THE F-TEST IN THE INTRABLOCK ANALYSIS OF

A CLASS OF PBIB D.ESIGNS 1

BY P. V. RAO (Received April 23, 1963)

1. Introduction

The F-test for comparing the group means of several groups of ob- servations is extensively used in experimental work. The test criterion used in this case is derived under certain assumptions about the joint probability distribution of the observations. In many practical situations, however, these assumptions do not always hold.

The object of the present paper is to investigate the robustness of F-test to departures from ANOVA assumptions for 2-associate PBIB de- signs with 2~=0 and 22=1. Similar investigations have been carried out for Randomized Block Designs by Welch [13] and Pitman [12], for Latin Square Designs by Welch [13] and for Balanced Incomplete Block Designs by Mitra [9].

For the definition of a 2-associate Partially Balanced Incomplete Block Design, the reader may refer to Bose, Clatworthy and Shrikhande [2]. Our discussion will be confined to designs with R~=0 and 22= 1. For these designs, the following relations between the parameters are known to hold.

(i.I) (1.2) (1.3) (i.4)

bk=vr, m + n = v - 1 , n = r ( k - 1 ) ,

P i , + P [ ~ + I =P~, + P~ .=m ,

P~, + P]~ + 1 = Ph + Ph = n ,

m P h - n P ~ , = m P h - n P ~ . . = O .

The experimental layout of a 2-associate Partially Balanced Incom- plete Block Design is determined by the plan of a 2-PBIBD and the follow- ing randomization procedure.

R~ : The b blocks of the plan are assigned to the b experimental blocks completely at random.

R~ : Within each experimental block, the k treatments of the rand- omly chosen block of the plan are assigned to the k plots completely at random.

Define Xv to be the yield of plot j in block i ( i = l , 2, . . . , b ; j = l ,

1 Journal Paper No. 291 of the College Experiment Station of the University of Georgia College of Agriculture Experiment Stations.

25

26 P . V . RAO

2, �9 �9 k) of the experimental material. Let t ~ Be imply that the t reat- ment t occurs in block i and write

net = 1 if t ~ B~,

= 0 otherwise,

X~(o=Yield of t reatment t in block i if t ~ Be,

= 0 otherwise,

X , . = k x , . = , t

X , ( . ) = k x , c . ) = , t

X.(o =rx . co = ZX (o ,

Z~(o = X<o -- ne, Z~..

Ze., Ze(.), Z.o) " " etc., are similarly defined. Also, let

2 1 2 I k J=n(n+l)--[n(P~.--P~)+P~2],

k4e,= - PI. ,

k4e~=P~. ,

kQ, = k X.(,) - Z n . X . ,

S,(Q,)-" Z Q . , ( r

where (t, u ) = i implies that t rea tment t and u are i th (i = 1, 2) associates. Noting tha t Q,=Z.(o, the ANOVA table for the intrablock analysis

of a 2-PBIBD with ~ = 0 and ~,=1 is obtained [2] as follows.

ANOVA TABLE

Source d.f. s.s. M.S

Blocks ignoring treatments b--I

Treatments adjusted for blocks v - I

Error bk -- b -- v + 1

Sa= ~ X%./k--Xl../bk

S~, .8=(k-e2) F,Q~/n + ( e l - e 2 ) ~ Q,St(Q~)/n Mr .B t t

.,ca ME

Total bk--I ~X2~j--X2../bk

The ratio U~-Mr.B/M8 is distributed as an F-distribution with v--1 and b k - b - v + l degrees of freedom if the ANOVA conditions are satis- fied and the null hypothesis Ho tha t the t rea tment effects are all equal is true. The hypothesis /-/0 can also be tested by considering the null

THE F-TEST IN T H E INTRABLOCK ANALYSIS 27

distribution of U, over all experimental layouts that could have been ob- tained under R~ and R~. This is possible because, under/ /0, the yield on a plot is independent of the effect of the t rea tment applied to it. If the value of U, obtained from the experiment is exceeded in a% of the cases for the permutat ion distribution of 0"1, we say tha t it is significant at the a percent level. This test, known as the permutation test, depends only on the assumption tha t the randomization procedures R, and R~ are car- ried out. Therefore, the permutat ion theory distribution of U, can be used to assess the robustness of the F-test to departures from ANOVA assumptions. This is done by evaluating the permutation theory moments of a related criterion U=Sr.B/(Sr.B~-SE), and comparing them with the corresponding moments of a ~((v-1), ( b k - b - v + l ) / 2 ) distribution, i.e., the distribution of U under ANOVA assumptions.

