For the binary matched-pairs model (9.5), consider a strictly fixed effects approach, replacing ????0 + ui in the model by ????0i. Assume independence of responses between and within subjects.

a. Show that the joint probability mass function is proportional to exp[

∑n i=1

????0i(yi1 + yi2) + ????1

(

∑n i=1 yi2

)] .

b. To eliminate {????0i}, explain why we can condition on {si = yi1 + yi2}

(Recall Section 5.3.4). Find the conditional distribution.

c. Let {nab} denote the counts for the four possible sequences, as in Table 9.1.

For subjects having si = 1, explain why ∑

i yi1 = n12 and ∑

i yi2 = n21 and

∑

i si = n∗ = n12 + n21. Explain why the conditional distribution of n21 is bin(n∗, exp(????1)∕[1 + exp(????1)]). Show that the conditional ML estimator is

????̂

1 = log (n21 n12 )

, with SE =

√ 1 n21

+

1 n12

d. For testing marginal homogeneity, the binomial parameter equals 1 2 .
Explain why the normal approximation to the binomial yields the test statistic z = n21 − n12 √n12 + n21 .
(The chi-squared test using z2 is referred to as McNemar’s test. Note that pairs in which yi1 = yi2 are irrelevant to inference about ????1.)