule:
Row At ∞ = 5% Reject Ho if Fc = MSR/MSE ≥ 4.76
At ∞ = 1% Reject Ho if Fc = MSR/MSE ≥ 9.78

Column At ∞ = 5% Reject Ho if Fc = MSC/MSE ≥ 4.76

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 At ∞ = 1% Reject Ho if Fc = MSC/MSE ≥ 9.78

Treatment At ∞ = 5% Reject Ho if Fc = MSTr/MSE ≥ 4.76

At ∞ = 1% Reject Ho if Fc = MSTr/MSE ≥ 9.78

Computations
Correction Factor
CF = (Grand Total)2/p2
CF = (836)2/16
CF = 43681

Total Sum of Squares

TSS = 𝛴𝛴𝑦𝑖𝑗𝑘 2 − 𝐶𝐹
TSS = y1112 +y1122+….+yppp2 -CF
TSS = (402+402+…+392) - 43681
TSS = 48738 – 43681
TSS = 5057

Row Sum of Squares

RSS = 1/p (𝛴𝛴𝑦𝑖 2 ) −𝐶𝐹
RSS = (1/4) (1952+2152+2082+2182) – 43681
RSS = (1/4) (175038) – 43681
RSS = 43759.5 – 43681
RSS = 78.5

Column Sum of Squares

CSS = 1/p (𝛴𝛴𝑦𝑗 2 ) −𝐶𝐹
CSS= (1/4) (2232+2162+1952+2022) – 43681
CSS = (1/4)(175214) – 43681
CSS = 43803.5 – 43681
CSS = 122.5

Treatment Sum of Squares

TrSS = 1/p (𝛴𝛴𝑦 … 𝑘 2 ) −𝐶𝐹
TrSS= (1/4) (3202+1962+1612+1592) – 43681

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TrSS = (1/4)(192018) – 43681
TrSS = 48004.5 – 43681
TrSS = 4323.5

Error SS
ESS = TSS – RSS – CSS -TrSS
ESS = 5057 – 78.5 – 122.5 – 4323.5
ESS = 532.5

Summary of ANOVA Table

SV df SS MS Fc Ftab5% Ftab1%
ns
Row 3 78.5 26.17 0.29 4.76 9.78

Column 3 122.5 40.83 0.46ns 4.76 9.78

Treatment 3 4323.5 1441.17 16.24** 4.76 9.78

Error 6 532.5 88.75

Total 15 5057

Decisions

Row
Since row Fc = 0.2948 is not > 4.76 then we fail to reject row Ho

Column
Since column Fc = 0.46 is not > 4.76 then we fail to reject column Ho

Treatment
Since treat Fc = 16.23825 is not > 9.78 then we fail to reject treat Ho at 1%

Conclusion

Row
The effects of rows (soil fertility gradient) to the yield are not significantly
different at 5% level of significance
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Column
The effects of column (sunlight exposure) to the yield are not significantly
different.

Treatment (brands of fertilizer)

The effects of different brands of fertilizer to the yield are significantly different
at 1% level of significance

Coefficient of Variation
𝑦…
𝑦̅.. = 𝑝2
𝑦̅.. = 836/42
𝑦̅.. = 52.25

√𝑀𝑆𝐸
𝐶𝑉 = 𝑥 100%
𝑦̅. .
√88. 75
𝐶𝑉 = 𝑥 100%
52.25
𝐶𝑉 = 44.16%

Notes:

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Assessment:

Problem 1.
Data below are yield of rice in kg/plot fertilized using different brands of
organic fertilizers arranged in Latin Square. The random assignments of the
different brands of organic fertilizers are shown by letters A, B, C, and D.

Row Columns
1 2 3 4
1 41(C) 41(D) 65(A) 50(B)
2 41(D) 95(A) 42(B) 38(C)
3 58(B) 36(C) 41(D) 75(A)
4 85(A) 46(B) 48(C) 39(D)

Problem 2
Four treatments on lanzones were compared in four stores: treatment A =
mixed sized in paper bag, B = mixed sized in clear plastic bag, C = small size
in plastic bag at reduced price, D = big size fruits in plastic bag at higher
price. The data in kg of fruit sold/day is presented below.

Row Store Number

1 2 3 4
Monday 15(A) 7(B) 42(C) 37(D)
Tuesday 20(B) 22(A) 48(D) 20(C)
Wednesday 24(D) 12(C) 12(B) 32(A)
Thursday 31(C) 16(D) 32(A) 15(B)

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 WORKSHEET

Problem No. ____

Number
Treatment: __________________ ____________
Row: _______________________ ____________
Column: _____________________ ____________
Experimental Unit: _____________ _____________

Responsible variable: ___________ _____________

The Null Hypothesis (Ho):

Row
Ho: _______________________________________________________
Column
Ho: _______________________________________________________
Treatment
Ho: _______________________________________________________

Row and Column Totals

Rows Columns Row
1 2 3 4 Totals
1
2
3
4
Column
Totals

Treatment
Totals
A
B
C
D

Computing the Degrees of Freedom

SV df
Row df row =
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 Column df column =
Treatment df treatment=
Error df error =
Total df Total =
Critical values of F
From F-table, using df Numerator = df row/column/treat =
and df Denominator =dferror =
Decision Rule:

Row

Column

Treatment

Correction Factor
CF =
CF =
CF =

Total Sum of Squares

TSS =
TSS =
TSS =
TSS =
TSS =

Row Sum of Squares

RSS =
RSS =
RSS =
RSS =
RSS =

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Column Sum of Squares
CSS =
CSS =
CSS =
CSS =
CSS =

Treatment Sum of Squares

TrSS =
TrSS=
TrSS =
TrSS =
TrSS =

Error SS
ESS = TSS – RSS – CSS -TrSS
ESS =
ESS =

Summary of ANOVA Table

SV df SS MS Fc Ftab5% Ftab1%
Row

Column

Treatment

Error

Total

Decisions

Row

Column

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Treatment

Conclusion

Row

Treatment

Coefficient of Variation
𝑦…
𝑦̅.. =
𝑝2
𝑦̅.. =
𝑦̅.. =

√𝑀𝑆𝐸
𝐶𝑉 = 𝑥 100%
𝑦̅. .
𝐶𝑉 =
𝐶𝑉 =

References:

CMU Statistics Manual, 2014

APEC Agricultural and Technical Cooperation Working Group. (2013).
Agricultural Statistics Best Practice Methodology handbook
DAVIS, BOB.(2000). Introduction to Agricultural Statistics. Delmar Cengage
Learning; 1st Edition.
IDAIKKADAR M. N. (2001). Agricultural Statistics. A handbook for
Developing Countries. 1st Edition
RANGASWAMY.R. (2009). Agricultural Statistics. New Age
International Publisher. 8122425925

Prepared by:

JESSA D. PABILLORE
jessapabillore916@gmail.com
09179869017

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