2. Preliminary results

In this section we will state some Lemmas which are used in the derivation of the permutation moments of U. In order to be able to write down certain algebraic expressions in a neat form, we will some- times replace multiple summation signs by a single one and denote the range of summation by a set. For example, theexpress ion

will be wri t ten as

P q

tCk

ZY~: , where A = { i , k l i c k , l ~ i ~ p , l ~ k ~ q } . A

When the unrestricted range of the subscripts is clear from the context, we will not write it down explicitly in the definition of the range. Thus A may sometimes be defined as A = {i, k l i C k } .

The Lemmas 2.1 and 2.2 will be stated without proof. Lemma 2.1 follows directly from the combinatorial properties of 2 - P B I B D with ~- - 0 and 2~=1. Lemma 2.2 can be proved by using elementary combinato- rial probability.

LEMMA 2.1. In a 2 - P B I B D with ~,=0 and 2..=1, let 2 , ,=~ i f ( t , u) = i ( i = l , 2), and 2,~=r. i f t = u . Then,

(2.1) 5"I, 2,1,1= nv , ,401

(2.2) 5], .~l,,~=nvP~ , A02

28 P . V . RAO

(2.3) Z ~,l.l~,~.2---v[m(P~)2 + n(P~2)~] , A0$

where .4o,= {t~, t~ I t~r A ~ - {t, u~, u2 I (t, u~ )=l - ( t , u2)} ,

and Ao3= {t,, t2, u,, u~ I ttCu,, t~r (t,, t2)= 1 - (u , , u..)}.

L E n A 2.2. When RI and R~ are carried out for a 2 - P B I B D ,

P(t ~ B<, u e B,)-2,,/b i f i =] ,

if i c j ,

where P(t ~ Bu u ~ Bj) is the probability that t ~ B~ and u ~ Bj . Under Ho, the observed yields X, , X~, . . - , X~ in block i can be re-

garded as a random permutation of the k numbers X~, X~, . - . , X~. Hence if t ~ B~, each of the observed values Z~,, Z~, . . . , Z~ are equally likely for Z~(o. Thus we have,

E(Z,(o)=P(t e B~)E(Z~r It e B,)+P(t ~ B,)E(Z~<o It ~ B~)

= (r/b)~ Z,,P(Z,r Z,j) + 0 y

-- (r/bk)Z~. = O .

Also, since randomization is carried out independently in different blocks, we have, if iC j , E(Z<oZjr162162 Thus, we have proved Lemma 2.3.

LF_JIMA 2.3. I f i r 3, then Z~(o and Zj(~) are uizcorrelated. More gener- ally, suppose that ~ is a product of a finite number of Z~(o's. I f ~ contains a letter, say Z<o , and no other letter in ~ has a block subscript i, and i f Z<o is repeated only once (i.e., i f it occurs with the exponent one) in ~, then E ( ~ ) - O.

Since we consider only designs with ~,=0, no block of the design will contain two treatments which are first associates. Therefore E(Z<oZ<,>)=O if (t, u )= l . Hence we have proved Lemma 2.4.

LE~MA 2.4. I f ~ contains two letters, say Z<o and Z~(,), such that they have the same block subscripts in common, and (t, u ) - 1 , then E

Let S , = ~ Z2~, S = b S - - Z S,, V = Z ( S , - S ) " / b ~ , M, , - -Z Z~, M, -X? M,,, .i ~, l t i,

and ~ = [ 1 - I r / ( b - 1 ) ] ~ . The expected values given below follow directly from Lemmas 2.1, 2.2 and 2.3, and by direct calculation.

