This paper introduces hierarchical selection models and illustrates how they may be used in meta-analysis. Our approach combines the use of hierarchical models, which allow one to investigate variability both within and between units (e.g., studies), and weight functions, which allow one to model non-randomly selected data. We show how Markov Chain Monte Carlo methods may be used to estimate the hierarchical selection model. This approach is illustrated first for known weight functions, and then extended to allow for estimation of the weight function. Hierarchical selection models are shown to be especially useful in meta-analysis, where one often needs to account for both between-study variability and bias involved in the collection of the studies. However, the methods presented are very general and can be used in a variety of other situations, e.g., in a multi-center clinical trial where there is bias involved in the selection of centers.
Weight functions provide an approach for examining sensitivity of results to bias in the way studies are obtained. However, this is shown to be different from examining sensitivity to unobserved studies directly. In order to investigate sensitivity of results to unobserved studies, while still accounting for between-study variability and bias in the collection of the observed studies, the hierarchical selection model approach is combined with data augmentation to account for unobserved studies. Again, Markov Chain Monte Carlo methods may be used to estimate the model. This is illustrated for both known and unknown (i.e., estimated) weight functions.