Consider the third of the three sampling schemes for a 2 × 2 × K table discussed in Section 4.8, and the two hypotheses H1 : 1 =···= K = 1 and H2 : 1 =···= K .

(i) Obtain the likelihood ratio test statistic for testing H1.

(ii) Obtain equations that determine the maximum likelihood estimates of the parameters under H2. (These equations cannot be solved explicitly.)

(iii) Determine the number of degrees of freedom of the limiting χ2-distribution of the likelihood ratio test for testing

(a) H1,

(b) H2.

[For a discussion of these and related hypotheses, see for example Shaffer (1973), Plackett (1981), or Bishop, Fienberg, and Holland (1975), and the recent study by Liang and Self (1985).]