Bias-reduced and separation-proof GEE with small or sparse longitudinal binary data

Generalized estimating equation (GEE) is a popular approach for analyzing correlated binary data. However, the problems of separation in GEE are still unknown. The separation created by a covariate often occurs in small correlated binary data and even in large data with rare outcome and/or high intra-cluster correlation and a number of influential covariates. This paper investigated the consequences of separation in GEE and addressed them by introducing a penalized GEE, termed as PGEE. The PGEE is obtained by adding Firth-type penalty term, which was originally proposed for generalized linear model score equation, to standard GEE and shown to achieve convergence and provide finite estimate of the regression coefficient in the presence of separation, which are not often possible in GEE. Further, a small-sample bias correction to the sandwich covariance estimator of the PGEE estimator is suggested. Simulations also showed that the GEE failed to achieve convergence and/or provided infinitely large estimate of the regression coefficient in the presence of complete or quasi-complete separation, whereas the PGEE showed significant improvement by achieving convergence and providing finite estimate. Even in the presence of near-to-separation, the PGEE also showed superior properties over the GEE. Furthermore, the bias-corrected sandwich estimator for the PGEE estimator showed substantial improvement over the standard sandwich estimator by reducing bias in estimating type I error rate. An illustration using real data also supported the findings of simulation. The PGEE with bias-corrected sandwich covariance estimator is recommended to use for small-to-moderate size sample (N ≤ 50) and even can be used for large sample if there is any evidence of separation or near-to-separation.