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 A C C C 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 A C C C 2 2 3 1 A C C C 3 3 1 2

4. Example with 4 Levels per Factor
FACTORS
VARIABLE
Automobiles A four levels
Tire positions B four levels
Tire treatments C four levels
Lifetime of a tire (days) A 1 2 3 4 B C 8 5 7 9 6

5. The Model for (Unreplicated) Latin Squares
Example:
Three
factors r , t
and g
each at m
levels  i =
1,... m y      = + + + + j = 1,
... , m
ijk j k
ijk i
Y = A + B + C + e k
... , = 1, m
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 – 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. An “interaction-like” term. (After all, there’s no replication!)
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, there’s 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   R o w s m  m – 1  2 + V m ( y – y ) 2
Rows i ..
...  i = 1  C o l u m n s m  m – 1  2 + V m ( y – y ) 2
Col . j .
...  j = 1  I n s i d e m  m – 1  2 + V m ( y – y ) 2
Inside
factor f a c t o r .. k
... 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    ( y – y ) 2 m 2 – 1
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. (Inside factor = Tire Treatment)
Our Example:
(Inside factor = Tire Treatment)
Tire Position
Auto.

11. General Linear Model: Lifetime versus Auto, Postn, Trtmnt Factor Type Levels Values Auto fixed Postn fixed Trtmnt fixed Analysis of Variance for Lifetime, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Auto Postn Trtmnt Error Total Unusual Observations for Lifetime Obs Lifetime Fit SE Fit Residual St Resid R

13. SPSS/Minitab DATA ENTRY VAR1. VAR2. VAR3. VAR4 855. 1. 1. 4 962. 2. 1

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
row 1 2 3 4 A B C D
column
row 1 2 3 4 A B C D
Row = 4 tire positions; column = 4 cars

17. Row = clinics; column = patients; letter = drugs for flu
Case Two
column
row 1 2 3 4 A B C D
column
row 5 6 7 8 B C D A
Row = clinics; column = patients; letter = drugs for flu

18. Row = 4 tire positions; column = 8 cars
Case Three
column
row 1 2 3 4 A B C D 5 6 7 8 B C D A
Row = 4 tire positions; column = 8 cars

19. ANOVA for Case 1 SSBR, 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:
SS: please refer to the book, Statistical Principles of research Design and Analysis by R. Kuehl.
DF: # of levels – 1 for all terms except error. DF of error = total DF – the sum of the rest DF’s.
Using Minitab in the same way can give Anova tables for all cases.

21. Graeco-Latin Squares m – 1 = m + 1 m – 1
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 1: m=3; four factors can be accommodated

23. Example 2: m = 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:
b b b b b5 a1
c1d1
c2d2
c3d3
c4d4
c5d5 a2
c2d3
c3d4
c4d5
c5d1
c1d2 a3
c3d5
c4d1
c5d2
c1d3
c2d4 a4
c4d2
c5d3
c1d4
c2d5
c3d1 a5
c5d4
c1d5
c2d1
c3d2
c4d3

26. We test 4 different Hypotheses.
ANOVA TABLE
SOURCE
SSQ df A B C D
Error
SSBA
SSBB
SSBC
SSBD
SSW 4 8 •
by Subtraction
TSS