Three or More Factors: Latin Squares

1
Three or More Factors Latin Squares
Example
Three factors, A (block factor), B (block
factor), and C (treatment factor), each at
three levels. A possible arrangement
B
B
B
1
2
3
C
C
C
A
1
1
1
1
C
C
C
A
2
2
2
2
A
C
C
C
3
3
3
3
2
Notice, first, that these designs are squares
all factors are at the same number of levels,
though there is no restriction on the nature of
the levels themselves. Notice, that these
squares are balanced each letter (level) appears
the same number of times this insures unbiased
estimates of main effects. How to do it in a
square? Each treatment appears once in every
column and row. Notice, that these designs are
incomplete of the 27 possible combinations of
three factors each at three levels, we use only 9.
3
Example
Three factors, A (block factor), B (block
factor), and C (treatment factor), each at
three levels, in a Latin Square design nine
combinations.
B
B
B
1
2
3
C
C
C
A
1
2
3
1
C
C
C
A
2
3
1
2
A
C
C
C
3
1
2
3
4
Example with 4 Levels per Factor
FACTORS
VARIABLE
Lifetime of a tire (days)
Automobiles A four levels Tire positions
B four levels Tire treatments C four levels

5
The Model for (Unreplicated) Latin Squares
Example
?


m
i

1,...
?
y
?
?
?
?
1,
...
m





j

,
ijk
j
k
ijk
i
...
,
k
m

1,
Y A B C e
AB, AC, BC, ABC
Note that interaction is not present in the model.
Same three assumptions normality, constant
variances, and randomness.
6
Putting in Estimates



y

(
y

y
)

(
y

y
)

(
y

y
)

R
y
ijk
...
i
..
...
.
j
.
...
..
k
...
or
bringing
y
to
the
left

hand
side
,


(
y

y
)

(
y

y
)

(
y

y
)

(
y

y
)

R
,
ijk
...
i
..
...
.
j
.
...
..
k
...
Variability among yields associated with Rows
Variability among yields associated with Columns
Variability among yields associated with Inside
Factor
Total variability among yields





y

y

y

y

2
y
where R
ijk
i
..
.
j
.
..
k
...
7
Actually, R



y
-
y
-
y
-
y

2
y
...
ijk
i
..
.
j
.
..
k

(
y
-
y
)
(
y
-
y
)
-
ijk
...
i
..
...
y
y
)
-
-
(
.
j
.
...
-
(
y
-
y
),
..
k
...
An interaction-like term. (After all, theres
no replication!)
8
The analysis of variance (omitting the mean
squares, which are the ratios of second to third
entries), and expectations of mean squares
S
o
u
r
c
e

o
f
S
u
m

o
f
D
e
g
r
e
e
s

o
f
E
x
p
e
c
t
e
d
v
a
r
i
a
t
i
o
n
s
q
u
a
r
e
s
f
r
e
e
d
o
m
v
a
l
u
e

o
f
m
e
a
n

s
q
u
a
r
e
?





?
m
R
o
w
s

?
m

1

V
2
?
m
(
y

y
)
2
Rows
i
..
...
i

1
?





?
m
C
o
l
u
m
n
s

?
m

1

V
2
?
m
(
y

y
)
2
Col
.
j
.
...
j

1
?





?
m
I
n
s
i
d
e

?
m

1

V
2
?
m
(
y

y
)
2
Inside
factor
..
k
...
f
a
c
t
o
r
k

1
?


b
y

s
u
b
t
r
a
c
t
i
o
n

?
(
m

1)(
m

2)
2
Error
?


?
?
?

