Consider a meta-analysis of studies with varying proportions of patient-level missing data, and assume that each primary study has made certain missing data adjustments so that the reported estimates of treatment effect size and variance are valid. These estimates of treatment effects can be combined across studies by standard meta-analytic methods, employing a random-effects model to account for heterogeneity across studies. However, we note that a meta-analysis based on the standard random-effects model will lead to biased estimates when the attrition rates of primary studies depend on the size of the underlying study-level treatment effect. Perhaps ignorable within each study, these types of missing data are in fact not ignorable in a meta-analysis. We propose three methods to correct the bias resulting from such missing data in a meta-analysis: reweighting the DerSimonian-Laird estimate by the completion rate; incorporating the completion rate into a Bayesian random-effects model; and inference based on a Bayesian shared-parameter model that includes the completion rate. We illustrate these methods through a meta-analysis of 16 published randomized trials that examined combined pharmacotherapy and psychological treatment for depression.