Analyzing Matched 2 × 2 Tables from all Corners

Squared 2 × 2 tables with binary data from matched pairs are typically analysed using Cochran-Mantel-Haenszel methodology, conditional logistic regression, or random intercepts logistic regression. These are all "pair-specific" type of approaches. However, many more methods and models for clustered binary data, including marginal models and marginalisable pair-specific models, can be applied. We provide a comprehensive overview of methods and apply them all to two well-known example datasets, the prime minister's performance and the myocardial infarction datasets. The simple setting of matched binary data allows us to compare and relate different models, methods and their estimates. A technical explanation is given for why in some settings boundary estimates are obtained.