A NOTE ON CONFIDENCE BOUNDS CONNECTED WITH ANOVA AND MANOVA FOR BALANCED AND PARTIALLY BALANCED INCOMPLETE BLOCK DESIGNS. v. P.

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1 I. A NOTE ON CONFIDENCE BOUNDS CONNECTED WITH ANOVA AND MANOVA FOR BALANCED AND PARTIALLY BALANCED INCOMPLETE BLOCK DESIGNS by :}I v. P. Bhapkar University of North Carolina. '".". This research was supported partly by the Office of Naval Research under Contract No. Nonr-855(06) for research in probability and statistics at Chapel Hill and partly by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. AF 49(638)-213. Reproduction in whole or in part is permitted for any purpose of the United States Government. Institute of Statistics M~eograph Series No. 216 January 1959 \ \ '-0:'.

2 A NOTE ON CONFIDENCE BOUNDS CONNECTED WITH ANOVA AND FOR BALAOCED AND PARTIALLY BALANCED INCOMPLETE BLOCK DESIGNS1 MANOVA By V. P. Bhapkar University of North Carolina 1. Introduction and summary It is well known (23J how in the case of any general "testable ll linear hypothesis for either ANOVA or MANOVA one can put simultaneous confidence bounds on a set of parametric functions which might be regarded as measures of deviation from the total hypothesis and its various components. The parametric functions are such that in each problem one of these can be appropriately called the "total" and the rest "partials" of various orders. For each problem the "total" one (1) is related to but not quite the same as the noncentrality parameter of the usual F-test of the total hypothesis in ANOVA and (ii) is the largest characteristic root of a certain parametric matrix which is related to but not quite the same as another parametric matrix whose nonzero characteristic roots occur as a set of noncentrality parameters in the power function for the test (no matter which one of the standard tests we use) of the total hypothesis in MANOVA. The same remark applies to "partials" of various orders considered in the proper sense. In this note for both ANOVA a~d MANOVA the hypothesis considered is that of equality of the treatment effects--vector equality in the 'This research was supported partly by the Office of Naval Research under Contract No. Nonr-855(06) for research in probability and statistics at Chapel Hill and partly by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. AF 49(638)-213. Reproduction in whole or in part is permitted for any purpose of the United States Government.

3 2 case of MANOVA. Starting from such a hypothesis explicit algebraic expressions are obtained for the total and partial parametric functions that go with the simultaneous confidence statements in the case of both ANOVA and MANOVA and for balanced and partially balanced designs. It is also indicated how to obtain in a convenient form the algebraic expression for the confidence bounds on each such parametric function without a derivation of these expressions in an explicit form. 2. Notation and preliminaries i) Let ~ denote a column-vector of n independent normal variables with a common variance 0 2 and the means given by (1) where A is a matrix of known constants and e is a vector of unknown parameters. Suppose - Xo: B e =0 -sxrn--- [Rank B = 5] is the "testable" hypothesis to be tested. Be=~ - - -sxl then the "totall' parametric function ~ If we write associated with de o is: where ~02 is the variance-covariance matrix of the best unbiased linear estimates of~. It may be observed that 6/0 2 is the noncentralityparameter of the F-test for ')loa The confidence bounds on 6 with a confidence coefficient greater than or equal to (1 - a) are then given by [23]:

4 3 where r =Rank A F is the 10ea% significance point of F with d.f. s a and n - r respectively SH o is the sum of squares due to 3{o and SE is the sum of squares due to error. We also have the simultaneous confidence statements (6) where 11 (a) = ~(a) ~(:) ~(a) ~(a) is any subvector of ~ ~(a) is the corresponding submatrix of ~ and S(a)H o is the corresponding sum of squares due to the partial hypothesis ~(a)o: ~(a) =0 In the case of experimental designs we have (7) CXa = t i + b j if the athobservation belongs to the i = l2 v j = l2...b i-th treatment and j-th block. The hypothesis of equality of treatment effects may be given by: (8) )t 0: (!v-l' -lv-i1h. = 0 where and J - f l } -rs t rxs We shall write J -rxr as J We shall assume that -r the design is connected. Then it is well known [lj that the equations for:!:. are C t = Q ~- - in the usual notation. Also (10) Cov(g) =o~ r.a ~ Then from (3) and (8) ~i =t i - tv' i =l2 v-l. We may express 6 in a symmetrical form by taking ~ =(lv-i' -lv-il)~ where ~i = 1... ti-v(t1+t2+ +tv)' i = 12 v. From (4) then we get

