CHI SQUARE TEST WITH BOTH MARGINS FIXED 33 E(x)=P;” j1,.·, Var(x:)=IN2/(N-1)JP (1-P)V j=1,·,c (2.5) Cov(x)=-[N2/(N-1)V Let x =(x,...,x)'.The expectation and covariance matrix of x are E(x)=NEm=μ Var(x)=[N2/(N-1)[P-MEVE'M]=Z (2.6) where r=diagfP V,....PV) M=diag(P.PeI (2.7) g=(红' The matrix E is rcxr,diag(A],...,A}denotes a block diagonal matrix with blocks A,...,A,and I denotes the identity matrix of order r. As will be proved in the appendix,the vector x is asympto- tically normal.Therefore a quadratic form x'Ax is asymptoti- cally a chi square if AZA A and the chi square will be central ifAμ=0，One such matrix is A=[(N-1)/N2][r-EvE'1 where r--diagipilv,..v) y=D-1-ee' euryo] and e is a column of r one's. It can be easily verified that AEA A and that Au =0. Moreover,Pearson's statistic (1.1)is simply [N/(N-1)]x'Ax
CHI SQUARE TEST WITH BOTH MARGINS FIXED -- n are where The ~atrix E is rcxr, diag{~, , . . . ,R1 den~tes a block diagonal - It matrix with blocks A1,. . . ,A,, and 11- denotes the identity matrix of order r. As will be proved in the appendix, the vector x is asymptotically normal. Therefore a quadratic form x'Ax is asymptotically a chi square if ACA = A and the chi square will be central if A'o = 0, One such matrix fs and e is a colmm of r one's. It call be ea~ILy verified that, AiA = A ai;d zhzz Ap = I?, Moreover, Pearson's statistic (1.1) is sinply r~/(N-l) Ix'Ax, Downloaded by [China Science & Technology University] at 17:36 14 September 2015
34 ALALOUF which is asymptotically equivalent to x'Ax and hence is asympto- tically a chi square.The number of degrees of freedom is tr(AZ) which can be easily shown to be (r-1)(c-1). 3.DISCUSSION As we noted before,(1.4)is the conditional distribution of (1.1)given the marginal totals in either a multinomial model with probability function (1.2)or a product-multinomial model with probability function (1.3).Thus the asymptotic chi square distribution obtained in Section 2 is the asymptotic conditional distribution under the product and product-multi- nomial models.Since the chi square distribution is independent of the marginal totals,this is also the unconditional asymptotic distribution. Thus ihe one proof given here for the case of fixed margi- nals can be easily extended to the other two models. There is a neat way of summarizing the situation:we may think of the frequencies as a single multinomial and study the distribution of (1.1)under constraints on the two sets of margins.In some cases,these constraints are imposed by the experiment itself;in other cases one set of constraints is imposed by the experiment,the other by the statistician as a mathematical device;and in yet other cases,the statistician 入.Ka imposes all constraints.Whatever the case,the result is the same. The splitting of degrees of freedom for testing various hypotheses follows easily.As a simple example,consider a set of independent multinomials with numbers of trials N,...Nc and probability vectors ,...Suppose we want to test simultaneously 0:1=…=万c90 (3.1) where Io is a given probability vector.The usual test
3k ALALOUF -.LJ W~:LLL~ -L is aejmptoticaliy equ<vaient to x'hx and hence is asymptoticaiiy a chi square. The nunber of degrees of freedom is tr(AC) wFirh car? be easily shc::~? tc be (r-1.) <a--1) 3; DISCUSSION . 4% ~- we nptes hrfnre3(l.&) Is ths c-zditis>zl dia~Li~nt~a~E uf (z, 1) given ths margiiial to :als iii either a mulcinoniai nlodel .._I L? wrLn probability function (1.2 j or a prsduct-rmltinomial model 22th probability fu-..ction (1.3). %us the asymptotic chi square distributi-nr? obtaked Secticr, 2 is the as;/rptotic conditional distribution under the product and product-nuitinomial. models. Since the chi sQuare distribution is independent of the marginal totals, this is also the iincondltional asym9totic di str-n~~t jnp. mr- ~ ~ Ailus ille me proof given here for the case of fixed marginal~ can be easily extended fo the other two models. -- mere is a neat way of summarizing the situation: we may -Li-l. ;;;x. ~f the frequencies as a single muitinsmiai and study the distribution of (1.1) under constraints on the two sets of margins. In some cases, these constraints are imposed by the experiment itself; in other cases one set of constraints is imp~sed by the experiment, the other by the statistician as a mathematicai device; 2nd in yet sther cases, the statistician imposes all constraints. Whatever the case, the result is the same. The splitting of degrees of freedom for testing various hypotheses follows easily. As a simple example, consider a set of independent multinomials with numbers of trials N1, ... XI C and probability vectors 5, ..., n . Suppose we want to test - simul taneousiy 7. - no: 11~ = '.. =q =q c 0' (3.1) where ITo is a given probability vector. The usual test Downloaded by [China Science & Technology University] at 17:36 14 September 2015