An investigation of penalization and data augmentation to improve convergence of generalized estimating equations for clustered binary outcomes

Abstract

Abstract Background In binary logistic regression data are ‘separable’ if there exists a linear combination of explanatory variables which perfectly predicts the observed outcome, leading to non-existence of some of the maximum likelihood coefficient estimates. A popular solution to obtain finite estimates even with separable data is Firth’s logistic regression (FL), which was originally proposed to reduce the bias in coefficient estimates. The question of convergence becomes more involved when analyzing clustered data as frequently encountered in clinical research, e.g. data collected in several study centers or when individuals contribute multiple observations, using marginal logistic regression models fitted by generalized estimating equations (GEE). From our experience we suspect that separable data are a sufficient, but not a necessary condition for non-convergence of GEE. Thus, we expect that generalizations of approaches that can handle separable uncorrelated data may reduce but not fully remove the non-convergence issues of GEE. Methods We investigate one recently proposed and two new extensions of FL to GEE. With ‘penalized GEE’ the GEE are treated as score equations, i.e. as derivatives of a log-likelihood set to zero, which are then modified as in FL. We introduce two approaches motivated by the equivalence of FL and maximum likelihood estimation with iteratively augmented data. Specifically, we consider fully iterated and single-step versions of this ‘augmented GEE’ approach. We compare the three approaches with respect to convergence behavior, practical applicability and performance using simulated data and a real data example. Results Our simulations indicate that all three extensions of FL to GEE substantially improve convergence compared to ordinary GEE, while showing a similar or even better performance in terms of accuracy of coefficient estimates and predictions. Penalized GEE often slightly outperforms the augmented GEE approaches, but this comes at the cost of a higher burden of implementation. Conclusions When fitting marginal logistic regression models using GEE on sparse data we recommend to apply penalized GEE if one has access to a suitable software implementation and single-step augmented GEE otherwise.

Keywords

• Clustered data
• Firth’s logistic regression
• Generalized estimating equations
• Logistic regression
• Non-convergence
• Separation