A set of recent papers attempts to explain the size and book-to-market anomalies with conditional CAPM or CCAPM models with economically motivated conditioning variables, or with factor models with economically motivated factors. The tests of these models, as presented, fail to reject the proposed model. We argue that these tests fail to reject the null hypothesis because they have very low statistical power against what we call the characteristics alternative. Specifically, the low power of these tests arises because they use as test portfolios, characteristic-sorted portfolios that do not have sufficient independent variation in the factor loadings and the characteristics. We propose several methods for constructing more appropriate test portfolios and for designing more powerful tests. We show that with these more powerful tests the models we examine are rejected at high levels of statistical significance.