Case studies appear prominently in political science, sociology, and other social science fields. A scholar employing a case study research design in an effort to estimate causal effects must confront the question, how should cases be selected for analysis? This question is important because the results derived from a case study research program ultimately and unavoidably rely on the criteria used to select the cases. While the matter of case selection is at the forefront of research on case study design, an analytical framework that can address it in a comprehensive way has yet to be produced. We develop such a framework and use it to evaluate nine common case selection methods. Our simulation-based results show that the methods of simple random sampling, influential case selection, and diverse case selection generally outperform other common methods. And, when a research design mandates that only a very small number of cases, say one or two, be selected in the course of a research program, the very simple method of sampling from the largest cell of a 2 x 2 table is competitive with other, more complicated, case selection methods. We show as well that a number of common case selection strategies work well only in idiosyncratic situations, and we argue that these methods should be abandoned in favor of the more powerful and robust case selection methods that our analytical framework identifies.