(2.4)

(2.5) E(Z,(oZj(,))=0 if i C j ,

=-2,S~/[bk(k-1)] , if i=3" and t C u ,

THE F-TEST IN THE INTRABLOCK ANALYSIS 29

4 I / (2.6) E(Z<o)-M~,/v ,

~- ~ - - ~ I 1 , (2.7) E(Z,(oZ,oo)-I,~(S,-M~,) ,[bk(k- )] if t C u

2 2 - - 2 / �9 ? (2.8) E(Z<,)ZJ(,a))--(r -2,,)S~S~j [b(b-1)k ] if i C j .

3. The first two permutat ion moments of U

Since

ST.B + Sr= [Z X~j-- X!./bk ] -- [Z X~./b- X!./bk ] t J t

= ~, z ~ = s , l,J

Sr.B+Se is a fixed quantity for all permutations of Z,j. Hence E(U)= E(Sr.~/S)=E(Sr.B)/S, and Vat(U)= V(Sr.,)/S". Thus the problem reduces to tha t of evaluating the moments of ST.B.

I t is easy to see tha t k"A=vP~2. Hence,

nE(Sr.,) = (k-e~)Z E(Q~) +(e~-e2)~ E(Q,S~(@))

( t , u ) f f i t

E(Q,Q,,) .

Thus

nkAE(Sr.B)=(k2A-pi2)EE(Q~)-n E E(Q,Q.) t t ,u

( t , u ) = l

=(v-1)P~,~.~ E(Q~)-n ~ E(Q,Q,,) t t , u

( t , u ) = 1

=(v--1)P~..Sq E(Z~.(o)--n Z E(Z.e)Z.(,)). ( t , u ) = l

From (2.5) it follows that the second te rm on the r ight hand side above is zero and

E(Z~.o)) = E [ Z Z,(o] ~

= ~ E(Z~(,)) + ~ E(Z,(,)Zi(,>).

On using (2.4) and (2.5) we get

E(Z~.(o) = ~, S,/v + O= S l y . t

Hence nkA E(Sr.B)=(v-1)P~S, so tha t

(3.1) E( U) = E(Sr.B)/S- ( v - 1)P~J(nkJ)

= k(v- 1)P&/nk'A

30 P . V . RAO

=k(v- -1) /nv

= k ( v - 1 ) / r ( k - - 1 ) v

=k(v -1 ) /bk (k - -1 )

= ( v - 1 ) / b ( k - 1 ) ,

which is same as the corresponding normal theory moment of U. To find the expected value of S~.s under permutation theory we first

consider the following expression:

(nkA)"S~r.~= [ (v- 1)Ph21Z.~(,)-- n ~ Z.~,)Z.(,)] ~ , t ( t , ~ O = ~

i.e.,

(3.2)

where,

(nvP~):S~r.~= [k ( v - 1)Ph~ Z~.(o--nk 21 Z.(,)Z.(,)] ~ t ( t , u ) = l

= [k(v-- 1)P~]"a, -- 2nk~P~a~ + n~k'-a~

and,

2 2 ~ 4 2 2 a ~ = [ ~ Z.(,)] - 21Z.(,) + 21 Z.(,)Z.(~) , t t t ~ t

2 a., = [~ , Z.( ,)][21(, , , ) .~Z(,~Z.(,)]

$

Z3 =N~,.,,)~ .(,)Z.~)+ 21 Z:(,)~.(,,)Z.(u)~.(,,),

a~ = [21c,.~)=~Z.c,)Z.o,)] ~ =Zc,-)=l=c,.~')Z~.(,)Z.cu~Z.(=,)+ 21 Z.coZ.(,,)Z.c~)Z.(~,).

t ~ t " ( t , u ) = l = ( t ' , u ' )

We will now proceed to evaluate the expected value of each az( i=l , 2, 3). We first have

~, E(Z~.{,))=?~. E[21 Z,(,)]'

,- 2 2 =~E(Z~{o+4 Z E(Z~l(~)Z~{,))+ 321 E(Z,v.cZ,z(o) i l ~ t 2 ii@~ 2

+6 ~ E '~"~*,, (Z,,coZ,,c,~Z,,(,))+ Z E(Z~(oZ,,c,)Z,~(,)Z,,(o).

