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

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