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 Col1 Col2 Col3 Col4 Square2 Col1 Col2 Col3 Col4Row1 A B C D Row1 A B C DRow2 B C D A Row2 B A D C Row3 C D A B Row3 C D A BRow4 D A B C Row4 D C B A

Latin Square Model

2

2

Note: Although there are 3 subscripts, there are only cases (defined by rows/cols)

1,..., ; 1,..., ; 1,..., ; ~ 0, independent

overall mean effect of ro

ijk i j k ijk ijk

i

r

Y i r j r k r N

1 1 1

1 1

w effect of column Effect of treament

0

Row, Column, Treatment Sums and Means:

Rows: Columns:

Treatments:

j k

r r r

i j ki j k

r rji

i ji ijk j ijkj i

i j k

YYY Y Y Y Y Y

r r

Y

2, 1 1

^ ^ ^ ^

^ ^ ^ ^ ^

Overall:

Least Squares Estimates:

Predicted Values and Residuals:

r rk

kk ijk ijki j i j

i j j kki

ijk j iki

Y YY Y Y Y Y

r r

Y Y Y Y Y Y Y

Y Y Y

^

2 2j k ijk i j kijk ijk ijkY Y e Y Y Y Y Y Y Y

Analysis of Variance

22

1 1

2

22 1

1

2

2 12

1

Total Sum of Squares: 1

Row Sum of Squares: 11

Col Sum of Squares: 1

r r

ijk TOi j

r

iri

i ROWi

r

jrj

j COLj

SSTO Y Y df r

rSSROW r Y Y df r E MSROW

r

r

SSCOL r Y Y df r E MSCOL

2

22 1

1

2

1 1

2Re

1

Trt Sum of Squares: 11

Remainder (Error) Sum of Squares: Re 2

1 2 Re

Testing for Treatment Effect

r

krk

k TRk

r r

i j kijki j

m

r

rSSTR r Y Y df r E MSTR

r

SS m Y Y Y Y Y

df r r E MS m

0 1

* *0

s: : ... 0 : Not all 0

Test Statistic: Reject H if 0.95; 1, 1 2Re

r A kH H

MSTRF F F r r r

MS m

Post-Hoc Comparison of Treatment Means & Relative Efficiency

^

1

ReTukey's HSD: 0.95; ; 1 2

1 2 ReBonferroni's MSD C= : 1 ; 1 2

2 2

Relative Efficiency of Latin Square to Completely Randomized Design:

1 Re

1

ij

ij

MS mHSD q r r r

r

r r MS mMSD t r r

C r

MSROW MSCOL r MS mE

r

^

2

^

3

Re

Relative Efficiency of Latin Square to Randomized Block Design:

1 ReRBD(Rows):

Re1 Re

RBD(Columns): Re

MS m

MSCOL r MS mE

rMS mMSROW r MS m

ErMS 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