Maximum likelihood regression methods for paired binary data

We discuss maximum likelihood methods for analysing binary responses measured at two times, such as in a cross-over design. We construct a 2 x 2 table for each individual with cell probabilities corresponding to the cross-classification of the responses at the two times; the underlying likelihood for each individual is multinomial with four cells. The three dimensional parameter space of the multinomial distribution is completely specified by the two marginal probabilities of success of the 2 x 2 table and an association parameter between the binary responses at the two times. We examine a logistic model for the marginal probabilities of the 2 x 2 table for individual i; the association parameters we consider are either the correlation coefficient, the odds ratio or the relative risk. Simulations show that the parameter estimates for the logistic regression model for the marginal probabilities are not very sensitive to the parameters used to describe the association between the binary responses at the two times. Thus, we suggest choosing the measure of association for ease of interpretation.