Fortwocategoricalvariables X and Y , let ij = P(X = i, Y = j), i = 1, ...,r, j = 1, ...,c. Consider H0: ij

Fortwocategoricalvariables X and Y , let πij = P(X = i, Y = j), i = 1, ...,r, j = 1, ...,c.

Consider H0: πij = P(X = i)P(Y = j) for all i and j with n observationshavingcellcounts

{yij}. Usingthemultinomialdistributionfor {yij} and theMLestimatefor πij of ˆπij,0 =

(yi+~n)(y+j~n) under H0 and ˆπij = yij~n under Ha, showthatthelikelihood-ratiostatisticfor testing H0 is

for ˆμij,0 = nˆπij,0. Specifyitslarge-sampledistribution.

2 ., log (3/43,0), 1 J