Box 2 | Connections between Bayes factors and p-values

Given the ubiquity of p-values, it is natural to seek relationships between them and the Bayes factor (BF). For example, given a p-value from a published study, is it possible to compute a corresponding BF? We have emphasized that an advantage of BFs over p-values is that the strength of the evidence that the BF conveys does not vary with factors that affect power, such as sample size or minor allele frequency (MAF). As this is not true of p-values, it follows that any translation from p-values to BFs has to depend on further assumptions. Nevertheless, some interesting connections exist between BFs and p-values.

Optimistic BFs

Under general assumptions¹, the following result holds for p-values that satisfy p < 1/e, in which e 2.72:

For example, a SNP with p = 10^-6 has BF<2.7 10⁴, and hence if = 10^-4 the SNP has a substantial probability of being unassociated (posterior probability of association (PPA) < 0.72). By contrast, if p = 10^-7 then BF<2.3 10⁵ and the PPA could be >0.95. Note also that when p = 0.05, we obtain BF<2.5, which is at best only modest evidence against H₀. This is striking given the widespread adoption of a significance level of 0.05.

As equation 7 provides only an upper bound on the BF, it can be thought of as providing an 'optimistic' BF for a given p-value. It seems unlikely that any useful lower bound exists, and so there is no corresponding 'pessimistic' BF.

The implied prior of p-values

The BF under a strictly additive model, with a N(0,) distribution for the effect size under H₁, produces approximately the same ranking as p-values from standard additive-model tests, providing that ² is chosen to be proportional to a factor that depends on sample size⁴² but is asymptotically proportional to 1/MAF(1 – MAF). A similar result holds for non-additive models⁴².

Therefore, for a given sample size, ranking SNPs by their p-values is equivalent to a Bayesian analysis that makes some very specific assumptions. In particular, it assumes that truly associated low-MAF SNPs tend to have larger effect sizes than SNPs with a larger MAF. Broadly speaking, this assumption may be reasonable^{43, 44}, but there is no apparent justification⁴⁵ for the mathematical form 1/MAF(1 – MAF). Bayesians are free to choose the dependence of on MAF according to whatever formula they believe best fits the available background information, and hence they can in principle develop better ways to prioritize SNPs for follow up. Furthermore, and perhaps more importantly, this example shows that frequentist analyses can make implicit assumptions of which the user is unaware — see the 'Imputation' subsection in the main text for another example.