Latin Square Design
Price Valuations for 8 Varieties of Tea,
Prepared on 8 Days in 8 Orders
Harrison and Bose (1942), Sankhya, Vol6,#2,pp151-166
Data Description
• Response: Mean Price Valuation among 6 judges
• 8 Tea Varieties (Crossing of Seed and Field Type):
 4 jats (tea seed) – BJ,KK,PG,CH
 2 Field Types - Pruned previous December, Unpruned
• 8 Dates of Manufacture (7/28 – 9/29)
• 8 Orderings of Preparation
• All varieties prepared on each date
• All varieties receive each order position of
preparation
Order of Manufactured Samples, Valuations, Means
Order\Date
1
2
3
4
5
6
7
8
Order\Date
1
2
3
4
5
6
7
8
DateMean
1
PG/P (5)
CH/P (7)
BJ/U (2)
CH/U (8)
KK/U (4)
PG/U (6)
BJ/P (1)
KK/P (3)
1
14.6
13.7
14.2
13.4
14.2
14.3
14.8
13.6
14.1000
2
13.7
13.0
13.4
14.6
14.2
13.7
13.9
14.1
13.8250
2
KK/P (3)
CH/U (8)
CH/P (7)
BJ/P (1)
PG/P (5)
KK/U (4)
BJ/U (2)
PG/U (6)
3
14.2
13.9
13.2
12.7
12.9
13.6
13.4
12.5
13.3000
3
BJ/U (2)
PG/P (5)
PG/U (6)
KK/U (4)
CH/P (7)
BJ/P (1)
KK/P (3)
CH/U (8)
4
12.7
14.1
13.7
13.9
13.1
13.9
13.4
12.9
13.4625
4
CH/U (8)
BJ/P (1)
KK/U (4)
BJ/U (2)
KK/P (3)
PG/P (5)
PG/U (6)
CH/P (7)
5
12.9
13.1
13.6
13.5
13.8
12.6
13.4
13.7
13.3250
5
CH/P (7)
KK/P (3)
PG/P (5)
PG/U (6)
BJ/U (2)
CH/U (8)
KK/U (4)
BJ/P (1)
6
12.5
12.5
13.0
12.6
12.1
12.8
12.2
12.8
12.5625
7
12.6
12.7
12.7
12.8
12.8
12.3
12.1
12.5
12.5625
6
KK/U (4)
PG/U (6)
BJ/P (1)
KK/P (3)
CH/U (8)
BJ/U (2)
CH/P (7)
PG/P (5)
7
PG/U (6)
BJ/U (2)
KK/P (3)
PG/P (5)
BJ/P (1)
CH/P (7)
CH/U (8)
KK/U (4)
8
BJ/P (1)
KK/U (4)
CH/U (8)
CH/P (7)
PG/U (6)
KK/P (3)
PG/P (5)
BJ/U (2)
Order
Variety
8
Mean
Mean
13.1
13.2875
13.7125
12.6
13.2000
13.5125
12.1
13.2375
13.0625
12.4
13.2375
13.1625
12.5
13.2000
13.5250
12.3
13.1875
13.2625
12.4
13.2000
12.8375
12.6
13.0875
12.5625
12.5000 AllMean 13.2046875
Date Mean
14.2500
Valuation - By Jat/Pruning Status
14.1250
14.0000
14.2500
13.8750
13.7500
14.1250
13.6250
13.5000
Mean Valuation
14.0000
13.8750
13.3750
13.2500
13.1250
13.0000
13.7500
12.8750
12.7500
13.6250
12.6250
12.5000
12.3750
12.2500
13.3750
1
2
3
4
5
6
7
8
7
8
Date Number
P
U
13.2500
Order Mean
13.1250
14.2500
13.0000
14.1250
14.0000
12.8750
13.8750
13.7500
12.7500
13.6250
13.5000
12.6250
Mean Valuation
Valuation (Pence)
13.5000
12.5000
13.3750
13.2500
13.1250
13.0000
12.3750
12.8750
12.7500
12.2500
0
1
2
3
4
5
12.6250
12.5000
Jat
12.3750
12.2500
1
2
3
4
5
Order Number
6
Latin Square Design - Model
• Model (8 Varieties, Dates, Orders, N=82=64) :
yijk     k   i   j   ijk
Note there is only one k i, j
  Overall Mean
^
  y ...
^
 k  Effect of Variety k  k  y .. k  y ...
^
 i  Effect due to Order i  i  y i..  y ...
^
 j  Effect due to Date j  j  y . j .  y ...
 ijk  Random Error Term
Latin Square Design - ANOVA & F-Test
8

