• Meta-analysis;
  • Publication bias;
  • Selection bias;
  • Selection model;
  • Sensitivity analysis

Summary.  A common conjecture in the study of publication bias is that studies reporting a significant result are more likely to be selected for review than studies whose results are inconclusive. We envisage a population of studies following the standard random-effects model of meta-analysis, and a selection probability given by a function of the study's ‘t-statistic’. In practice it is difficult to estimate this function, and hence difficult to estimate its associated bias correction. The paper suggests the more modest aim of a sensitivity analysis in which the treatment effect is estimated by maximum likelihood constrained by given values of the marginal probability of selection. This gives a graphical summary of how the inference from a meta-analysis changes as we allow for increasing selection (as the marginal selection probability decreases from 1), with an associated diagnostic plot comparing the observed treatment effects with their fitted values implied by the corresponding selection model. The approach is motivated by a medical example in which the highly significant result of a published meta-analysis was subsequently overturned by the results of a large-scale clinical trial.