A penalization approach to random-effects meta-analysis

Abstract

Systematic reviews and meta-analyses are principal tools to synthesize evidence from multiple independent sources in many research fields. The assessment of heterogeneity among collected studies is a critical step when performing a meta-analysis, given its influence on model selection and conclusions about treatment effects. A common-effect (CE) model is conventionally used when the studies are deemed homogeneous, while a random-effects (RE) model is used for heterogeneous studies. However, both models have limitations. For example, the CE model produces excessively conservative confidence intervals with low coverage probabilities when the collected studies have heterogeneous treatment effects. The RE model, on the other hand, assigns higher weights to small studies compared to the CE model. In the presence of small-study effects or publication bias, the over-weighted small studies from a RE model can lead to substantially biased overall treatment effect estimates. In addition, outlying studies may exaggerate between-study heterogeneity. This article introduces penalization methods as a compromise between the CE and RE models. The proposed methods are motivated by the penalized likelihood approach, which is widely used in the current literature to control model complexity and reduce variances of parameter estimates. We compare the existing and proposed methods with simulated data and several case studies to illustrate the benefits of the penalization methods.

Keywords: common-effect model; heterogeneity; meta-analysis; penalized likelihood; random-effects model.

FIGURE 1

Forest plots of the four examples of meta-analyses.

FIGURE 1

Forest plots of the four examples of meta-analyses.

FIGURE 2

Plots of study-specific standardized residuals of the four examples of meta-analyses under the common-effect (denoted by filled triangles) and random-effects (denoted by unfilled dots) settings. The plus signs represent truncated standardized residuals whose absolute values are greater than 5.

FIGURE 3

Illustration of the penalization methods using the meta-analysis by Bohren et al. (a) The estimated between-study standard deviation against the tuning parameter λ for the penalized likelihood. (b) The loss function against the tuning parameter λ. (c) The loss function against the estimated between-study standard deviation by tuning λ. (d) The loss function against the tuning parameter τ_t. Each vertical dashed line in panels (b)–(d) represents the optimal value that minimizes the loss function; the horizontal and vertical dashed lines in panel (a) correspond to the optimal values from panels (b) and (c).

FIGURE 4

Loss functions for the meta-analyses by Storebø et al. (upper panels), by Carless et al. (middle panels), and by Bjelakovic et al. (lower panels). The left panels show the loss functions by tuning λ against the estimated between-study standard deviation; the right panels show the loss functions by tuning τ against τ_t. Each vertical dashed line represents the optimal value that yields the minimum loss.