Cell counts y ij in an I J contingency table have a multinomial (n ij ) distribution Show that P(Y ij n ij ), i 1, , I, j 1, , J can be expressed as where ij ij IJ IJ IJ and d is a constant independent of the data Find an alternative expression using local odds ratios ij , by showing that 1 1 J 1 d n (n ')' ep nlog(,) i 1 j 1 1 I 1 J 1 n4 og( ) n log( ) j log( u) i 1 j 1

Question: Cell counts {y ij } in an I J contingency table have a multinomial (n; { ij }) distribution. Show that {P(Y ij =

Cell counts {y_ij} in an I × J contingency table have a multinomial (n; {π_ij}) distribution. Show that {P(Y_ij = n_ij), i = 1,..., I, j = 1,..., J} can be expressed as

1-1 J-1 d" n! (n;')' ep| nlog(,) i=1 j=1 -1 I-1 J-1

+ n4 og( / ) + n.log( / ) j log( /

where α_ij = π_ij π_IJ/π_IJ π_IJ and d is a constant independent of the data. Find an alternative expression using local odds ratios {θ_ij}, by showing that

u) i=1 j=1

1-1 J-1 d" n! (n;')' ep| nlog(,) i=1 j=1 -1 I-1 J-1 + n4 og( / ) + n.log( / ) j log( / u) i=1 j=1

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