Estimating equations for measures of association between repeated binary responses

Moment-based methods for analyzing repeated binary responses using the marginal odds ratio as a measure of association have been proposed by a number of authors. Carey, Zeger, and Diggle (1993, Biometrika 80, 517-526) have recently described how the marginal odds ratio can be estimated using generalized estimating equations (GEE) based on conditional residuals (deviations about conditional expectations). In this paper, we show that other measures of association between pairs of binary responses, e.g., the correlation, can also be estimated using conditional residuals. We demonstrate that the estimator of the correlation based on conditional residuals is nearly efficient when compared with maximum likelihood or second order estimating equations (GEE2) except when the correlation is large. This estimator also yields more efficient estimates of the correlation than the usual GEE estimator that is based on unconditional residuals. Furthermore, the gains in efficiency can be quite considerable when some of the responses are missing or incomplete, or, alternatively, when cluster sizes are unequal (in the clustered data setting).