Latin Square Designs (§15.4)
• Lecture Objective
– Introduce basic experimental designs that
account for two orthogonal sources of
extraneous variation.
• Terminology
– Square design
– Orthogonal blocks
– Randomizations
Latin Square -1
Examples
• A researcher wishes to perform a yield experiment under field
conditions, but she/he knows or suspects that there are two
fertility trends running perpendicular to each other across the
study plots.
• An animal scientists wishes to study weight gain in piglets but
knows that both litter membership and initial weights
significantly affect the response.
• In a greenhouse, researchers know that there is variation in
response due to both light differences across the building and
temperature differences along the building.
• An agricultural engineer wishing to test the wear of different
makes of tractor tire, knows that the trial and the location of the
tire on the (four wheel drive, equal tire size) tractor will
significantly affect wear.
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Latin Square Design
• A class of experimental designs that allow for two
sources of blocking.
• Can be constructed for any number of treatments, but
there is a cost. If there are t treatments, then t2
experimental units will be required.
• If one of the blocking factors is left out of the design,
we are left with a design that could have been obtained
as a randomized block design.
• Analysis of a Latin square is very similar to that of a
RBD, only one more source of variation in the model.
• Two restrictions on randomization.
Latin Square -3
Cold Protection of Strawberries
• Three different irrigation methods (treatment
levels) are used on strawberries:
1. drip,
2. overhead sprinkler,
3. no irrigation.
• We wish to determine which of these is most
effective in protecting strawberries from
extreme cold.
• All strawberries grown through plastic mulch.
• Measure weight of frozen fruit (lower values
indicate more protection).
Latin Square -4
high
Moisture
Field
Layout
Nitrogen Level
none drip
drip
over
CANAL
high
Moisture
Which design will
best allow us to
account for both soil
moisture and
nitrogen gradients?
over
none over none
drip
Moisture and Soil Nitrogen
are two sources of
extraneous variation that
we wish to simultaneously
control for.
low
Nitrogen Level
none drip
drip
low
over
over none
over none drip
CANAL
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Advantages and Disadvantages
Advantages:
• Allows for control of two extraneous sources of
variation.
• Analysis is quite simple.
Disadvantages:
• Requires t2 experimental units to study t treatments.
• Best suited for t in range: 5  t  10.
• The effect of each treatment on the response must be
approximately the same across rows and columns.
• Implementation problems.
• Missing data causes major analysis problems.
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Constructing a Latin Square Design for t
Treatments
• Treatments designated by first t capital letters in the
alphabet (A,B,C, etc.)
• Number the levels of blocking factor 1 (call it “Rows”) as
R1, R2, … Rt.
• Number the levels of blocking factor 2 (call it “Columns”)
as C1, C2, … Ct.
• Assign the treatment letters in alphabetic order,
beginning with A, to the t units in the first row.
• For the second row, start with the letter B and assign
treatment letters to the t-th letter then follow with A.
• For rows 3 through t, simply shift the treatment letters up
one at a time, placing the shifted letter in the last unit of
Latin Square -7
the row.
Basic Square
C1
C2
C3
C4
R1
A
B
C
D
R2
B
C
D
A
R3
C
D
A
B
R4
D
A
B
C
Latin Square -8
Randomization
Get a random
ordering of the
rows.
1234
replaced by
2143
C1
C2
C3
C4
R1
B
C
D
A
R2
A
B
C
D
R3
D
A
B
C
R4
C
D
A
B
Reorder the rows according to randomization.
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Randomization
Get a random
ordering of the
columns.
1234
replaced by
4231
C1
C2
C3
C4
R1
A
C
D
B
R2
D
B
C
A
R3
C
A
B
D
R4
B
D
A
C
Reorder the columns according to randomization.
Two Blocking Factors = Two Randomizations
= Two Constraints on Randomization
Latin Square -10
Latin Square Linear Model: A Three-Way AOV
yij(k )  m  ri  j t k  e ij(k ) , (i, j, k  1,, t )
t
yij(k)
m
ri
j
tk
eij(k)
= number of treatments, rows and columns.
= observation on the unit in the ith row, jth column given the kth treatment.
The indicator k is in parenthesis to remind us that specifying i and j
effectively determines the treatment k.
= the general mean common to all experimental units.
= the effect of level i of the row blocking factor. Usually assumed
N(0,sr2), a random effect.
= the effect of level j of the column blocking factor. Usually assumed
N(0,s2), a random effect.
= the effect of level k of treatment factor, a fixed effect.
= component of random variation associated with observation ij(k).
Usually assumed N(0,se2).
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Latin Square Analysis of Variance
Source
Rows
Columns
Trtments
Error
Total
df
t-1
t-1
t-1
(t-1)(t-2)
t2-1
SSQ
SSRow
SSCol
SSTrt
SSE
TSS
MSQ
MSRow
MSCol
MSTrt
MSE
F
MSRow/MSE
MSCol/MSE
MSTrt/MSE
Latin Square -12
Sums of Squares
TSS 
t
(y
i , j 1
ij ( k )
 y )
2
t
SSRow t  ( yi.(.)  y )
2
i 1
t
SSCol  t  ( y. j (.)  y )
2
j 1
t
SSTrt  t  ( yij (.)  y )
2
k 1
SSE  TSS  SSRow SSCol  SSTrt
Latin Square -13
Experimental Error
Experimental error = response differences between two experimental
units that have experienced the same treatment. In this case though,
the “replicates” for each treatment are spread across the t row and t
column blocks in a specific fashion.
Even more so than with randomized block designs, the variability
among treatment replicates includes the row and column block effects.
In similar fashion as for RCBDs, the specific latin square layout will
filter out the extraneous (row & col) sources of variability when
performing comparisons of treatment means.
Note that this would not have been the case if the experiment had
erroneously been laid out as a CRD or RBD…
Latin Square -14
Latin Square Mean Squares and F Statistics
MSRow =
MSCol
=
MSTrt
=
MSE
=
SSRow
(t - 1)
SSCol
(t - 1)
SSTrt
(t - 1)
SSE
((t - 1)(t - 2)
Ftrt
Frow
Fcol
MSTrt

