A simulation-based approach to Bayesian sample size determination for performance under a given model and for separating models

Sample size determination (SSD) is a crucial aspect of experimental design. Two SSD problems are considered here. The first concerns how to select a sample size to achieve specified performance with regard to one or more features of a model. Adopting a Bayesian perspective, we move the Bayesian SSD problem from the rather elementary models addressed in the literature to date in the direction of the wide range of hierarchical models which dominate the current Bayesian landscape. Our approach is generic and thus, in principle, broadly applicable. However, it requires full model specification and computationally intensive simulation, perhaps limiting it practically to simple instances of such models. Still, insight from such cases is of useful design value. In addition, we present some theoretical tools for studying performance as a function of sample size, with a variety of illustrative results. Such results provide guidance with regard to what is achievable. We also offer two examples, a survival model with censoring and a logistic regression model. The second problem concerns how to select a sample size to achieve specified separation of two models. We approach this problem by adopting a screening criterion which in turn forms a model choice criterion. This criterion is set up to choose model 1 when the value is large, model 2 when the value is small. The SSD problem then requires choosing $n_{1}$ to make the probability of selecting model 1 when model 1 is true sufficiently large and choosing $n_{2}$ to make the probability of selecting model 2 when model 2 is true sufficiently large. The required n is $\max(n_{1}, n_{2})$. Here, we again provide two illustrations. One considers separating normal errors from t errors, the other separating a common growth curve model from a model with individual growth curves.