Efficiency of generalized estimating equations for binary responses

Using standard correlation bounds, we show that in generalized estimation equations (GEEs) the so-called `working correlation matrix'R(alpha) for analysing binary data cannot in general be the true correlation matrix of the data. Methods for estimating the correlation param-eter in current GEE software for binary responses disregard these bounds. To show that the GEE applied on binary data has high efficiency, we use a multivariate binary model so that the covariance matrix from estimating equation theory can be compared with the inverse Fisher information matrix. But R(alpha) should be viewed as the weight matrix, and it should not be confused with the correlation matrix of the binary responses. We also do a comparison with more general weighted estimating equations by using a matrix Cauchy-Schwarz inequality. Our analysis leads to simple rules for the choice of alpha in an exchangeable or autoregressive AR(1) weight matrix R(alpha), based on the strength of dependence between the binary variables. An example is given to illustrate the assessment of dependence and choice of alpha.