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THE IJTNI VERSITY 0 01 963 SC/) APPLICATIONS OF THE FACTORIAL CALCULUS TO GENERAL UNEQUAL NUMBERS ANALYSES W. T. Fedierer and M. Zolon MRC Technical Summary Rcport 11393 MATHEMATICS RESEARCH CENTER
MATHEMATICS RESEARCH CENTER, UNITED STATES ARMY THE UNIVERSITY OF WISCONSIN Contract No.: DA-11-022-ORD-2059 APPLICATIONS OF THE FACTORIAL CALCULUS TO GENERAL UNEQUAL NUMBERS ANALYSES W. T. Federer and M. Zelen MRC Technical Summary Report #393 April 19 63 Madion, Wiconin
ABS TRACT The neceary element of the calculu for factorial a developed by'kurkjian and Zelen are decribed, modified where neceary, and applied to the analyi of unbalanced n-way claification with fixed effect. Etimator for all main effect and interaction effect parameter are obtained along with the aociated variance. The um of quare for each effect eliminating all other effect i preented in a form uitable for direct computation. Thi form reult in coniderable computational aving over the method of fitting contant ued in general regreion theory. The reult are applied to the particular cae of proportional frequencie in the ubclae.
APPLICATIONS OF THE FACTORIAL CALCULUS TO GENERAL UNEQUAL NUMBERS ANALYSES/I W. T. Federer 'a nd M. Zelen/3 1. Introduction Thi paper i the econd in a erie of paper which applie the calculu for factorial arrangement developed by Kurkjian and Zelen [1962] to variou problem in the analyi of experiment deign. The firt paper dealing with application [1963] wa devoted to the analyi of block and direct product deign. The main object of thi paper i to apply thi pecial calculu to an alternate way of treating the analyi of variance with unequal number. The uual way thi i done i by the method of fitting contant; cf Federer [1957], Yate [1934]. The ue of the pecial method developed here lead to ubtantial computational aving over the method of fitting contant. Section two of thi paper contain the neceary part of the factorial calculu which i the tarting point of our invetigation. Section three develop the general theory for unequal number and ection four how the reulting implification when the frequencie are proportional. Sponored by the Mathematic Reearch Center, United State Army, Madion, Wiconin under Contract No.: DA-ll-022-ORD-2059. 8 On abbatic leave from Cornell Univerity. SMathematic Reearch Center, U. S. Army.
-2- #393 2. Element of the Factorial Calculu The notation and pecial operation ued in thi paper will be a modified verion of the calculu for factorial arrangement introduced in Kurkjian and Zelen [196Z]. The modification are traightforward generalization which are ueful in treating the cae of unequal number. Conider a factorial experiment with the n factor {A} uch that factor A ha m level for = 1, 2,..., n. The ith treatment combination conit of the n-tuple i = (il, y 2,... i ) where i denote a particular n level from factor A. The number of treatment combination i v = II m S =l S Let Y denote a v x 1 random vector following a multivariate normal ditribution with (Z.1) E(Y) 1i + t 2-1 (2.Z) V(Y) = c N The quantity 1 i a v x vector having unity element, ý± i a calar, t i a v x 1 vector of (fixed) treatment effect; and the matrix N i a v x v diagonal matrix having only non-zero diagonal element n. which denote the number of 1 th obervation on the i treatment. The element of t are not linearly independent, but atify a ingle linear retraint which will be decribed later. Alo define {a}s = i1 2,..., n, to denote vector, termed primitive element, uch that a' [a(l), a(?),. a(m)]