Using Lemma 2.3, the right hand side above is equal to

2

which on applying (2.6) and (2.7) reduces to

(3.3) M, + [3vr(r - 1)/b(b -- 1)k ~] ~ S~,S~2

= M, +[3(r-1)/bk]~ .

THE F-TEST IN TH E INTRABLOCK ANALYSIS 31

Also,

By Lemma 2.3, this is equal to

N E(Z~(,)Z~(,))+ N E(Z~v,)Z~..(,))+ N E(Z,,(oZqc=)Z,:(,)Z~(~)). t~eu i l~ef2 ~ u

t ~u i l~ i ~

But,

E (Zq(oZq(,,)Zi:o)Z~:(~)):],J~,~-l)( ~ SqS~2)/b(b--1)k:(k-1) ~ t l ~ t I i l ~ i $

=0 since ~,~=0 or 1.

Hence using (2.7) and (2.8) we get

�9 . ~ 9 ! E(Z"(oZ"(,,))=[Z , J S ~ - M ~ ) / b k ( k - 1 ) ] § ~ ( r -~ ,~ ) t ~ t t@b~ $~eU

i l~ i 2

S~,S~Jb(b--1)k'] .

By using Lemma (2.1) the r ight hand side of the equation above can be shown to be equal to

(3.4) [nv(Z S ~ - M~)/bk(k-- 1)] + Iv(v-- 1)r"- nv](S ~- ~ S~)/b(b-- 1)k'

= ( Z Z~ -- M,) + [r(v -- 1) -- (k - 1)](S 2- Z S ~),/k(b- 1)

= S ' - M ~ - ( r - 1 ) ~ / b k .

From (3.3) and (3.4) we get

(3 .5) E ( a , ) : S ' -t- 2(r - 1)6/bk.

Direct application of Lemmas 2.3 and 2.4 yields E(a,)=0. We will now evaluate E(a3) as follows.

E(Z2.(:)Z.(:~,Z.(~,,,) = Z E(Z:.,,Z:.<~:,) ($, Ul)=l=(~, ~$) (e, u)=l

A,t

where Ao~--{t, u~, u~[u~r (t, u~)-- l=(t , u,)} . Putting ~,~,--0 in Z E(Z?c,)Z~.c~)) we get

(3.6) .Z E(Z~.(,)Z2.c~))=mr~/bk.

Also,

2 Z . . . . �9 Z E(Z.o)Z.,=~, . (~))=Z 5"I, E(Zi,c,,Z,~(~,Z~(=,,Z,,(~2,) AOt AOt ~lZ~ZStt

32 P . V . RAO

Using Lemma 2.3 and Lemma 2.4, the above expression can be shown to be equal to

no~ A05

where Aos= {i , i:, 3",, 3".,., 3~ li:r j~r .

On using Lemma 2.1 this reduces to

(3.7) --(nvrP~, ~ S~S~2)/k2(k-1)b(b--1) = -nP~fi , b k ( k - 1 ) . il~l ~

Hence, using (3.6) and (3.7) we get

(8.8) E (t, U l ) f l = ( t , u 2)

E(Z,(,~)Z.(~pZ.(,,)) = [ m r - - n P ~ f f ( k - 1)]8, bk

= mrP~2~/nbk.

