AN EASY METHOD OF CONSTRUCTING LATIN SQUARE

DESIGNS BALANCED FOR THE IMMEDIATE RESIDUAL

AND OTHER ORDER EFFECTS

V . K . S H A M I n s t i t u t e o f A g r i c u l t u r a l R e s e a r c h S t a t i s t i c s

N e w D e l h i , I n d i a

ABSTRACT

B r a d l e y ( 1 9 5 8 ) p r o p o s e d a v e r y s i m p l e p r o c e d u r e f o r c o n s t r u c - t i n g l a t i n s q u a r e d e s i q n s t o c o u n t e r b a l a n c e t h e i m m e d i a t e s e q u e n t i a l

e f f e c t f o r a n e v e n number o f t r e a t m e n t s . When t h e number of t r e a t -

m e n t s i s o d d , b a l a n c e i n a s i n q l e l a t i n s q u a r e i s n o t p o s s i b l e . I n t h e p r e s e n t n o t e w e h a v e d e v e l o p e d a n a n a l o g o u s method f o r t h e

c o n s t r u c t i o n o f s u c h d e s i g n s w h i c h may b e u s e d f o r a n e v e n or o d d

number o f t r e a t m e n t s . A p r o o f h a s a l s o b e e n o f f e r e d t o a s s u r e t h e

g e n e r a l v a l i d i t y o f t h e p r o c e d u r e .

1. INTRODUCTION

I n many f i e l d s o f e x p e r i m e n t a t i o n w h e r e a n e x p e r i m e n t i s c o n -

t i n u e d o v e r s e v e r a l p e r i o d s , sometimes t h e r e s p o n s e o f t h e s u b j e c t s i s n o t o n l y i n f l u e n c e d by t h e t r e a t m e n t a p p l i e d i n t h e p e r i o d o f

o b s e r v a t i o n b u t a l s o by t h e p o s i t i o n o f t h e t r e a t m e n t i n t h e o r d e r o f p r e s e n t a t i o n a n d / o r by t h e i m m e d i a t e l y p r e c e d i n g t r e a t m e n t s . I n s u c h s i t u a t i o n s b a l a n c e d l a t i n s q u a r e s , a s d e f i n e d by W i l l i a m s

( 1 9 4 9 ) I may b e u s e d i n o r d e r t o h a v e t i d y a n a l y s i s a n d t o h a v e

-119-

SHARMA

estimates wi th h igh p r e c i s i o n . An easy method of c o n s t r u c t i o n of such d e s i g n s i n a s i n g l e

l a t i n s q u a r e w a s proposed by Bradley (1958) when t h e number of t r e a t m e n t s , t, i s even. But when t is odd, ba lance i n a s i n g l e l a t i n s q u a r e is e a s i l y s e e n t o be imposs ib le ; however, i n such s i t u a t i o n s ba lance can be achieved i n two l a t i n s q u a r e s (Wil l iams

Bradley ' s work by s u g g e s t i n g a n analogous e a s y method of con- s t r u c t i o n of these d e s i g n s ba lanced w i t h r e s p e c t t o immediate predecessors which can be used whe the r t i s even o r odd. To a s s u r e t h e g e n e r a l v a l i d i t y of t h e method, a proof has a l s o been provided.

1949 ) . The purpose of t h e p r e s e n t n o t e i s t o supplement

2. METHOD

( a ) C o n s t r u c t t w o t x t t a b l e s i n which rows refer t o i n d i v i d u a l s , and columns from l e f t t o r i g h t r e f e r t o t h e o r d e r of p r e s e n t a t i o n of exper imenta l c o n d i t i o n s .

( b l I n both t h e s q u a r e s , number t h e o r d e r of p r e s e n t a t i o n

(c) Number t h e t r e a t m e n t s i = 0, I, 2, . . .., t - 1 . ( d ) Assign t h e s e t r e a t m e n t symbols s u c c e s s i v e l y t o t h e t

from 1 t o t s u c c e s s i v e l y .

cel ls i n t h e f i r s t row of b o t h t h e s q u a r e s by proceeding from l e f t t o r i g h t , e n t e r i n g o n l y i n odd-numbered ce l l s i n t h e f i r s t and even-numbered ce l l s i n t h e second s q u a r e , and t h e n r e v e r s i n g t h e d i r e c t i o n , f i l l i n g i n even-numbered ce l l s i n t h e f i r s t and odd-numbered cells i n t h e second s q u a r e .