MSE
MSRow

MSE
MSCol

MSE
~ F( t 1),( t 1)( t 2 )
~ F( t 1),( t 1)( t 2 )
~ F( t 1),( t 1( t 2 )
We reject the null hypothesis of no main effect if the value of
the F-statistic is greater than the 100(1-a)th percentile of the F
distribution with degrees of freedom specified above.
Latin Square -15
Latin Square Example
The strawberry irrigation cold protection study data are given
below. The effectiveness of the three irrigation methods was
measured by the weight of the frozen fruit, with lower weights
representing more effective protection. The study question is “
Which irrigation method provided the most protection?”
Row
1
1
1
2
2
2
3
3
3
Column
1
2
3
1
2
3
1
2
3
Treatment
Drip
Over
None
None
Drip
Over
Over
None
Drip
Weight
51
119
60
98
43
31
99
87
49
Latin Square -16
Data strawb;
input row column irrig $ weight @@;
datalines;
1 1 drip
51 1 2 over 119 1 3 none
60
2 1 none
98 2 2 drip
43 2 3 over
31
3 1 over
99 3 2 none
87 3 3 drip
49
; run;
proc glm;
class row column irrig;
model weight = row column irrig;
title 'Strawberry Irrigation Latin Square Exp'; run;
Source
Model
Error
Corrected Total
DF
6
2
8
R-Square
0.782679
Sum of
Squares
5840.000000
1621.555556
7461.555556
Coeff Var
40.23037
Latin Square in
SAS
Mean Square
973.333333
810.777778
Root MSE
28.47416
F Value
1.20
Pr > F
0.5205
weight Mean
70.77778
Source
row
column
irrig
DF
2
2
2
Type I SS
817.555556
2616.222222
2406.222222
Mean Square
408.777778
1308.111111
1203.111111
F Value
0.50
1.61
1.48
Pr > F
0.6648
0.3826
0.4026
Source
row
column
irrig
DF
2
2
2
Type III SS
817.555556
2616.222222
2406.222222
Mean Square
408.777778
1308.111111
1203.111111
F Value
Pr > F
0.50
0.6648
1.61
0.3826
Latin1.48
Square -17 0.4026
Input Data
Analyze > General Linear Model > Univariate
Latin Square in SPSS
Note: You must use a custom model and
only ask for main effects.
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SPSS Ouptut
Latin Square -19
Latin Square in Minitab
Stat > ANOVA > General Linear Model
Latin Square -20
MTB: ANOVA and Sums of Squares
General Linear Model: weight versus row, column, irrig
Factor
row
column
irrig
Type
fixed
fixed
fixed
Levels
3
3
3
Values
1, 2, 3
1, 2, 3
drip, none, over
Analysis of Variance for weight, using Adjusted SS for Tests
Source
row
column
irrig
Error
Total
DF
2
2
2
2
8
S = 28.4742
Seq SS
817.6
2616.2
2406.2
1621.6
7461.6
Adj SS
817.6
2616.2
2406.2
1621.6
R-Sq = 78.27%
Adj MS
408.8
1308.1
1203.1
810.8
F
0.50
1.61
1.48
P
0.665
0.383
0.403
R-Sq(adj) = 13.07%
Latin Square -21
Latin Square with R
> straw <- read.table("Data/latin_square.txt",header=TRUE)
> straw.lm <- lm(weight ~ factor(row) + factor(column) +
factor(irrig), data=straw)
> anova(straw.lm)
Analysis of Variance Table
Response: weight
Df Sum Sq Mean Sq F value
factor(row)
2 817.56 408.78 0.5042
factor(column) 2 2616.22 1308.11 1.6134
factor(irrig)
2 2406.22 1203.11 1.4839
Residuals
2 1621.56 810.78
Pr(>F)
0.6648
0.3826
0.4026
Latin Square -22
Medical Example of a Latin Square
Latin Square -23
Randomized, controlled,
double-blinded, NICE!
Design extracts out
differences due to time and
patients!
Latin Square -24