30 ALALOUF X-克82 (1.1) 1=jj=1+N1N is a chi square uder either one of the following models:(1)N is fixed and the xij's have a joint multinomial distribution;(2) the column totals N1,..,Nc are fixed and for j=1,...,c,the random vectors (x1j)are independent multinomials. Specifically,let (pij,i=1,....r,j=1,..c)denote the cell probabilities under the multinomial model (1)Pi+= P1jr1F1,,r and p+.对？P1j寸s1,,c.The nu11 1 hypothesis is that Pij=Pit P.and under this hypothesis,the joint probability function of the cell frequencies is ,PP+产 1349:43 (1.2) Under the product multinomial model(2),the null hypothesis is that each of the c multinomials has the same probability vector,(P1+,...,P)say.Under this hypothesis and model,the joint probability function of the cell frequencies is ) 州 19 (1,3) KojouyaL Finally,it may happen that both sets of margins are fixed. One example of where this might occur in practice is the follo- wing.Suppose N individuals belonging to c classes (ethnic groups,for example)are hired for N summer jobs which fall into r categories.Both the number of individuals of each class and the number of jobs of each category are fixed.The hypothesis of independence in this context is the hypothesis that individuals are assigned to different job categories without regard to their papeojuMoALALOUF is a chi square under eliher one of rhe foilowing models: (lj ?.! is fixed and the x 's have a joint multinomial distribution; (2) ij *L. =- - - LL~C COI;C~~FI~~ iotais Ti, 9.. j[* p_;:<p_& fOr j z i- ? cj t-~ i C random vectors (xlj,. . . ,X .) are independent multinomials. rj Specifically, iet (p.. i = 1, . . . , r, j = 1, . ..: c} denote 1J ' the cell probabilities under the multi~oxial iiiodel p,, = A, Under the product multinomial mode:! 12), the nldl hvcotlrlesis . - is that each of the c multinomials has the same probability vector, (pl+, . . . ,p*) say. IJnder this hypothesis and model, the joint probability function of the cell frequencies is Finally, it may happen that - beth sets of margins are fixed. One example of where this might occur in practice is the follo￾wing. Suppose N individuals belonging to c classes (ethnic groups, for example) are hired for N summer jobs which fall into r categories. Both the number of individuals of each class and th2 number of jobs of each category are fixed. The hypothesis of independence in this context is the hypothesis that individuals are assigned to different job categories without regard to their Downloaded by [China Science & Technology University] at 17:36 14 September 2015