Fixed- and random-effects models in meta-analysis.

TLDR

Meta‑analysis summarizes multiple studies by combining effect‑size estimates, and two main statistical models—fixed‑effects and random‑effects—were developed to infer either the observed study parameters or the distribution of parameters across a population of studies. The study evaluates how confidence intervals and hypothesis tests perform under fixed‑ and random‑effects procedures, confirming that each is optimal for its intended inference. Fixed‑effects models estimate a single common effect size assuming homogeneity, whereas random‑effects models treat effect sizes as a random sample from a distribution, allowing for between‑study variability. The authors confirm that fixed‑effects procedures are best for inference about observed study parameters, random‑effects procedures for population‑level inference, and that conditionally random‑effects procedures display intermediate properties. The article does not consider the mixed‑effects model, which is relevant when study‑level covariates are included (see Hedges, 1992).

Abstract

There are 2 families of statistical procedures in meta-analysis: fixed- and randomeffects procedures. They were developed for somewhat different inference goals: making inferences about the effect parameters in the studies that have been observed versus making inferences about the distribution of effect parameters in a population of studies from a random sample of studies. The authors evaluate the performance of confidence intervals and hypothesis tests when each type of statistical procedure is used for each type of inference and confirm that each procedure is best for making the kind of inference for which it was designed. Conditionally random-effects procedures (a hybrid type) are shown to have properties in between those of fixed- and random-effects procedures. The use of quantitative methods to summarize the results of several empirical research studies, or metaanalysis, is now widely used in psychology, medicine, and the social sciences. Meta-analysis usually involves describing the results of each study by means of a numerical index (an estimate of effect size, such as a correlation coefficient, a standardized mean difference, or an odds ratio) and then combining these estimates across studies to obtain a summary. Two somewhat different statistical models have been developed for inference about average effect size from a collection of studies, called the fixed-effects and random-effects models. (A third alternative, the mixedeffects model, arises in conjunction with analyses involving study-level covariates or moderator variables, which we do not consider in this article; see Hedges, 1992.) Fixed-effects models treat the effect-size parameters as fixed but unknown constants to be estimated and usually (but not necessarily) are used in conjunction with assumptions about the homogeneity of effect parameters (see, e.g., Hedges, 1982; Rosenthal & Rubin, 1982). Random-effects models treat the effectsize parameters as if they were a random sample from