Choice of Link Functions for Generalized Linear Mixed Models in Meta-Analyses of Proportions

Two-step approaches for synthesizing proportions in a meta-analysis require first transforming the proportions to a scale where their distribution across studies can be approximated by a normal distribution. Commonly used transformations include the log, logit, arcsine, and Freeman-Tukey double-arcsine transformations. Alternatively, a generalized linear mixed model (GLMM) can be fit directly on the data using the exact binomial likelihood. Unlike popular two-step methods, this accounts for uncertainty in the within-study variances without a normal approximation and does not require an ad hoc correction for zero counts. However, GLMMs require choosing a link function; we illustrate how the AIC can be used to choose the best-fitting link when different link functions give different results. We also highlight how misspecification of the link function can introduce bias; using an empirical sandwich estimator for the standard error may not sufficiently avoid undercoverage due to link function misspecification. We demonstrate the application of GLMMs and choice of link function using data from a systematic review on the prevalence of fever in children with COVID-19.