randomeffects

Random effects refer to a modeling approach in which certain effects are treated as random variables rather than fixed constants. In meta-analysis, a random-effects model assumes that the true effect size varies across studies and is drawn from a distribution with mean μ and between-study variance τ^2. Each study’s observed effect is modeled as the sum of its true effect and within-study sampling error. This framework acknowledges that differences in populations, interventions, or settings can yield different underlying effects.

Compared with fixed-effect models, random-effects models account for heterogeneity beyond sampling error. The random effects are typically assumed to be independent of within-study errors and to follow a normal distribution with mean μ and variance τ^2. This leads to wider confidence intervals and different weighting of studies, particularly when heterogeneity is substantial.

Estimation methods for random-effects models include the DerSimonian-Laird approach (a method-of-moments estimator for τ^2), maximum likelihood (ML), and restricted maximum likelihood (REML). Bayesian methods, using prior distributions for μ and τ^2, are also common. In multilevel or mixed-effects contexts, random effects represent subject- or group-level deviations (such as random intercepts or slopes) and are estimated via ML or REML with software such as lme4 in R or equivalent packages in other languages.

Applications of random-effects models span meta-analysis of diverse studies and hierarchical/multi-level data with clustering. They enable estimation of the average effect across a population of studies and allow prediction of effects in new settings through prediction intervals that incorporate τ^2. Model choice often depends on the degree of heterogeneity and the research question.