The paltry power of priors versus populations

Biological experiments often involve hypothesis testing at the scale of thousands to millions of tests. Alleviating the multiple testing burden has been a goal of many methods designed to boost test power by focusing tests on the alternative hypotheses most likely to be true. Very often, these methods either explicitly or implicitly make use of prior probabilities that bias significance for favored sets thought to be enriched for significant finding. Nevertheless, most genomics experiments, and in particular genome-wide association studies (GWAS), still use traditional univariate tests rather than more sophisticated approaches. Here we use GWAS to demonstrate why unbiased tests remain in favor. We calculate test power assuming perfect knowledge of a prior distribution and then derive the population size increase required to provided the same boost without a prior. We show that population size is exponentially more important than prior, providing a rigorous explanation for the observed avoidance of prior-based methods.