a problem far removed from that which faces us in genome-wide association studies, where we face the problem of testing ‘multiple hypotheses’ (for a particular disease, one hypothesis for each SNP, or region of correlated SNPs, in the genome) and we thus do not subscribe to the view that one should correct significance levels for the number of tests performed to obtain ‘genome-wide significance levels’. Nonetheless, our aim is to keep the false positive rate within acceptable bounds and this still leads to the view that very low P values are needed for strong evidence of association. But the factor determining the threshold is not the number of tests performed, but the a priori probability that there is likely to be a true association at any specified location in the genome. Of course, we cannot know this prior probability from objective evidence, but we can perhaps estimate an order of magnitude.There are two linked questions. The first concerns the choice of an appropriate ‘threshold’ for reporting possible associations as likely to be genuine. Here the mathematics is quite straightforward if we make the simplifying assumption that we have the same power to detect all true associations. Then we have18 Posterior
