Latin Square Designs
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Transcript Latin Square Designs
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: Yk 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