Question: Consider a three-way contingency table with probabilities pijk for cell (i, j, k) for i = 1, . . . , I, j = 1,

Consider a three-way contingency table with probabilities pijk for cell (i, j, k) for i = 1, . . . , I, j = 1, . . . , J, and k = 1, . . . , K such that P i P j P k pijk = 1. Suppose a sample of size N is taken and let nijk be the number of observations that fall in cell (i, j, k). Based on the 2 test, H0 is rejected is 2 = P i,j,k (nijkNpijk) 2 Npijk 2 df . Find the degrees of freedom for the 2 test for the following hypotheses:

(i) H0 : pijk = piqjrk (Three-way independence)

(ii) H0 : pijk = piqjk (Independence of the first variable from the other two)

(iii) H0 : pijk = pi|kqj|krk (Conditional independence of the first two variables given the third)

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