Multivariate meta-analysis of mixed outcomes: a Bayesian approach

Abstract

Multivariate random effects meta-analysis (MRMA) is an appropriate way for synthesizing data from studies reporting multiple correlated outcomes. In a Bayesian framework, it has great potential for integrating evidence from a variety of sources. In this paper, we propose a Bayesian model for MRMA of mixed outcomes, which extends previously developed bivariate models to the trivariate case and also allows for combination of multiple outcomes that are both continuous and binary. We have constructed informative prior distributions for the correlations by using external evidence. Prior distributions for the within-study correlations were constructed by employing external individual patent data and using a double bootstrap method to obtain the correlations between mixed outcomes. The between-study model of MRMA was parameterized in the form of a product of a series of univariate conditional normal distributions. This allowed us to place explicit prior distributions on the between-study correlations, which were constructed using external summary data. Traditionally, independent 'vague' prior distributions are placed on all parameters of the model. In contrast to this approach, we constructed prior distributions for the between-study model parameters in a way that takes into account the inter-relationship between them. This is a flexible method that can be extended to incorporate mixed outcomes other than continuous and binary and beyond the trivariate case. We have applied this model to a motivating example in rheumatoid arthritis with the aim of incorporating all available evidence in the synthesis and potentially reducing uncertainty around the estimate of interest.

Keywords: Bayesian analysis; multiple outcomes; multivariate meta-analysis; rheumatoid arthritis.

Figure 1

Structure of the data and the role of the data elements in the model.

Figure 2

Examples of constructing prior distribution for the between-study correlation using independent noninformative prior distributions for the standard deviations and the regression coefficient, which can lead to implausible prior distribution for the correlation.

Figure 3

Examples of constructing prior distributions for the between-study model parameters using interdependent prior distributions for the parameters.

Figure 4

Forest plots for Health Assessment Questionnaire (HAQ): from univariate random-effects meta-analysis (URMA; left) and from bivariate random-effects meta-analysis (BRMA) of HAQ and Disease Activity Score (DAS-28; middle) and for DAS-28 also from BRMA (right). Graph shows estimates from the systematic review with 95% confidence intervals (grey solid lines), predicted missing estimates from BRMA with 95% credible intervals (CrIs; grey dashed lines), ‘shrunken’ estimates with 95% CrIs (black solid lines), and the pooled estimates with 95% CrIs (black solid lines for pooled effect from each of the meta-analyses and black dashed lines representing results from URMA for comparison).

Figure 5

Forest plots for estimates of (from left to right) 20% response according to the American College of Rheumatology (ACR) criteria, Disease Activity Score (DAS-28), and Health Assessment Questionnaire (HAQ) from trivariate random-effects meta-analysis (TRMA) of HAQ, DAS-28, and ACR20. Graph shows estimates from the systematic review with 95% confidence intervals (grey solid lines), predicted missing estimates from TRMA with 95% credible intervals (CrIs; grey dashed lines), ‘shrunken’ estimates with 95% CrIs (black solid lines), and the pooled estimates with 95% CrIs (black solid lines for pooled effect from each of the TRMAs and black dotted (dashed) lines representing results from bivariate random-effects meta-analysis (univariate random-effects meta-analysis) for comparison).