Latin Square Designs
KNNL – Sections 28.3-28.7
Description
• Experiment with r treatments, and 2 blocking factors:
rows (r levels) and columns (r levels)
• Advantages:
 Reduces more experimental error than with 1 blocking factor
 Small-scale studies can isolate important treatment effects
 Repeated Measures designs can remove order effects
• Disadvantages




Each blocking factor must have r levels
Assumes no interactions among factors
With small r, very few Error degrees of freedom; many with big r
Randomization more complex than Completely Randomized
Design and Randomized Block Design (but not too complex)
Randomization in Latin Square
• Determine r , the number of treatments, row blocks, and
column blocks
• Select a Standard Latin Square (Table B.14, p. 1344)
• Use Capital Letters to represent treatments (A,B,C,…) and
randomly assign treatments to labels
• Randomly assign Row Block levels to Square Rows
• Randomly assign Column Block levels to Square Columns
• 4x4 Latin Squares (all treatments appear in each row/col):
Square 1
Row1
Row2
Row3
Row4
Col1
A
B
C
D
Col2
B
C
D
A
Col3
C
D
A
B
Col4
D
A
B
C
Square2
Row1
Row2
Row3
Row4
Col1
A
B
C
D
Col2
B
A
D
C
Col3
C
D
A
B
Col4
D
C
B
A
Latin Square Model
Note: Although there are 3 subscripts, there are only r 2 cases (defined by rows/cols)
Yijk    i   j   k   ijk
 ijk ~ N  0,  2  independent
i  1,..., r ; j  1,..., r ; k  1,..., r ;
  overall mean i  effect of row i  j  effect of column j  k  Effect of treament k
r
r
r
      
i 1
i
j 1
j
k 1
k
0
Row, Column, Treatment Sums and Means:
r
Rows: Yi   Yijk
Y i
j 1
Treatments: Yk   Yijk
r
Y
 i
r
Y k
i, j
Columns: Y j    Yijk
Y  j 
i 1
Y
 k
r
r
r
Overall: Y   Yijk
Y j 
r
Y  
i 1 j 1
Y
r2
Least Squares Estimates:
^
^
   Y 
 i  Y i  Y 
^
 j  Y  j   Y 
^
 k  Y k  Y 
Predicted Values and Residuals:
^
^
^
^
^
Y ijk      i   j   k  Y i   Y  j   Y k  2Y 
^
eijk  Yijk  Y ijk  Yijk  Y i   Y  j   Y k  2Y 
Analysis of Variance
r
r

Total Sum of Squares: SSTO   Yijk  Y 
i 1 j 1

2
dfTO  r 2  1
r
r

Row Sum of Squares: SSROW  r  Y i  Y 
i 1

2
df ROW  r  1
E MSROW    2 
r  i2
i 1
r 1
r
r

Col Sum of Squares: SSCOL  r  Y  j   Y 
j 1

2
df COL  r  1
E MSCOL   2 
r
r

Trt Sum of Squares: SSTR  r  Y k  Y 
k 1

2
dfTR  r  1
r
r
E MSTR   2 

Remainder (Error) Sum of Squares: SS Re m   Yijk  Y i  Y  j   Y k  2Y 
i 1 j 1
df Re m   r  1 r  2 
E MS Re m   2
Testing for Treatment Effects: H 0 :  1  ...   r  0
Test Statistic: F * 
MSTR
MS Re m
H A : Not all  k  0
Reject H 0 if F *  F  0.95; r  1,  r  1 r  2  

2
r   2j
r  k2
k 1
r 1
j 1
r 1
Post-Hoc Comparison of Treatment Means &
Relative Efficiency
Tukey's HSD: HSDij  q  0.95; r;  r  1 r  2  
MS Re m
r

r  r  1 


 2 MS Re m
Bonferroni's MSD  C=
:
MSD

t
1

;
r

1
r

2




ij


2
2
C
r




Relative Efficiency of Latin Square to Completely Randomized Design:
^
E1 
MSROW  MSCOL   r  1 MS Re m
 r  1 MS Re m
Relative Efficiency of Latin Square to Randomized Block Design:
^
RBD(Rows): E 2 
^
MSCOL   r  1 MS Re m
RBD(Columns): E 3 
rMS Re m
MSROW   r  1 MS Re m
rMS Re m
Comments and Extensions
• Treatments can be Factorial Treatment Structures
with Main Effects and Interactions
• Row, Column, and Treatment Effects can be Fixed or
Random, without changing F-test for treatments
• Can have more than one replicate per cell to increase
error degrees of freedom
• Can use multiple squares with respect to row or
column blocking factors, each square must be r x r.
This builds up error degrees of freedom (power)
• Can model carryover effects when rows or columns
represent order of treatments