  Meta-analysis is a statistical method used in evidence synthesis for
combining, analyzing and summarizing studies that have the same target endpoint
and aims to derive a pooled quantitative estimate using fixed and random
effects models or network models. Differences among included studies depend on
variations in target populations (i.e. heterogeneity) and variations in study
quality due to study design and execution (i.e. bias). The risk of bias is
usually assessed qualitatively using critical appraisal, and quantitative bias
analysis can be used to evaluate the influence of bias on the quantity of
interest. We propose a way to consider ignorance or ambiguity in how to
quantify bias terms in a bias analysis by characterizing bias with imprecision
(as bounds on probability) and use robust Bayesian analysis to estimate the
overall effect. Robust Bayesian analysis is here seen as Bayesian updating
performed over a set of coherent probability distributions, where the set
emerges from a set of bias terms. We show how the set of bias terms can be
specified based on judgments on the relative magnitude of biases (i.e., low,
unclear and high risk of bias) in one or several domains of the Cochrane's risk
of bias table. For illustration, we apply a robust Bayesian bias-adjusted
random effects model to an already published meta-analysis on the effect of
Rituximab for rheumatoid arthritis from the Cochrane Database of Systematic
Reviews.
