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A Bayesian Missing Data Framework for Multiple Continuous Outcome Mixed Treatment Comparisons

Research Report

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Structured Abstract


Bayesian statistical approaches to mixed treatment comparisons (MTCs) are becoming more popular due to their flexibility and interpretability. Many randomized clinical trials report multiple outcomes with possible inherent correlations, but there is little previous work in modeling them statistically. We aimed to build on existing hierarchical modeling and missing data methods to obtain novel and improved Bayesian approaches to MTCs for multiple continuous outcomes.

Data Sources

We reviewed randomized clinical trials published in English after 1979 that examined physical therapy interventions for community-dwelling adults with knee pain secondary to osteoarthritis (OA). After screening, 84 randomized trials met the inclusion/exclusion criteria, reporting variously on knee pain, disability, quality of life, and functional outcomes.


After a review of existing hierarchical Bayesian methods for MTCs with a single continuous outcome, we introduce novel Bayesian approaches for multiple continuous outcomes (here, pain and disability) simultaneously, rather than in separate MTC analyses, by generalizing existing models to treat missing data the same as unknown parameters and to incorporate correlation structure between outcomes. We also introduce an arm-based model that is less constrained than existing models. We produce Bayesian treatment ranks based on a sensible scoring system incorporating weights for the multiple outcomes. We also offer simulation studies to check our method's Type I error, power, and the probability of incorrectly selecting the best treatment.


In our OA data analysis, while all the models gave similar goodness of fit, they yielded different best treatments, with aerobic exercise emerging as best according to the older models, but proprioception exercise being preferred by our weighted ranking models. Still, few statistically significant differences between treatments were observed. Our missing data approaches had better power and Type I error than previous Bayesian methods in our simulation study. Ignoring missing data or correlation between outcomes can produce biased MTC estimates leading to high Type I error and low power, especially when the data from missing treatments depend on the observed data.


Our missing data approaches appear preferable for incorporating missing data and correlation structure in MTC modeling, to traditional contrast-based approaches, and thus in obtaining more precise and robust parameter estimates.

Key Messages

  • Since researchers often choose study arms based on previous trials, it is important to consider any unobserved treatment arms in an MTC as missing data and subsequently use Bayes' Rule to learn about the treatments' relative relationships. This makes it easier to assign prior distributions on random effects and delivers better statistical inference.
  • Our arm-based models are less constrained than previous contrast-based models and can thus yield parameters with more straightforward interpretations, especially in the presence of correlations between outcomes.