Meta-analysis of effect sizes reported at multiple time points: a multivariate approach

Abstract

Background: Many comparative studies report results at multiple time points. Such data are correlated because they pertain to the same patients, but are typically meta-analyzed as separate quantitative syntheses at each time point, ignoring the correlations between time points.

Purpose: To develop a meta-analytic approach that estimates treatment effects at successive time points and takes account of the stochastic dependencies of those effects.

Methods: We present both fixed and random effects methods for multivariate meta-analysis of effect sizes reported at multiple time points. We provide formulas for calculating the covariance (and correlations) of the effect sizes at successive time points for four common metrics (log odds ratio, log risk ratio, risk difference, and arcsine difference) based on data reported in the primary studies. We work through an example of a meta-analysis of 17 randomized trials of radiotherapy and chemotherapy versus radiotherapy alone for the postoperative treatment of patients with malignant gliomas, where in each trial survival is assessed at 6, 12, 18, and 24 months post randomization. We also provide software code for the main analyses described in the article.

Results: We discuss the estimation of fixed and random effects models and explore five options for the structure of the covariance matrix of the random effects. In the example, we compare separate (univariate) meta-analyses at each of the four time points with joint analyses across all four time points using the proposed methods. Although results of univariate and multivariate analyses are generally similar in the example, there are small differences in the magnitude of the effect sizes and the corresponding standard errors. We also discuss conditional multivariate analyses where one compares treatment effects at later time points given observed data at earlier time points.

Limitations: Simulation and empirical studies are needed to clarify the gains of multivariate analyses compared with separate meta-analyses under a variety of conditions.

Conclusions: Data reported at multiple time points are multivariate in nature and are efficiently analyzed using multivariate methods. The latter are an attractive alternative or complement to performing separate meta-analyses.