36 ALALOUF +41-司611-司+22-61(2- C1 re c is an integer less than c,M=2 Ni,M2 j=1 c1 ,3X:)21=212·Tisp11t8 the continge如 j=1 table into 2 subtables of c and c-c]colums,respectively. Each of the first two terms tests for the equality of 's within the corresponding subtable;and the last two terms toge- ther test for equality between subtable averages. APPENDIX We show here that x =(x,...,x)'as defined by (2.2)and (2.1),has a limit normal distribution.Our proof follows close- ly that of Lehmann (1975,p.352)for convergence to normality in finite population sampling.Let m'=(mf,...,me)be an rc- component fixed vector,with m=(m),1,....c. will show that m'x=mx,is asymptotically normal pr j=1 J ded m lies in a certain vector space to be characterized later. Note that 点点 where the xi's are the contingency table entries. Let U,...Uy be independent random variables,each distri- buted uniformly on (0,1).Define the sets of integers j-1 B与nl2。。<n≤乏。g3,1…,c (A.1) 8=0 =0 and the intervals 入Ko j-1 A=x18P。<x≤。Pg,j1,,c(a.2) s=0 8=0ALALOUF . - - -. . - - . . firkJ-'NNUlX rr &e -L--- LA-,. ,.L-4. -- = ;-*: -*:-, 7 Z,L~%JW iic~z LLL~L A\A7 j 0 -;A , as def tne2 by (2 -2) a.'d - (2.1). has a limlt normal distribution. Our pmsf follows close- - .. . -z '7 ..----- f7C7C - - . . '2L.I: -I___T_I_____.-,-. *- ?iL__ -, LLIGL V1 LTIII:ICIII: jl>: J p J-'L/ bClL LIIIIVCL'CIIZLG 0 LLJ LIULIII-IILJ $2 finite nnaulzt'ln -r snx~licn. . .2 let m'= (mi5.1.1~') he ?n rs- ' c component fixed ~ector. with m: = (rnr,9+,.,n~..2); j=i 3-.. >c. 3 IJ Lj We will show that m'x = m!x is asymptotically normal provi- j=l J j ded m lies in a certain vector space to be characterized later. nr mte that where the xiiTs are the contingency table entries. - J Let TuT l,...EN be independent random variables, each distri￾buted uniformly on (O,l). Define the sets of integers Downloaded by [China Science & Technology University] at 17:36 14 September 2015