( e l Obtain t h e s u c c e s s i v e rows of t h e s q u a r e s by adding i n t e g e r 1 t o each e lement of t h e p r e v i o u s row and reducing t h e elements , if n e c e s s a r y , by module t ( i n s h o r t mod t ) .

I t is t o be noted t h a t i n each of t h e c o n s t r u c t e d s q u a r e s every t r e a t m e n t symbol o c c u r s i n each r o w and i n each column p r e c i s e l y once. Moreover, when t is even, each t r e a t m e n t symbol is preceded e x a c t l y once by o t h e r t r e a t m e n t symbols and once by none of t h e t r e a t m e n t symbols i n e i t h e r of t h e t w o s q u a r e s . Thus, i n t h i s case e i t h e r of t h e t w o s q u a r e s may be used . T h i s s i t - u a t i o n o c c u r s i n n e i t h e r of t h e t w o s q u a r e s i f t i s odd. However, when both t h e s q u a r e s a r e c o n s i d e r e d t a g e t h e r , each t r e a t m e n t

-120-

LATIN SQUARES

symbol i s p r e c e d e d by e v e r y o t h e r ( symbol) e x a c t l y twice and t w i c e by none of them. C o n s e q u e n t l y , b o t h t h e s q u a r e s must be used i n t h i s case.

T a b l e s 1 and 2 i l l u s t r a t e t h e method f o r t = 4 and 1; = 5

r e s p e c t i v e l y .

TABLE 1

L a t i n s q u a r e s for t = 4

1st S q u a r e 2nd S q u a r e

I n d i v i - Orde r of P r e s e n t a t i o n I n d i v i - Order o f P r e s e n t a t i o n dua 1 d u a l

1 2 3 4 1 2 3 4

A 0 3 1 2 A * 3 0 2 1

B 1 0 2 3 B * 0 1 3 2

c 2 1 3 0 c * 1 2 0 3

D 3 2 0 1 D* 2 3 1 0

TABLE 2

L a t i n s q u a r e s f o r t = 5

1st S q u a r e 2nd S q u a r e

I n d i v i - Orde r o f P r e s e n t a t i o n dua 1

1 2 3 4 5

A 0 4 1 3 2

B 1 0 2 4 3

C 2 1 3 0 4

D 3 2 4 1 0

E 4 3 0 2 1

I n d i v i - Orde r of P r e s e n t a t i o n dua 1

1 2 3 4 5

3 . PROOF

A * 4 0 3 1 2

B * 0 I 4 2 3

C* 1 2 0 3 4

D* 2 3 1 4 0

E * 3 4 2 0 1

I t w i l l be s u f f i c i e n t t o show t h a t t h e s t e p s o u t l i n e d unde r

-121-

SHA RMA

the above procedure give rise to squares satisfying the following:

( i l Each one is a latin square irrespective of even o r odd value of t .

l i i l Each treatment is preceded by every other treatment equally of ten.

Let the set of t treatments be represented by the elements of mod t including zero. To prove the first property, we consider any treatment, say i , occurring in either of the initial rows of the two squares. Now as the treatments in the initial row repre- sent all the elements of mod t and the successive rows are obtained by adding 1 to each element of the preceding row and reducing the elements by mod t , evidently the treatment i will occur once and only once in each row and in each column.

We shall prove the second property first for even values of t and then for odd values.

Condition f i i l in terms of the differences between the values for two adjacent treatments implies that these differences should take all values from 1 to t - 1 equally often. The square will satisfy the required conditions if each difference occurs once in the initial row of both the latin squares and the succes- sive rows are obtained by adding 1 to each element of the pre- ceding row, for each element of any column will differ by a constant amount (which will be different for each pair of adjacent columns) from the corresponding element of the preceding column. As each treatment occurs once in each column, it will be preceded on each occasion by a different treatment.

Now, the elements in the first row of the two squares from left to right are

0 , t - 1 , I, t - 2 , 2, t - 3 )..., t / d + l , t / 2 - 1 , t / 2

: - 1 , o , t - 2 , 1 , t - 3 , 2 ,..., t / 2 - 2 , t / z , t / 2 - 1

t - I, 2, t - 3, 4 , t - 5, ..., t - 2, 1

I, t - 2 , 3, t - 4 , 5 ,..., 2, t - 1

and

respectively giving successive differences as

and

respectively.

