On the conditional and mixture model approaches for matched pairs

Matched-pair studies are often used in various applications such as in medical and biological research. In some experiments, blocks of size 2 exist naturally, examples include two eyes or two feet of a person. In these studies, the conditions within each blocks are assumed to be identical but differ across blocks. When the latter variation is fully accommodated, a large number of nuisance parameters have to be introduced. In the case of binary response, both conditional and mixture model approaches can be used to eliminate these nuisance parameters. Under the mixture model assumption, an explicit expression for the maximum likelihood estimator of the structure parameter is found in various situations. Its limiting distributions are studied in details. It reveals that, under the logistic link, the mixture model approach is asymptotically more efficient than the conditional approach. Under the identity link and the log link, mixture or non-mixture model approaches produce the same estimate. Hence, using the mixture model approach does not lose efficiency when the non-mixture model is valid. On the other hand, it protects us from certain model departures. New and more efficient testing methods are also proposed based on new discoveries.