Robust covariance estimation in mixed-effects meta-regression models

Abstract

In this PhD thesis we consider robust (sandwich) variance-covariance matrix estimators in the context of univariate and multivariate meta-analysis and meta-regression. The underlying model is the classical mixed-effects meta-regression model. Our goal is to enable valid statistical inference for the model coefficients. Specifically, we employ heteroscedasticity consistent (HC) and cluster-robust (CR) sandwich estimators in the univariate and multivariate setting. A key aim is to provide better small sample solutions for meta-analytic research and application. Tests based on the original formulations of these estimators are known to produce highly liberal results, especially when the number of studies is small. We therefore transfer results for improved sandwich estimation by Cribari-Neto and Zarkos (2004) to the meta-analytic context. We prove the asymptotic equivalence of HC estimators and compare them with commonly suggested techniques such as the Knapp-Hartung (KH) method or standard plugin covariance matrix estimation in extensive simulation studies. The new versions of HC estimators considerably outperform their older counterparts, especially in small samples, achieving comparable results to the KH method. In a slight excursion, we focus on constructing confidence regions for (Pearson) correlation coefficients as the main effect of interest in a random-effects meta-analysis. We develop a beta-distribution model for generating data in our simulations in addition to the commonly used truncated normal distribution model. We utilize different variance estimation approaches such as HC estimators, the KH method and a wild bootstrap approach in combination with the Fisher-z transformation and an integral z-to-r back-transformation to construct confidence regions. In simulation studies, our novel proposals improve coverage over the Hedges-Olkin-Vevea-z approach and Hunter-Schmidt approaches, enabling reliable inference for a greater range of true correlations. Finally, we extend our results for the HC estimators to construct CR sandwich estimators for multivariate meta-regression. The aim is to achieve valid inference for the model coefficients, based on Wald-type statistics, even in small samples. Our simulations show that previously suggested CR estimators such as the bias reduced linearization approach, can have unsatisfactory small sample performance for bivariate meta-regression. Furthermore, they show that the Hotelling’s T^2-test suggested by Tipton and Pustejovsky (2015) can yield negative estimates for the degrees of freedom when the number of studies is small. We suggest an adjustment to the classical F -test, truncating the denominator degrees of freedom at two. Our CR extensions, using only the diagonal elements of the hat matrix to adjust residuals, improve coverage considerably in small samples. We focus on the bivariate case in our simulations, but the discussed approaches can also be applied more generally. We analyze both small and large sample behavior of all considered tests / confidence regions in extensive simulation studies. Furthermore, we apply the discussed approaches in real life datasets from psychometric and medical research.