-122-

LATIN SQlJARES

W e n o t e h e r e t h a t i n t h e case of t h e f i r s t s q u a r e t h e odd

v a l u e s i n d e c r e a s i n g o r d e r a re i n t e r l a c e d w i t h t h e even v a l u e s i n i n c r e a s i n g o r d e r w h i l e i n t h e case o f t h e second t h e odd v a l u e s i n a s c e n d i n g o r d e r are i n t e r l a c e d w i t h even v a l u e s i n d e s c e n d i n g o r d e r and t h u s t h e d i f f e r e n c e s are i n c l u d e d o n l y once i n each r o w . T h e r e f o r e , t h e l a t i n s q u a r e deve loped from e i t h e r o f t h e i n i t i a l r o w s w i l l s a t i s f y t h e c o n d i t i o n l i i l .

A s i m i l a r argument a p p l i e s f o r t h e odd v a l u e o f t. I n t h i s case t h e i n i t i a l row o f t h e f i r s t l a t i n s q u a r e

a , t - I , I , t - 2 , 2 , t - 3 , 3 , t - 4 ,..., I t - 3 ) / 2 , I t + 1 1 / 2 , It - 1 ) / 2

g i v e s o n l y even d i f f e r e n c e s , e a c h o f them o c c u r r i n g twice. The i n i t i a l r o w o f t h e second l a t i n s q u a r e

t - 1 , 0 , t - 2 , I, t - 3 , 2 , t - 4 , 3 , . . . , It + 1 ) / 2 , I t - 3 1 / 2 , (t - 1 ) / 2

g i v e s o n l y odd d i f f e r e n c e s , e a c h o c c u r r i n g t w i c e i n t h i s case as w e l l . The p a i r o f l a t i n s q u a r e s g e n e r a t e d from t h e s e two i n i t i a l rows t h e r e f o r e p r o v i d e s a b a l a n c e d d e s i g n .

I t i s wor thwhi l e t o ment ion t h a t B r a d l e y ’ s m o d i f i c a t i o n

can s t i l l be a p p l i e d , w i t h o u t l o s i n g i t s u t i l i t y , t o e a c h o f t h e l a t i n s q u a r e s to c o u n t e r - b a l a n c e t h e o r d e r e f f e c t s i n columns as

w e l l a s r o w s . As an i l l u s t r a t i o n w e t a k e t = 5.

TABLE 3

M o d i f i c a t i o n f o r t = 5 t o c o u n t e r b a l a n c e o r d e r e f f ec t s i n columns as w e l l a s rows

1st Square 2nd Square

1 2 3 4 5 1 2 3 4 5

0 4 1 3 2 A * 4 0 3 1 2

4 3 0 1 B * 0 . I 4 2 3

1 0 2 4 3 c * 3 4 2 0 i

3 2 4 I 0 D * 1 2 0 3 4

2 1 3 0 4 E * 2 3 1 4 0

,

-123-

SHARMA

ACKNOWLEDGEMENTS

The author is very much grateful to Dr. M.N. Das, Director, Institute of Agricultural Research Statistics, for his valuable guidance in the preparation of this paper. The financial assis- tance provided by the Indian Council of Agricultural Research is also gratefully acknowledged.

Bradley a proposd un procede trss simple pour l'6laboration de plans d'expdrience en carre latin afin de contrebalancer l'effet sdquentiel immediat pour un nombre pair de traitements. Quand le nombre de traitements est impair, l'gquilibre n'est pas possible dans un seul carre latin. Nous avons dCveloYp5 dans cette note une mgthode analogue d'6laboration de tels plans, qui peut 6tre utilised autant pour un nombre pair que pour un nombre impair de traitements. Nous prgsentons aussi une preuve pour garantir la validit6 g6ndrale du procdd6.

REFERENCES

[ 1 1 Bradley, J.V. (1958). Complete counterbalancing of immediate sequential effects in a latin square design. J .

Amer. Statist. Assoc., 53, 525-528.

the estimation of residual effects of treatments: Australiun J . Sci. Res.,Series A, 2, 149-168.

1 2 1 Williams, E.J. (1949). Experimental aesigns balanced for

Received IS Febrmary 1974 DR. V.K. SHARMA Institute of Agricultural Research Statistics

New Delhi, India

-124-