The Capital Asset Pricing Model (CAPM) and its various multi-factor extensions are generally viewed as the central paradigm of asset pricing research. In this paper we describe a formal statistical methodology to evaluate the validity of these models. Our approach is couched in the context of the "Size" and "Value" anomalies and thus is based on a well specified alternative hypothesis. We explore the finite sample properties of our approach by working with a fully Bayesian linear model and conclude that most of the "fail to reject'' results that recently appear in the finance literature are a consequence of the lack of power of the statistical procedures commonly used.