Finally,

E $1:~$2

(t 1, U l ) = l = ( t ~, U 2)

E( Z.a~,Z.c,~,Z.cu~,Z.(,,~J = E E ( Z.(,~,Z.c,2,Z".~,J A06

+ ~ E( Z.(,1)Z.(,3)Z.(,1)Z.(.,)) ,

where Ao,= {t,, t2, u It,:/:t2, ( t . u )=l=( t2 , u)}, and

A07= {t,, t,,ul, u2 ]tl--/:t2, utCu~, (tt, u J = l = ( t 2 , u2)}. It is easy to see that the first expression on the right hand side above is same as expression (3.7). Hence it is equal to - n P ~ / b k ( k - - 1 ) . The second expression can be written as

[E + E + E + E]E(Z.,,1,I,,,Z.,.~,Z.,...,) , A'08 "4"09 '410 A l l

where

A0,= {tl, t~, ul, u2 I(t,, t,)=(u,, u,)=(tl, u,)=(t,, u , ) = l } , A ~ = {t,, t2, u,, u, I(t,, t , )=( t , , u l ) = ( t , , u , ) = l , (u,, u . ) = 2 } ,

A,o= {t . t,, u . us I(tl, t,)=2, (tl, Ul)=(t,, u,)=(ul, u~)= l} , A1,----{t~, tz, ul, u2 [(t~, t2)'-(u~,u2):2, (t~, u~)-(t~, u : ) - l } ,

and the above notation implies that the expression outside the square brackets must be summed wifh respect to .each of the summation signs inside the square brackets and the results added. Let the four compo- nents of the sum above be denoted by /)1, D2,/)3 and D, respectively. Now consider

THE F-TEST IN TH E INTRABLOCK ANALYSIS 33

D~= ~ E(Z!(,)Z~(.)) +2 ]E E(Z~.o,)Z.<,,)Z.(,~,) ($, ~)=I A06

+6 Z] E(Z.,,~,Z.,,,)Zx.~)Z.(~,)), -41~

where

A, ,= {t,, t2, u~, u~ [t~r t~q:u~, ( t , t~)--(u, u , )= ( t , u~)=(t~, u , ) - - l } .

Using (3.6) and (3.7), the first two expressions above are respectively equal to mr6/bk and 2nP[~6/bk(k--1). The third expression is equal to

5-], E(Z~(n)Z~,~)Z~(.~)Z,,(~)). From Lemma 2.2 and Lemma 2.3 it A.}A I +I I, '+,~, '~:~, ~I~l

follows that the only non-.~anishing terms of this sum are those for which i~=i,, i~=is and i~: /~ . Hence it is equal to

Z ~ E(Z,~(,,,Z,~(.~)Z,,(,,)Z,:<:~,) AI~ ~1~

=~. ~, 2,~,R~,S~S~Jb(b-1)k2(k-1) ~

Therefore, we have,

D, = [ m r - 2(k - 1)-~nP~ -b (bk)-'(k - 1) '~ R,,.,~,~]~/bk . A08

Similar considerations will show that

D~ = [ Z R,~,,~R,2~l~/[bk(k-- 1)12 , and A13

D3---- [ Z 2n~R,2~,]~/[bk( k - 1)] 2 , A14

where,

A~3= {t~, t~, u~, u~ [ (u~, u~)=2, (t,, t , ) - - ( t , , u~)-(t2, u ~ ) - - - 1 , ~

A~,-{t~, t,, u~, u, I (t~, t,)--2, (u~, u,)=(t~, u~)=(t2, u , ) = l ,

t~r t~r .

Finally,

D, = ~ Z E ( Z,I(,I)Z~j(,~,Zis(~I,Z,,(~)) . All ~1" +~' "$8" ~i

The non-vanishing terms on the right hand side above are those for which il=i~, is=i, and i~r or i~ - i , , i~=i3 and i~r Hence

D , - ~ ~ [E(Z,,(,,)Z,I(,,)Z,,,:I,Z,,,:2,)+E(Z,,r

34 P . V . RAO

As in the previous cases, these expressions can be simplified using the Lemmas proved earlier and the relations between the parameters of the design. It can be shown that