#393-3- New element can be formed with the operation of the ymbolic direct product (SDP) which i denoted by The SDP between a P and a g i defined to be (2.3) [ap a %] --[ap (1,1), ap (1, 2),..., ap (lmg, [ap q [apg( pq pq 1 apq(2,1), a pq..., apq (mp apq pg (m mq)] Note that the ubcript refer to the primitive element involved and the argument i a vector of two element which are ordered lexographically. The lexographical order i to hold the firt element of the argument fixed at 1 and run through the level 1, 2,..., mq of Aq; then change the firt argument to level 2 of Ap and run through the level of A ; etc. The SDP i alo defined for more than two primitive element in the ame way; i.e., a PD aq () ar, etc. The element of a P D a q denote the vector whoe element are the parameter aociated with the two factor interaction between factor A and Ag; the element of a a denote the vector aociated with the three factor interaction among factor Ap, Aq, and A, etc.' Let x be a variable which take on the value 0 or 1. We define a for x 1 x a a 1 for x =0 and ue the convention that a 1= a. Then, if x =(x x., "'", x) n a
-4- #393 generalized interaction may be denoted by x ax1 xn 21a and will have m = IInm component. i=l a xi x 2 axn ai 1 0 (x Z '" ann The model relating the treatment effect to the interaction parameter can be written by defining I for x =I SS 1,for x = 0; x I x x S1I 1xIXZ 12 X... XI n where I i m xm identity matrix and 1 i an m x 1 column vector having all element equal to unity. Then we can write (2.4) t ix ax x where x Xz. '(x Xn) and the ummation refer to all the Z2 1 x n-digit binary number x excluding x = (0, 0, 0... 0). The component of t are taken in the ame lexographical ordering a the n-factor interaction. x The interaction parameter a are not linearly independent. Let u be a th 1 xn vector with all element equal to zero except for the element which i equal to unity and define r by
#393-5- I I I vr =(iix 1 X... X 1) N(I 1 X1i 2 X... X 1n) We then define the m x m diagonal matrix 5 S m u u (2.5) W : - (I S) 2,, n S rv The element of W 5 are proportional to the number of time the variou level of factor A appear. Then a convenient et of retraint among the parameter may be taken to be W a 0 p =1 2,... n; p p p p q [W XW [a a =0,, p0 q =1, Z,... n SI p X 1l qq. (2.6) 1 xi:x... xi 2 n " 2 ' n [W1... Wq [a a.(. I XI X... X1 'With thee retraint, "it can ea.ly be hown that (Z. 7) [ 1 x 1 x... X n 1 w] 20n x... XW nt
-6- #393 We hall find it convenient to ue the following notational convention. Let Z ( = 1, Z, ri) be matrice and x = (x X 2... I Xn) be an n-digit binary number. Then, we hall alway write Z and Z(x) to denote SxI x z xn S Zx =ZI Xz2 X. XZnn (2.8) Z(x) =Z I X Z 2 X... XZ for thoe x =x... x P When the Z are calar quantitie (ay) Z : z then 5.5 x (2.9) z = z(x) 3. The General Theory of Unequal Number In thi ection we. hall develop the general theory for unequal number. Our procedure will be to firt find etimate for the variou interaction parameter and their"variance and then derive the aociated um of quare for ue in the analyi of variance. The etimation of the variou interaction contitute no real problem; the difficult problem i to determine the appropriate um of quare. Let W = W X W X.. X W ; then it can eaily be hown that the etimable n function for t. may be etimated from 1 (3.1) t= [I1-,W ]Y = I v ]y where 1 i a v x I column vector having all element unity, " = 1 1 ; and I i the v x v identity matrix.
#393-7- Conequently we have (3.2) var 2 [1 1 - Define the m x m matrix M by S S S. (3. 3) M S W S]W1 Alo let. 4 *.M for X I= { S M= 1 for x =0 (3.4) S S. "-x x I x2z M M XM X... X M 1 2 n x We note that M may alo be written a xx Mx = M(x) (IX)' where M(x)=M XM X... XM and x =x... x =1 I.' P i S with the remaining x = O; i.e. M(x) i the direct product of thoe M for which x = 1. We alo record for reference M M W =m I -I W (3.5) M W a = m a SS S S S M W 1 =0 M W1.S
-8- #393 After ome algebra one can how that (3.6) a wtýv W- and (3.7) var a x [MX WN- w(mx)) 2 /vi.x Note that the var a may be written a ^-1 x 2 z vara =M(x)[(i)' WN WI M(x) IV The variance can be further implified. For thi purpoe define x W if x =1 5 Iifx=O0 = Ix if X 0 S~l Then the quantity [ (IX W] may be written. n x n x -x -x HI)W:I X (I ), WS = W(x) H X (I W W(x)(I:)w- S=l The diagonal matrix [(Ix) W N. 1 WIx] can alo be written S=l Wx N- WIx W(x) R(x) W(x) where (3.8) R(x) = (IX)' Wl-x N-1 Wl-x Ix
