Approaches to parameter and variance estimation in generalized linear models

In many studies of clustered binary data, it is reasonable to consider models in which both response probability and cluster size are related to unobserved random effects. Two resampling methods have been recently proposed in the literature for mean parameter estimation in this setting: within-cluster resampling (WCR) and within-cluster paired resampling (WCPR). These procedures are believed to provide valid estimates in the presence of nonignorable cluster size. We identify the parameters estimated under WCR and under unweighted generalized estimating equations and elaborate on their differences and validity. We propose a simple weighted generalized estimating equations strategy that is asymptotically equivalent to WCPR but avoids the intensive computation involved in WCPR. We investigate the parameter estimated by WCPR for a generalized mixed model. We show that the parameter estimated by WCPR may be affected by factors other than the actual effects of exposure and propose an alternative strategy for the analysis of correlated binary data with cluster-specific intercepts based on simple generalized estimating equations for random intercept-matched pairs. We study the problem of variance estimation in small samples using robust or sandwich variance estimators. Robust variance estimators are widely used in linear regression with heteroscedastic errors, generalized linear models with possibly misspecified variance model, and generalized estimating equations. In these settings, the robust variance estimator provides asymptotically consistent estimates of the covariance matrix of mean parameters. However, the robust variance estimator may severely underestimate the true variance in studies with small sample size. Bias-corrected versions of the robust variance estimator have been proposed to improve its small sample performance. We introduce a new class of corrected robust variance estimators with an emphasis on variance reduction and small sample performance. These estimators are applicable to linear regression, generalized linear models and generalized estimating equations. We show in simulations that the new estimators perform better in terms of variance and confidence interval coverage than many current estimators, while maintaining comparable average confidence interval width.Doctor of Philosoph