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.2018 Mar;9(1):73-88.
doi: 10.1002/jrsm.1274. Epub 2017 Dec 7.

An improved method for bivariate meta-analysis when within-study correlations are unknown

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An improved method for bivariate meta-analysis when within-study correlations are unknown

Chuan Hong et al. Res Synth Methods.2018 Mar.

Abstract

Multivariate meta-analysis, which jointly analyzes multiple and possibly correlated outcomes in a single analysis, is becoming increasingly popular in recent years. An attractive feature of the multivariate meta-analysis is its ability to account for the dependence between multiple estimates from the same study. However, standard inference procedures for multivariate meta-analysis require the knowledge of within-study correlations, which are usually unavailable. This limits standard inference approaches in practice. Riley et al proposed a working model and an overall synthesis correlation parameter to account for the marginal correlation between outcomes, where the only data needed are those required for a separate univariate random-effects meta-analysis. As within-study correlations are not required, the Riley method is applicable to a wide variety of evidence synthesis situations. However, the standard variance estimator of the Riley method is not entirely correct under many important settings. As a consequence, the coverage of a function of pooled estimates may not reach the nominal level even when the number of studies in the multivariate meta-analysis is large. In this paper, we improve the Riley method by proposing a robust variance estimator, which is asymptotically correct even when the model is misspecified (ie, when the likelihood function is incorrect). Simulation studies of a bivariate meta-analysis, in a variety of settings, show a function of pooled estimates has improved performance when using the proposed robust variance estimator. In terms of individual pooled estimates themselves, the standard variance estimator and robust variance estimator give similar results to the original method, with appropriate coverage. The proposed robust variance estimator performs well when the number of studies is relatively large. Therefore, we recommend the use of the robust method for meta-analyses with a relatively large number of studies (eg, m≥50). When the sample size is relatively small, we recommend the use of the robust method under the working independence assumption. We illustrate the proposed method through 2 meta-analyses.

Keywords: correlation; multivariate meta-analysis; random effects; robust variance estimator.

Copyright © 2017 John Wiley & Sons, Ltd.

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Figures

Figure 1:
Figure 1:
Coverage of confidence intervals forβ1 –β2 estimated from the Riley method with standard variance estimator and the Riley method with robust variance estimator under scenarios 1–9 (denoted as S1–S9) for different settings of within-study correlation (ρWi) and between-study correlation (ρB). The between-study variations(τ12,τ22) are set at 0.25, representing a relatively small between-study/within-study variation ratio (small VR). The red line refers to Riley method with robust variance estimator and the blue line refers to the Riley method with standard variance estimator.
Figure 2:
Figure 2:
Coverage of confidence intervals forβ1 –β2 estimated from the Riley method with standard variance estimator and the Riley method with robust variance estimator under scenarios 1–9 (denoted as S1–S9) for different settings of within-study correlation(ρWi) and between-study correlation (ρB). The between-study variations(τ12,τ22) are set at 0.5, representing a relatively moderate between-study/within-study variation ratio (moderate VR). The red line refers to Riley method with robust variance estimator and the blue line refers to the Riley method with standard variance estimator.
Figure 3:
Figure 3:
Coverage of confidence intervals forβ1 –β2 estimated from the Riley method with standard variance estimator and the Riley method with robust variance estimator under scenarios 1–9 (denoted as S1–S9) for different settings of within-study correlation (ρWi) and between-study correlation (ρB). The between-study variations(τ12,τ22) are set at 5, representing a relatively large between-study/within-study variation ratio (large VR). The red line refers to Riley method with robust variance estimator and the blue line refers to the Riley method with standard variance estimator.
Figure 4:
Figure 4:
Coverage of confidence intervals ofβ1 estimated from the Riley method with standard variance estimator and the Riley method with robust variance estimator under scenarios 1–9 (denoted as S1–S9) for different settings of marginal correlation (ρS)· The between-study variations(τ12,τ22) are set at 0.25, representing a relatively large between-study/within-study variation ratio (large VR). The red line refers to Riley method with robust variance estimator and the blue line refers to the Riley method with standard variance estimator.
Figure 5:
Figure 5:
Coverage of confidence intervals ofβ1 –β2 estimated from the Riley method with standard variance estimator and the Riley method with robust variance estimator under scenarios 10–15 (denoted as S10–S15) for different settings of marginal correlation (ρS). The between-study variations(τ12,τ22) are set at 0.25, representing a relatively moderate between-study/within-study variation ratio (moderate VR). The red line refers to Riley method with robust variance estimator and the blue line refers to the Riley method with standard variance estimator.
Figure 6:
Figure 6:
Coverage of confidence intervals ofβ1 –β2 estimated from the Riley method with robust variance estimator when the true underlying distribution is bivariatet distribution. The red line refers to Riley method with robust variance estimator.
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