(a) Let (X1,... , Xn) ~ multinomial (m, pi,..., pn). Consider testing H0: p1 = p2 versus H1: p1 ‰  p2. A test that is often used, called McNemar's Test, rejects H0 if

Show that this test statistic has the form (as in Exercise 10.31)

where the XiS are the observed cell frequencies and the expected cell frequencies are the MLEs of mpi, under the assumption that p1 = p2.
(b) McNemar's Test is often used in the following type of problem. Subjects are asked if they agree or disagree with a statement. Then they read some information about the statement and are asked again if they agree or disagree. The numbers of responses in each category are summarized in a 2 x 2 table like this:

The hypothesis H0: p1 = p2 states that the proportion of people who change from agree to disagree is the same as the proportion of people who change from disagree to agree. Another hypothesis that might be tested is that the proportion of those who initially agree and then change is the same as the proportion of those who initially disagree and then change. Express this hypothesis in terms of conditional probabilities and show that it is different from the above H0. (This hypothesis can be tested with a X2 test like those in Exercise 10.31.)

Xi,a (observed-expected)2 expected Afer AgreeAgrore DisagreeXi Agree Disagree Agree Xa After X4