Bayesian hierarchical models for network meta-analysis incorporating nonignorable missingness

Abstract

Network meta-analysis expands the scope of a conventional pairwise meta-analysis to simultaneously compare multiple treatments, synthesizing both direct and indirect information and thus strengthening inference. Since most of trials only compare two treatments, a typical data set in a network meta-analysis managed as a trial-by-treatment matrix is extremely sparse, like an incomplete block structure with significant missing data. Zhang et al. proposed an arm-based method accounting for correlations among different treatments within the same trial and assuming that absent arms are missing at random. However, in randomized controlled trials, nonignorable missingness or missingness not at random may occur due to deliberate choices of treatments at the design stage. In addition, those undertaking a network meta-analysis may selectively choose treatments to include in the analysis, which may also lead to missingness not at random. In this paper, we extend our previous work to incorporate missingness not at random using selection models. The proposed method is then applied to two network meta-analyses and evaluated through extensive simulation studies. We also provide comprehensive comparisons of a commonly used contrast-based method and the arm-based method via simulations in a technical appendix under missing completely at random and missing at random.

Keywords: Bayesian hierarchical models; Network meta-analysis; nonignorable missingness; selection models.

Figure 1

Diagram of model assumptions. The arm-based method assumes studies at random (SR) while the contrast-based methods assume effects at random (ER). Homogeneous/heterogeneous refer to variances and perfect/exchangeable/unstructured refer to correlation matrices. Perfect refers to a correlation of 1 among random effects for different treatments within the same trial. Finally, HOM and ID are the homogeneous and heterogeneous models in Lu and Ades.

Figure 2

Graphical representations for the networks of the smoking cessation and prevention of pain on injection with Propofol data sets. The size of each node is proportional to the number of trials including the respective intervention, and the thickness of the link is proportional to the numbers of trials investigating the relation.

Figure 3

Population-averaged event rates variations with changes in α_1k. Posterior medians and their 95% credible intervals are presented for the smoking cessation data.

Figure 4

Performance comparisons for the AB and CB methods. Biases and MSEs under both MCAR and MAR mechanisms are shown.