Hence,

D , = ~ ( ~,,,,~l.~ + 2,~,~,,.,)~/[bk( k - 1 ) ] 2 . d.ll

Z E(Z.,,pZ.,,~,Z.c~)Z.(~,) = [mr - 2 ( k - 1)- 'nP~ + (bk ) - ' (k - 1) -5

( Z 2,~22,2~ + 5-].2,x~,2~) l~/bk ,

where the first summation is subject to the condition that tlCus, t~:/:ul, (tl, u~)= 1= (t2, u2) and the second is subject to (t~, u2) =2 = (t~, u~) and (t~, u~)=l=(t2, u~). A little consideration will show that the results of two summations are identical. Using this fact and Lemma 2.1, the ex- pression can be shown to be equal to

Hence,

and

[ m r - 2 ( k - 1 ) - l n P [ l + 2 n - l ( k - - 1 ) -~ {m(P~2) 2+n(P12) 2} ]~/bk.

Z (t 1, ~i)~1=(~2, u 2)

E( Z.,,I,Z.(:I)Z.(,2)Z.(:s, ) - [ m r - 3n(k-- 1)-IP~1

+ 2n-l(k--1) -1 {m(P~2)' +n(P~2) 2} ]~/bk ,

E (a3 )=2[mr-2n (k -1 ) - 'P~ l +n- ' ( k - -1 ) -l{m(P[2) 2 +n(P[2) ~} ]8/bk .

Therefore

and

(nvP~2)"E(S2r.,) = [k(v-- 1)P~2]2E(al) + (nk)2E(a3) ,

E( U 2) = S -2E(S ~.a) = (nv)-2k:(v - 1) 2 + {2( r - 1)[k(v- 1)P~]'

+ 2(nk):mr-4(nk)2n(k--1)- lPll +n(k-1) -1(nk) ~

x [m(Pi2) 2 +n(P,~2)2]} [1-- V / ( b - 1)]/bk(nvP~) 2 .

By using a little straight forward algebra it can be shown that

Var( U)=2(v-1)[(v-1)(r-1)(k-1)+m][1- E'(b--1)]/kb~(k--1) ' = 2(v-- 1)k2[(v-- 1)(n-- k.+ 1) -I- m][1 -- V/'(b - 1)]/v~n s .

I t is interesting to note that the first two permutation moments of Lr do not depend on the so-called parameters of the second kind,

THE F-TEST IN THE INTRABLOCK ANALYSIS 35

4. Comparison of the distribution of U with the ~-distribution

The mean and variance of a /~-distribution with (v - l ) / 2 and ( b k - b --v-~1)/2 are given by

Mean=(v-1) /b(k- -1) , Variance-- 2(v-- 1)(bk- b--v + 1)/b'~(k- 1)-'[b(k- 1)-~ 2]

----2(v-- 1)k~[(v-1)(n-k)+k]/v:n:(vn+2k).

Thus the permutation and normal theory means of U are identical, but the variances differ in the two situations. The ratio of the permutation theory variance to the normal theory variance is

e= [ ( v - 1 ) ( n - - k + l)-t-m](vn-~-2k) [ 1 - V/(b--1)] [ ( v -1 ) (n -k ) - l -n]vn

( 2m ) ( 1 ; 2 k l [ l _ V / ( b _ t ) ] " = 1-~ (v--1)(n--k)-t-n x v n /

Thus if the block errors are homogeneous, that is, V--0, then the variances will also be almost equal provided b(k-1) is large as compared with 2 (i.e. vn is larger compared with 2k) and n is large as compared with k ~-1 or, equivalently, if V=O, then the variances are practically equal if v is large as compared with 2 and n is large as compared with k. In such cases, the F-test can be expected to perform reasonably well as an approximation to randomization test.

When V---0, e > l ; showing that the permutation theory variance of U is larger than the corresponding variance under normal theory. How- ever, heterogeneity of block variances reduces the value of e. Thus, in general," a slight heterogeneity of block variances may improve the F-test as an approximation to randomization test, but, large value of V will result in the F-test becoming somewhat conservative.

5. Acknowledgement

The author acknowledges with pleasure the help and guidance he received from Dr. A. C. Cohen, under whose direction this work was carried out.

UNIVERSITY OF GEORGIA

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