8
Total Sum of Squares : TSS   yijk  y ...

2
 31.4486 df  82  1  63
i 1 j 1
8

Variety Sum of Squares SSVariety  8 y .. k  y ...
k 1
8

Order Sum of Squares SSOrder  8 y i..  y ...
i 1
8

Date Sum of Squares SSDate  8 y . j .  y ...
j 1



2
 8.2223 dfV  8  1  7
2
 0.1848 df O  8  1  7
2
 20.7823 df D  8  1  7
Error Sum of Squares SSE  TSS  SST  SSR  SSC 
 y
8
8
i 1 j 1
ijk  y i ..  y . j .  y .. k  2 y ...

2
 2.2591
df E  (t 2  1)  3(t  1)  (t  1)(t  2)  42
Note: We can partition Variety SS into main effects for jat and pruning, and their
interaction (next slide)
Decomposing Variety Sum of Squares
Partitioni ng the Variety SS into Jat, Pruning and JatxPrunin g Interactio n :
y ..1  y ..2
y ..3  y ..4
Jat : J 1 
 13.6125 J 2 
 13.1125
2
2
y  y ..6
y  y ..8
J 3  ..5
 13.3938 J 4  ..7
 12.7000
2
2
4

SSJ  16 J m  y ...

2
m 1
 7.4442 df J  4  1  3
y ..1  y ..3  y ..5  y ..4
Pruning : P1 
 13.2844
4
y ..2  y ..4  y ..6  y ..8
P2 
 13.1250
4
2

SSP  32 P m  y ...
m 1

2
 0.4064 df P  2  1  1
SSJP  SSVar  SSJ  SSP  8.2223  7.4442  0.4064  0.3717 df JP  3(1)  3
Analysis of Variance
Source
Variety
Jat
Pruning
JxP
Order
Date
Error
Total
df
7
3
1
3
7
7
42
63
SS
8.2223
7.4442
0.4064
0.3717
0.1848
20.7823
2.2591
31.4486
MS
1.1746
2.4814
0.4064
0.1239
0.0264
2.9689
0.0538
F
21.8383
46.1338
7.5558
2.3036
P
0.0000
0.0000
0.0088
0.0908
Evidence of Jat and Pruning Main Effects, Interaction not significant at 0.05
significance level
Pairwise Comparison of Jat Means
Tukey’s - q from Studentized Range Dist. k=4,n = (t-1)(t-2)=42
• Note: Each Jat Mean is based on 2t=16 observations
MSE
0.0538
Wij  q.05 (k ,n )
 3.784
 0.2194
2t
16
• Bonferroni’s Method - t-values from table on class website
with n = (t-1)(t-2)=42 and C=4(4-1)/2=6
Bij  t.05 / 2,C ,v
Comparison
BJ-KK
BJ-PG
BJ-CH
KK-PG
KK-CH
PG-CH
Mean i
13.6125
13.6125
13.6125
13.1125
13.1125
13.3938
2MSE
2(0.0538)
 2.770
 0.2272
2t
2(8)
Mean j Difference Conclude
13.1125
0.5000
BJ>KK
13.3938
0.2188
NSD
12.7000
0.9125
BJ>CH
13.3938
-0.2813
KK<PG
12.7000
0.4125
KK>CH
12.7000
0.6938
PG>CH
Very close to
significant
difference
Relative Efficiency
• Relative Efficiency of LS to CRD (how many times as
many replicates would be needed for CRD to have as
precise of estimates of treatment means as LS does):
MSECR MSOrder  MSDate  (t  1) MSE
RE ( LS , CR) 

MSE LS
(t  1) MSE
0.0264  2.9689  7(0.0538) 3.3719


 6.96  7
9(0.0538)
0.4842
Would need approximately 56 reps per variety to have as precise of estimates
of variety means if experiment conducted as completely randomized design