Can we recover information from concordant pairs in binary matched pairs?

When possible values of a response variable are limited, distributional assumptions about random effects may not be checkable. This may cause a distribution-robust estimator, such as the conditional maximum likelihood estimator to be recommended; however, it does not utilize all the information in the data. We show how, with binary matched pairs, the hierarchical likelihood can be used to recover information from concordant pairs, giving an improvement over the conditional maximum likelihood estimator without losing distribution-robustness.