5 4 (11 ) I ~ = 51 (~-;-1 ) Q.-1 (.!v-1' -lv-11)5-1v-l ii) In all experimental situations what we capture under any design and sampling scheme is certain experimental units or individuals. According as just one character or many characters (or variates or responses) are observed on each such unit or individual the problem is said to be a univariate or a multivariate one. Let! denote a matrix of n independent p-dimensional normal variables with a common variance-covariance matrix~ p being the number of characters observed on each individual and let the means be given by (12) where e is a matrix of unknown parameters. C. r'-;'. Xo: B e U = pxn is the "testable" hypothesis to be tested. " c :) B8U= sxu Suppose [Rank U =u 5 pj If we write then the "total" parametric function 6 associated with %0 is given by [23J (15) 6 = c [ ' D-1~] max It may be observed that the characteristic roots of ~1Q.-1~(QI~U)-1 are the noncentrality parameters in the power function of the test (no matter whleh one of the standard tests we use) of the total hypothesis given by (13) I~ (16) t The confidence statement is given by..!.. J...!. C (5 ) - [-!- C ]2 c~ (SE) < ~2 < max -Ho n-r Cl max Ct (SH) + [-!- c ]~ c t m (SE) max - 0 n-r Cl ax -

6 .. 5 where H o and E are the sum of products matrices due to the hypothesis and error respectively and C is the looa% significance point of the a distribution of the largest characteristic root with d.f. u s and n- r. In this case we have simultaneous confidence statements similar to (6) given by (17) Ct [S ] _ [s c]t ct (S) <fj < max -(a)ho ~ a max -E - (a) ct [S ] + [~C ]t Ct (S) max -(a)ho n-r a max -E where 6(a) =cmax[~~~(~)~(a)] 1 ~(a) being a submatrix of ~ obtained by choosing some rows of~. In addition we have by dr~pping some columns of ~simultaneous confidence statements given by (18) ct (5 ) [s c J~ ct (S ) < dz < max -(b)ho - n:r a max -(b)e - (b) - ct (5 ) + [..L c ]t ct (S ) max -(b)ho n-r a max -(b)e where 6(b) =cmax[~(b)e-'~(b)] ; ~(b) being a submatrix of ~ obtained by choosing some columns'of ~ : (b)h o and 2(b)E are the corresponding submatrices of SHo and SE. where ~k) These two operations may be combined. In the case of experimental designs we have i =12 v j =12 b k = 12 p denotes the k-th character measured on the a-th experimental unit or individual that turns up for the i-th treatment and the j-th block; and ti k ) b;k) stand respectively for the contributions to the expectation of the k-th variate made by the i-th treatment and the j-th block. It may be observed from (1) and (12) that we have the same "structure

7 6 matrix" ~ in the multivariate situation. This "structure matrix" depends on the design as well as on what the experimental statisticians have been calling the mode1 e.g. (7) and (19). - - In this set-up so far as the hypothesis is concerned we shall take U =I for simplicity. 3. Balanced incomplete block designs i) Here C = r I _.![(r - /..) I + A3 ] = 6Y I - ~ 3 - -v k -v -v k-v k-v Imposing the usual condition J t = 0 to get unique solutions we have -v - Therefore and hence t=ji.q. - II.V- cp =~ (I -3 )Q - I\V -v-l -v-i! - (20) whence Thus D- 1 = 2'-V(1 _.! 3 ) - k -v-l v -v-1 6 = ~'! cpt (I -.! 3 )cp k - -v-l v -v-l - =~ ~t (1 _.! 3 )~ k_ -v v-v_ (from (11) ) = 6.'! k ~f~ v = L l;~ i=l Then we can have a confidence statement (5) with s =v-i and 6 given by (22). (since 3 ~ = 0) -v_ n= bk r =b+ v-i

8 7 For the "partial" statements (6) if we have then <p'() = (<P. <P. <P. ) (t < v -1) - a It (from (20).) Hence Then For a symmetrical form of expression we take ~(a) = (It' -ltl)s(a) where l;. (") =t i -t~l [to +t. + +t. +t ] -l j a j V t so that A AV t::' (1 1 J )t:: U(a) - 1C ~(a) =t+l - t+i -t+l ~(a) (since ~t+1~(a) = 0) ii) In the multivariate situation we shall have the confidence bounds (16) with n =bk r = b + v-i s =v-i and A = C [<PI~. {I -.!. J )<1>] max - k -v-l v -v-l - Here again we may write ~ =(!v-l' -~V-ll)~ where l; = (~(I) S(p» v and l;~j) = t~j) _.!. \' t$j). Then A = ~ C (l;'l;). 1 1 V L "V k max - - t=l

