Association analysis provides an opportunity to find genetic variants underlying complex traits. A principal components regression (PCR)-based approach was shown to outperform some competing approaches. However, a limitation of this method is that the principal components (PCs) selected from single nucleotide polymorphisms (SNPs) may be unrelated to the phenotype. In this article, we investigate the theoretical properties of such a method in more detail. We first derive the exact power function of the test based on PCR, and hence clarify the relationship between the test power and the degrees of freedom (DF). Next, we extend the PCR test to a general weighted PCs test, which provides a unified framework for understanding the properties of some related statistics. We then compare the performance of these tests. We also introduce several data-driven adaptive alternatives to overcome difficulties in the PCR approach. Finally, we illustrate our results using simulations based on real genotype data. Simulation study shows the risk of using the unsupervised rule to determine the number of PCs, and demonstrates that there is no single uniformly powerful method for detecting genetic variants.