For illustration, consider the two allele case and suppose we have a p-value based on the Wald statistic for f with MLE f^=4n11n22−n122(2n11+n12)(2n22+n12) and asymptotic variance (12)V(f)=var(f^)=12np1p2(1−f)[2p1p2(1−f)(1−2f)+f(2−f)]. The χ2 statistic is given by f^2∕V(0)=nf^2. As shown elsewhere (Wakefield, 2007, 2009) an asymptotic Bayes factor (ABF) may be obtained by combining the “likelihood” f^∣f∼N{f,V(0)} with prior f ~ N(0, W) to give (13)ABF=V(0)+WV(0)exp(−z22WV(0)+W), where z=f^∕V(0) and W is the prior variance for f. If we take W independent of n then ABF tends to ∞ and 0 under the null and alternative respectively, as n → ∞, as desired. The ABF (13) is dependent on z only, and hence the Wald p-value (or equivalently the χ2 statistic), when we take the prior variance W = K × V(0) = K/n, where K is a constant that does not depend on the data. This p-value prior gives ABFp=(1+K)1∕2exp(−nf^22K1+K) and under this prior identical rankings of significance will be achieved between ABFp and the χ2 statistic. The dependence of the prior on n is troubling and leads to the p-value Bayes factor being
