Testing equality of proportions with incomplete correlated data (Q1059344)

Let \((\psi_ i,\phi_ i)\) be independent, identically distributed pairs of zero-one random variables with (possible) dependence of \(\psi_ i\) and \(\phi_ i\) within the pair. For n pairs, both variables are observed, but for \(m_ 1\) additional pairs only \(\psi_ i\) is observed and for \(m_ 2\) others \(\phi_ i\) is observed. If \(\pi_{1\cdot}=P\{\psi_ i=1\}\) and \(\pi_{\cdot 1}=P\{\phi_ i=1\}\), the problem is to test \(\pi_{1\cdot}=\pi_{\cdot 1}.\) Maximum likelihood estimates of \(\pi_{1\cdot}\) and \(\pi_{\cdot 1}\) are obtained via the EM algorithm. A test statistic is developed whose null distribution is asymptotically chi-square with one degree of freedom (as n and either \(m_ 1\) or \(m_ 2\) tend to infinity). If \(m_ 1=m_ 2=0\) the statistic reduces to that of \textit{Q. McNemar}'s test [Psychometrika 12, 153-157 (1947)]; if \(n=0\), it is equivalent to the statistic for testing equality of two independent proportions. This test is compared with other tests by means of Pitman efficiency. Examples are presented.