T
o
t
a
l

m

1
2
2
(
y

y
)
ijk
...
i
j
k
9
The expected values of the mean squares
immediately suggest the F ratios appropriate for
testing null hypotheses on rows, columns and
inside factor.
10
Our Example
(Inside factor Tire Treatment)
Tire Position
Auto.
11
General Linear Model Lifetime versus Auto,
Postn, TrtmntFactor Type Levels Values
Auto fixed 4 1 2 3 4Postn fixed
4 1 2 3 4Trtmnt fixed 4 1 2 3
4Analysis of Variance for Lifetime, using
Adjusted SS for TestsSource DF Seq SS
Adj SS Adj MS F PAuto 3
17567 17567 5856 2.17
0.192Postn 3 4679 4679
1560 0.58 0.650Trtmnt 3 26722
26722 8907 3.31 0.099Error 6
16165 16165 2694Total 15
65132 Unusual Observations for LifetimeObs
Lifetime Fit SE Fit Residual St
Resid 11 784.000 851.250 41.034
-67.250 -2.12R
12
(No Transcript)
13
SPSS/Minitab DATA ENTRYVAR1 VAR2 VAR3 VAR4855 1
1 4962 2 1 1848 3 1 3831 4 1 2877 1 2 3817 2
2 2. . . .. . . .. . . .871 4 4 3
14
Latin Square with REPLICATION

• Case One using the same rows and columns for all
  Latin squares.
• Case Two using different rows and columns for
  different Latin squares.
• Case Three using the same rows but different
  columns for different Latin squares.

15
Treatment Assignments for n Replications

• Case One repeat the same Latin square n times.
• Case Two randomly select one Latin square for
  each replication.
• Case Three randomly select one Latin square for
  each replication.

16
Example n 2, m 4, trtmnt A,B,C,D
Case One
column column column column
row 1 2 3 4
1 A B C D
2 B C D A
3 C D A B
4 D A B C
column column column column
row 1 2 3 4
1 A B C D
2 B C D A
3 C D A B
4 D A B C

• Row 4 tire positions column 4 cars

17
Case Two
column column column column
row 1 2 3 4
1 A B C D
2 B C D A
3 C D A B
4 D A B C
column column column column
row 5 6 7 8
5 B C D A
6 A D C B
7 D B A C
8 C A B D

• Row clinics column patients letter drugs
  for flu

18
Case Three

5 6 7 8
B C D A
A D C B
D B A C
C A B D
column column column column
row 1 2 3 4
1 A B C D
2 B C D A
3 C D A B
4 D A B C

• Row 4 tire positions column 8 cars

19
ANOVA for Case 1SSBR, SSBC, SSBIF are computed
the same way as before, except that the
multiplier of (say for rows) m (Yi..-Y)2
becomes mn (Yi..-Y)2 and degrees of
freedom for error becomes(nm2 - 1) - 3(m - 1)
nm2 - 3m 2
?
?
20
ANOVA for other cases

1. SS please refer to the book, Statistical
   Principles of research Design and Analysis by R.
   Kuehl.
2. DF of levels 1 for all terms except error.
   DF of error total DF the sum of the rest DFs.

Using Minitab in the same way can give Anova
tables for all cases.
21
Graeco-Latin Squares
In an unreplicated m x m Latin square there are
m2 yields and m2 - 1 degrees of freedom for the
total sum of squares around the grand mean. As
each studied factor has m levels and, therefore,
m-1 degrees of freedom, the maximum number of
factors which can be accommodated, allowing no
degree of freedom for factors not studied, is A
design accommodating the maximum number of
factors is called a complete Graeco-Latin square
m

1

2

m

1

m

1
22
Example 1m3 four factors can be accommodated
23
Example 2m 5 six factors can be
accommodated
24
In an unreplicated complete Graeco-Latin square
all degrees of freedom are used up by factors
studied. Thus, no estimate of the effect of
factors not studied is possible, and analysis of
variance cannot be completed.
25
But, consider incomplete Graeco-Latin Squares
b1 b2 b3
b4 b5
c4d4
a1
c1d1
c5d5
c2d2
c3d3
a2
c2d3
c3d4
c4d5
c5d1
c1d2
c3d5
a3
c4d1
c5d2
c1d3
c2d4
c5d3
a4
c1d4
c2d5
c3d1
c4d2
c5d4
c1d5
c2d1
c3d2
a5
c4d3
26
We test 4 different Hypotheses.
ANOVA TABLE
df
SSQ
SOURCE
4 4 4 4 8
A B C D Error
SSBA SSBB SSBC SSBD SSW

by Subtraction
TSS 24