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

Abstract

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.

Keywords: AIC; generalized linear mixed model; link function; meta-analysis; prevalence; proportion.

Figure A.1

95% coverage across 2000 simulations when the median prevalence (π) was 0.05; the left side shows the results for the model-based SE estimator and the right side shows those for the sandwich estimator. Each panel represents the true link function and the x-axis represents the between-study standard deviation ($\tau$).

Figure A.2

Bias across 2000 simulations; the left side shows the results when the median prevalence (π) was 0.3 and the right side shows the results for when π=0.05. Each panel represents the true link function and the x-axis represents the between-study standard deviation ($\tau$).

Figure A.3

Probit, logit, and cloglog functions; vertical lines represent functions evaluated at π = 0.05 (black) and π = 0.3 (blue). The probit and logit functions tend to resemble each other more than the cloglog function, with the exception of when π is close to 0.

Figure A.4

Estimated distributions of study-specific prevalences of fever in children with COVID-19 using a GLMM with different link functions. The estimated distributions of study prevalences under the probit and logit links resemble each other more closely, while that under the cloglog link differs slightly.

Figure A.5

Sample size, number of participants with fever, and estimated prevalence (95% confidence interval) of fever in pediatric studies of COVID-19 symptoms. Overall estimate is from a GLMM with random intercepts for each study and cloglog link and can be interpreted as the estimated median prevalence of fever across the studies.

Figure A.6

Estimated distribution of study-specific prevalences of fever in children with COVID-19 using a GLMM with the cloglog link function. The red band represents the uncertainty in the median prevalence and the blue band represents the prediction interval for a new study prevalence.

Figure 1.

95% coverage across 2000 simulations when the median prevalence (π) was 0.3; the left side shows the results for the model-based SE estimator and the right side shows those for the sandwich estimator. Each panel represents the true link function and the x-axis represents the between-study standard deviation ($\tau$).

Figure 2.

Each bar shows the proportion of the 2000 simulations in which each link function was chosen by AIC when the true median prevalence (π) was 0.3. Each panel represents the true link function and the x-axis represents the between-study standard deviation ($\tau$).