The Latin Square design
The Latin square design is used where the researcher desires to
control the variation in an experiment that is related to rows and
columns in the field.
Field marks:
 Treatments are assigned at random within rows and columns, with each treatment once per row and once per column.
 There are equal numbers of rows, columns, and treatments.
 Useful where the experimenter desires to control variation in two different directions
Sample layout:
Different colors represent different treatments. There are
4 treatments (AD) assigned to 4 rows (IIV) and 4
columns (14).
Row I A B C D
Row II C D A B
Row III D C B A
Row IV B A D C
Column 1 2 3 4
ANOVA table format:
Source of variation 
Degrees of freedom^{a} 
Sums of squares (SSQ) 
Mean square (MS) 
F 
Rows (R) 
r1 
SSQ_{R} 
SSQ_{R}/(r1) 
MS_{R}/MS_{E} 
Columns (C) 
r1 
SSQ_{C} 
SSQ_{C}/(r1) 
MS_{C}/MS_{E} 
Treatments (Tr) 
r1 
SSQ_{Tr} 
SSQ_{Tr}/(r1) 
MS_{Tr}/MS_{E} 
Error (E) 
(r1)(r2) 
SSQ_{E} 
SSQ_{E}/((r1)(r2)) 

Total (Tot) 
r^{2}1 
SSQ_{Tot} 


^{a}where r=number of treatments, rows, and columns. 
Sample ANOVA table:
Source of variation 
Degrees of freedom 
Sums of squares (SSQ) 
Mean square (MS) 
F 
Rows 
3 
40.77 
13.59 
5.91^{a} 
Columns 
3 
125.39 
41.80 
18.16^{a} 
Treatments 
3 
160.57 
53.52 
23.26^{a} 
Error 
6 
13.81 
2.30 

Total 
15 
340.54 


^{a}F test with 3,6 degrees of freedom at P=0.05 is 4.76 
Sample SAS GLM statements:
PROC GLM;
CLASS ROWS COLUMNS TREATS;
MODEL WHATEVER = ROWS COLUMNS TREATS;
RUN;
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Wednesday, August 16, 2000
