On random-effects meta-analysis

Abstract

Meta-analysis is widely used to compare and combine the results of multiple independent studies. To account for between-study heterogeneity, investigators often employ random-effects models, under which the effect sizes of interest are assumed to follow a normal distribution. It is common to estimate the mean effect size by a weighted linear combination of study-specific estimators, with the weight for each study being inversely proportional to the sum of the variance of the effect-size estimator and the estimated variance component of the random-effects distribution. Because the estimator of the variance component involved in the weights is random and correlated with study-specific effect-size estimators, the commonly adopted asymptotic normal approximation to the meta-analysis estimator is grossly inaccurate unless the number of studies is large. When individual participant data are available, one can also estimate the mean effect size by maximizing the joint likelihood. We establish the asymptotic properties of the meta-analysis estimator and the joint maximum likelihood estimator when the number of studies is either fixed or increases at a slower rate than the study sizes and we discover a surprising result: the former estimator is always at least as efficient as the latter. We also develop a novel resampling technique that improves the accuracy of statistical inference. We demonstrate the benefits of the proposed inference procedures using simulated and empirical data.

Keywords: Clustered data; Evidence-based medicine; Genetic association; Heterogeneity; Individual patient data; Maximum likelihood estimation; Random-effects model; Research synthesis; Summary statistic.

Fig. 1

Estimated density functions of: (a) β̂_ML and β̂_MA; (b) ${\widehat{\sigma }}_{\text{ML}}^2$ and ${\widehat{\sigma }}_{\text{MA}}^2$ in simple linear regression. In each panel, the solid curve corresponds to the maximum likelihood estimator and the dashed curve to the meta-analysis estimator.

Fig. 2

Empirical coverage probabilities of nominal 95% confidence intervals plotted against τ² for (a) K = 10 and (b) K = 20, and plotted against K for (c) τ² = 0·03 and (d) τ² = 0·07. In each panel, the different curves correspond to the new resampling method (solid), the DerSimonian–Laird method (dashed), the Jackson–Bowden method (dotted), and the Hardy–Thompson method (dot-dash).

Fig. 3

Empirical coverage probabilities of nominal 95% confidence intervals when the random effects are from the τ × Ga(1, 1) distribution centred at its mean, plotted against τ² for (a) K = 10 and (b) K = 20, and plotted against K for (c) τ² = 0·03 and (d) τ² = 0·07. In each panel, the different curves correspond to the new resampling method (solid), the DerSimonian–Laird method (dashed), the Jackson–Bowden method (dotted), and the Hardy–Thompson method (dot-dash).