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. 2015 May 30;34(12):2062-80.
doi: 10.1002/sim.6475. Epub 2015 Mar 24.

A Bayesian framework to account for uncertainty due to missing binary outcome data in pairwise meta-analysis

N L Turner  1 , S Dias, A E Ades, N J Welton
Affiliations

A Bayesian framework to account for uncertainty due to missing binary outcome data in pairwise meta-analysis

N L Turner et al. Stat Med. .

Abstract

Missing outcome data are a common threat to the validity of the results from randomised controlled trials (RCTs), which, if not analysed appropriately, can lead to misleading treatment effect estimates. Studies with missing outcome data also threaten the validity of any meta-analysis that includes them. A conceptually simple Bayesian framework is proposed, to account for uncertainty due to missing binary outcome data in meta-analysis. A pattern-mixture model is fitted, which allows the incorporation of prior information on a parameter describing the missingness mechanism. We describe several alternative parameterisations, with the simplest being a prior on the probability of an event in the missing individuals. We describe a series of structural assumptions that can be made concerning the missingness parameters. We use some artificial data scenarios to demonstrate the ability of the model to produce a bias-adjusted estimate of treatment effect that accounts for uncertainty. A meta-analysis of haloperidol versus placebo for schizophrenia is used to illustrate the model. We end with a discussion of elicitation of priors, issues with poor reporting and potential extensions of the framework. Our framework allows one to make the best use of evidence produced from RCTs with missing outcome data in a meta-analysis, accounts for any uncertainty induced by missing data and fits easily into a wider evidence synthesis framework for medical decision making.

Keywords: Bayesian; bias; decision making; meta-analysis; missing data; pattern-mixture model.

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Figures

Figure 1
Figure 1
Directed acyclic graph (DAG) of our missing data framework: μ i,kis the study‐specific log‐odds of the outcome in treatment 1 (the control), δ i,kis the log‐odds ratio, πi,kall=p(success), πi,kobs=p(success | observed), r i,kis the number of observed events, c i,kis the number of observed outcomes, θ i,kis the missing data parameter, q i,k=p(missing), m i,kis the number of missing outcomes and n i,kis total number randomised. Ellipses denote stochastic parameters or observed data. Small boxes denote constants. The two large boxed (‘plates’) represent indexing of studies iand treatments k. Single line arrows denote stochastic relationships, and double‐line arrows denote logical relationships. Note that c i,k=n i,k−m i,k; however, we omit the logical relationship between these parameters in the DAG, as c i,kis constant conditional on the observed number of missing outcomes.
Figure 2
Figure 2
Box‐plots of posterior distribution for πi,kmissfor each study arm [i,k] for trial i, arm k. Plotted for scenarios 1 and 2, respectively. Boxes represent inter‐quartile range, the whiskers represent the 95% CI and line within the box represents the median.
Figure 3
Figure 3
Box‐plots of posterior distribution for πi,kmissfor each study arm [i,k] for trial i, arm kin the schizophrenia meta‐analysis 19. Plotted for fixed and random effect models where a missing data model is used with Beta(1,1) priors for πi,kmiss. Boxes represent inter‐quartile range, the whiskers represent the 95% CI and line within the box represents the median.
Figure C.1
Figure C.1
Caterpillar plots showing posterior mean (dot) and 95% CIs (line) of p(missing) for each trial arm in the artificial data scenarios. Indices [i,k] represent arm k(k= 1=placebo;k= 2=active treatment) of trial i. (a) Scenario 1 – trials 2, 3, 5, 6, 7, 8, 9 and 11 had 20% of outcomes removed (from the observed events) from each arm. (b) Scenario 2 – trials 2, 5, 6, 8, 9 and 11 had 20% of outcomes removed from each arm.
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