#393-9- Hence the variance of a i 22z (3.9) var a = [M(x)W(x)R(x)W(x) M(x)](r /v Note that R(x) and W(x) are diagonal matrice. It remain to find the um of quare aociated with a. We hall find thi making ue of a reult on quadratic form recently given by Rao [1962] Lemma. Let X = (X I X 2,... I Xn) have a ingular multivariate normal ditribution with E(X) = 0 and varx 2 Z where the rank of Z i f (f< n) 2 Then a neceary and ufficient condition for the quadratic form X'S X/0- have a chi-quare ditribution with f degree of freedom i that S be a real ymmetric n x n matrix having the propertie that to (i) S -- S (ii) 3=3 We now turn our attention to finding the matrix S of the above lemma when 23 M(x) W(x) R(x) W(x) M(x). We point out that the degree of freedom Xn x aociated with a i f(x) = fi(m-l) (rank of var a x). Hence there will S=l 1 exit at leat r(x) = m(x) - f(x) linearly independent non-etimable function of a. Thee non-etimable function will be denoted by (3.10) K'(x) W(x) ax where K'(x) i r(x) x m(x), ha rank r(x), and W(x) denote the direct product of thoe W for which x = 1
-10- #393 Define the m(x) x m(x) matrix V(x) by W(x) M(x) R- (x) K(x) V(x) K(x) U (x) (3.11) K'(x) R-1(x) 0 U(x) K'(x) 0 where (3.12) U(x) = [K'(x) R- (x) K(x)]-l It can be verified that the matrix S of the lemma i (3.13) S = v V(x) R-1 (x) V(x) Therefore, the required um of quare i (3.14) vz (ax)' {V(x) R- 1 (x) V(x)}(NX) The above um of quare till require knowledge of the matrix V(x) which i not known explicitely. Uing (3. 5), we can write and therefore m a =M W a S S S S S (3.15) m(x) ax = M(x) W(x) ax Subtituting (3.15) in (3.14) reult in v 2 -x W x- Mxx -) (a ) WxMxVxR(xVxMxWxl(
#393-11- Uing the relation of a matrix to it invere give W(x) M(x) V(x) + R- (x) K(x) U(x) K'(x) -I K'(x) R- (x) V(x) = O. Conequently, W(x) M(x) V(x) R- (x) V(x) M(x)W(x) = R- (x) [I - K(x) U(x) K'(x) R- (x)] v x{ 1} (3.16) ( m -- x) which follow from the fact that the matrix in quare bracket in idempotent. Therefore the um of quare can be written a )a() R- (x) [I - K(x) U(x)K'(x) R-(x)] } (x) Since R(x) i a diagonal matrix, the main computational labor i in computing the r(x) x r(x) invere matrix -1-1 U(x) = [K'(x) R (x) K(x)] The um of quare for main effect may be written explicitely a r(x) = 1 In thi cae for (ay) a K'(x) =, W(x) = W R 5 =R(x) =(W X1WX...XIx... X1 W)N (W1XWIX...xI x...xw 1 2 2 n n 1 n n u R- U:(
-12- #393 The um of quare for the main effect aociated with A (3.17) (-) a {R [I U ] i thu 4. Proportional Frequencie A i well known, the cae of proportional frequencie turn out to be particularly imple. The cae of proportional frequencie arie when N=NIXN X... X N n =H1 XN =l where the N are uch N i the direct product of all N I Define n = 1 N I ; then the W quantitie are S 5 S S 5 and Conequently we have m W ==- N n nn w=n xw =_ =l S r x' - x 1 )NI =v Nx W(x)R(x)W(x) =(Ix) WN- WI x _ N(x) r rn(x) and var ; -x 2 [MI NI(MI ) XM 2 N (M 2 ) X... XMn N(M r v
#393-13- Note that x S x MS N(MS)= n M forx =1 SS I n for x =0 Therefore the variance can be written a (4.1) var a =- M(x) rv 2 -x a n =rv The um of quare aociated with ax can eaily be written by noting that the econd term in (3.16) i a null matrix. Thi can be demontrated by writing (4.2) R- 1 (x) =M(x) N(x) and thu v n (x) K'(x) R -(x) ax -rm(x) K'(x) N(x) ax 0 v n (x) The um of quare aociated with a i then vx rv - x ( x) (4.3) (m-- ) (ax R-l(xY(a,= a Nx)a n (x)
-14- #393 REFERENCES 1. Federer, \V. T. (1957), Variance and covariance analyi for unbalanced alaification, Biometric 13, 333-62. Z. Kurkjian, B. and Zelen, M. (1962), A calculu for factorial arrangement, Ann. Math. Statit. 3, 600-619. 3. Kurkjian, B. and Zelen, M. (1963), Application of the calculu for factorial arrangement I: Block and direct product deign, Biometrika 50, 1-11. 4. Rao, C.?,. (1962), A note on the generalized invere of a matrix with application to problem in mathematical tatitic. Jour. Roy. Stat. Soc., B, 24, 152-158. 5. Yate, F. (1934), The analyi of multiple claification with unequal number in the different clae, T. Am. Statit. Aoc. 29, 51-66