9 8 W'e shall have one set of "partial ll statements given by (17) with I:::. AV C [(pi (1 1 J1 ) (from (24) ) (a) - k max -(a) -t - t+l -J-(a)' or (equivalently) = ~ Cmax[~(a)~(a)] (from (25) ) where (18) with Similarly we shall have another set of lipartial ll statements given by It. = C ["" ~ (1 -! J )'" ] (f (21» L.\(b ) max ~(b) 'k -v-l v -v-l ~(b) rom - 'A.v C (t: l l; ) - k max ~(b)_(b) As mentioned earlier we may have licombined partial" statements. 4. Partially balanced incomplete block designs i) Let us consider a PBIBD with m associate classes and association matrices B. (i = Ol m). Then it is well known [1] that -1 m = L \: ~k' where ~o =.!v ' k=o 1 = 12 m ; - I \' and on imposing the condition ~lv i = on (9) we have m i = "( L etc ~k)9 k=o = 9 say. It is well known that when the design is connected Rank = v-i so that the condition We have further J l t = is sufficient to give unique solutions. - v-

10 9 (26) and Let J C = 0 -Ivand J 1 Q =0 - v- = (C~ C) =.~-" - - d' - = (~~! ) = ~." ft - Then But 0 =- J 1 1 Q in view of (26). -V - v- - Therefore i = ~g -! ~1v-1 9 = ( ~ = (~; -!.llv-l :! ) [ ~lj - e J )Q - -lv-l - Hence whenc e t ~ J-' (_Et -ej 1I :x) - - v lv-i. - J -v -Iv Thus E' C - e J C + x J l =I lv..l v...v But (26) > ~1v-1 =- ' - I... ' Hence that is (28) _ E' _C + e C' =I - x J - - -v - -Iv Ee=! -xj - - -v - ~lv

11 10 Also _x) = 1 -v Hence C l x =o. But C l J 1 =0 and Rank C l =v-i Therefore tv - x =x J 1 Furthermore J l x =1 whence x J l J 1 =1 that is - -v -v- -v-v _ 1 x - - v (28) thus reduces to _ 1 EC-I --J -v v-v Now 1\ (I -J }t = (1 -J )E Q -v-l -v-ll - -v-l -v-ll - - o = (1 -J )E C E ~.!V-l J - -v-l -v-ll J -lv-l Therefore o = (!v-l' -J ) (1 _! J ) E ~.!V-l J (from (29) ) -v-ll -tv v -v --J -lv-l =(lv-i' =(lv-i' -J )(1 _! J )(E~ - e J ) -v-i1 -v v -tv - - -lv-l -J HE; - ej) (from (27) ) -v-ll - - -lv-l (30) =Ell - f J - J f' + J - - -lv-l -v-ll - eo -v-l Furthermore premultiplying by (lv-i' -lv-ll) both sides of the equation we have tn) Hence - -lv-l V -v-ll - = 1 f' 1 J J 1 -v - eo 1v-I: -v -1v-l Ell - f J! J 11 d Q 11 =!v-l and therefore

12 (32) 6 =~I 11 ~ =~I S (from (11) ). r (k-l) Here we may note that C ii =ao =~k-- d C ~t an ij - at - - k if 11 i-th and j-th treatments are t-th associates. Then we can have a confidence statement (5) with n =bk r =b + v-i s =v - land 6 given by (32). The "partial" statements (6) however cannot be made in a compact form unless we know the association scheme. For example if we have ~(a) = (~' ~2' ~t)' then where 11 = [~ ~] V~-l D-(') = X _ VW- 1 Z... a - -- and thus t v-t-l 6( ) =~(I ) [X - vw- 1 Z]<D( ) a -a a ii) In the multivariate situation we shall have the confidence bounds (16) with n = bk r = b + v-i s =v-i and 6 = C [~I C 1 ~] = C [~I C ~] max max _ -_ We shall have as before one set of "partial" statements given by (17) with 6( ) = C [~(I )[x _ YW-1Z]~( )] a max - a a The other set of "partial" statements given by (18) will be with As before we may have the "combined partial" statements.

13 '.. ill General "connected" incomplete block designs It is well known [1] that in general we have 21 =9 which on imposing the condition ~lv 1 =0 yields 1 = g. Then arguing as before from (26) to (32) we have!1 =<1)' C11 ~ =~' C ~ Then we can have a confidence statement (5) with v n = L: r i i=l b = L: k j j=l r =b + v _ 1 s =v-i and!1 given by.(33). We can have "partial" statements and confidence bounds in the multivariate situation analogous to those for PBlBD. Acknowledgment I am indebted to Professor S.N. Roy for suggesting this problem and for suggesting improvements. References [1] R.C. Bose "Least square aspects of analysis of variance" (mimeographed notes) University of North Carolina [2] S.N. Roy "Some aspects of multivariate analysis" John Wiley and Sons New York (3] S.N. Roy and J. Roy "Analysis of variance with univariate or multivariate fixed or mixed classical model s" Institute of Statistics University of North Carolina Mimeograph Series No