Fixed- and Random-Effects Models

Abstract

Deciding whether to use a fixed-effect model or a random-effects model is a primary decision an analyst must make when combining the results from multiple studies through meta-analysis. Both modeling approaches estimate a single effect size of interest. The fixed-effect meta-analysis assumes that all studies share a single common effect and, as a result, all of the variance in observed effect sizes is attributable to sampling error. The random-effects meta-analysis estimates the mean of a distribution of effects, thus assuming that study effect sizes vary from one study to the next. Under this model, variance in observed effect sizes is attributable to both sampling error (within-study variance) and statistical heterogeneity (between-study variance).

The most popular meta-analyses involve using a weighted average to combine the study-level effect sizes. Both fixed- and random-effects models use an inverse-variance weight (variance of the observed effect size). However, given the shared between-study variance used in the random-effects model, it leads to a more balanced distribution of weights than under the fixed-effect model (i.e., small studies are given more relative weight and large studies less). The standard error for these estimators also relates to the inverse-variance weights. As such, the standard errors and confidence intervals for the random-effects model are larger and wider than in the fixed-effect analysis. Indeed, in the presence of statistical heterogeneity, fixed-effect models can lead to overly narrow intervals.

In addition to commonly used, generalizable models, there are additional fixed-effect models and random-effect models that can be considered. Additional fixed-effect models that are specific to dichotomous data are more robust to issues that arise from sparse data. Furthermore, random-effects models can be expanded upon using generalized linear mixed models so that different covariance structures are used to distribute statistical heterogeneity across multiple parameters. Finally, both fixed- and random-effects modeling can be conducted using a Bayesian framework.