Modelling heterogeneity variances in multiple treatment comparison meta-analysis--are informative priors the better solution?

Abstract

Background: Multiple treatment comparison (MTC) meta-analyses are commonly modeled in a Bayesian framework, and weakly informative priors are typically preferred to mirror familiar data driven frequentist approaches. Random-effects MTCs have commonly modeled heterogeneity under the assumption that the between-trial variance for all involved treatment comparisons are equal (i.e., the 'common variance' assumption). This approach 'borrows strength' for heterogeneity estimation across treatment comparisons, and thus, ads valuable precision when data is sparse. The homogeneous variance assumption, however, is unrealistic and can severely bias variance estimates. Consequently 95% credible intervals may not retain nominal coverage, and treatment rank probabilities may become distorted. Relaxing the homogeneous variance assumption may be equally problematic due to reduced precision. To regain good precision, moderately informative variance priors or additional mathematical assumptions may be necessary.

Methods: In this paper we describe four novel approaches to modeling heterogeneity variance - two novel model structures, and two approaches for use of moderately informative variance priors. We examine the relative performance of all approaches in two illustrative MTC data sets. We particularly compare between-study heterogeneity estimates and model fits, treatment effect estimates and 95% credible intervals, and treatment rank probabilities.

Results: In both data sets, use of moderately informative variance priors constructed from the pair wise meta-analysis data yielded the best model fit and narrower credible intervals. Imposing consistency equations on variance estimates, assuming variances to be exchangeable, or using empirically informed variance priors also yielded good model fits and narrow credible intervals. The homogeneous variance model yielded high precision at all times, but overall inadequate estimates of between-trial variances. Lastly, treatment rankings were similar among the novel approaches, but considerably different when compared with the homogenous variance approach.

Conclusions: MTC models using a homogenous variance structure appear to perform sub-optimally when between-trial variances vary between comparisons. Using informative variance priors, assuming exchangeability or imposing consistency between heterogeneity variances can all ensure sufficiently reliable and realistic heterogeneity estimation, and thus more reliable MTC inferences. All four approaches should be viable candidates for replacing or supplementing the conventional homogeneous variance MTC model, which is currently the most widely used in practice.

Figure 1

Presents the treatment networks with the number of trials informing each treatment comparison in our two illustrative examples. The treatment network on the left is the network for our first illustrative example. The treatment network on the right side is the network for our second illustrative example. The circles represent the treatments in the network, the lines represent the comparisons where head-to-head (direct) evidence is available, and the numbers in the lines present the number of randomized clinical trials available per comparison. Abbreviations: PEG-2A (Peginterferon-2a); PEG-2B (Peginterferon-2b); INF (Interferon), RBV (Ribavirin); Trt (Treatment).

Figure 2

Presents the posterior distributions of the between-trial variance parameters in the first illustrative example under the six employed MTC models: the homogeneous variance model (row 1); the unrestricted variances model (row 2); the exchangeable variances model (row 3); the consistency variances model (row 4); the frequentistically informed variances model (row 5); and the empirically informed variances model (row 6). The two presented comparisons are: peginterferon-2a (PEG-2A) vs Interferon (INF) (column 1), and Peginterferon-2a (PEG-2A) vs Peginterferon-2b (PEG-2B) (column 2). The comparison of PEG-2B vs INF was selective excluded due to the posterior variance distributions being more similar across the five heterogeneous variance approaches.

Figure 3

Presents the posterior distributions of the between-trial variance parameters in the second illustrative example under the six employed MTC models: the homogeneous variance model (row 1); the unrestricted variances model (row 2); the exchangeable variances model (row 3); the consistency variances model (row 4); the frequentistically informed variances model (row 5); and the empirically informed variances model (row 6). The three presented comparisons are: Treatment 2 (Trt2) versus control (column 1); treatment 4 (Trt2) versus Control; and Trt4 versus Trt1. The remaining comparisons were selective excluded due to the posterior variance distributions being more similar across the five heterogeneous variance approaches.