#### Frequentist approaches

Frequentist approaches typically measure SNP associations using $$p$$-values. A $$p$$-value is the probability of observing the same result, or a more extreme result, under the null hypothesis. We reject the null hypothesis for very low $$p$$-values because there is a very low probability that the data (or more extreme data) came from the this hypothesis. Intuitively, in GWAS, $$p$$-values are interpreted as how likely a putative disease associated variant is due to random chance. The GWAS significance $$p$$-value threshold is set at $$5\times 10^{-8}$$ to control for the number of false-positive associations (type I error).

However, $$p$$-values are inadequate at quantifying true associations in genetic association studies because their interpretation should depend on the power of the study (sample size, effect size and MAFs) - in that the threshold value chosen should vary with power. The power is the complement of a type II error (false negative) and measures the probability that the null hypothesis is rejected, given that the alternative hypothesis is true, i.e. measuring true positives. If the power of the study is low then there will be fewer associations found. Indeed, 1 found that the $$5\times 10^{-8}$$ $$p$$-value threshold was valid for common ($$MAF>5%$$) genetic variation in the European population, however it should be more stringent for lower frequency variants (e.g.$$1\times 10^{-8}$$ for $$MAF>0.01%$$ at LD $$r^2<0.8$$).

A true association depends not just on how unlikely the $$p$$-value is under the null hypothesis (which will be the same amongst all tests) but also on how unlikely it is under the alternative hypothesis of a genetic effect, which will vary across tests. A very low $$p$$-value may indicate a lack of evidence for the null hypothesis, but this doesn’t really mean anything if the $$p$$-value is just as unlikely under the alternative (as it may be in low powered studies).

#### Bayesian approaches

Bayesian approaches typically measure SNP associations using posterior probabilities, derived from prior probabilities (typically small, ~$$10^{-4}$$) and Bayes factors. These posterior probabilities can be directly interpreted as probabilities and can be meaningfully compared both within and across studies (unlike $$p$$-values).

To get strong evidence of association (i.e. a posterior probability close to 1), the Bayes factor needs to be very big to compensate for the very small prior probability it is multiplied by. This is analogous to the requirement of $$p$$-values being very small to reach genome-wide significance but differs in the reasoning. Bayes factors are required to be big because of the small number of SNPs that are expected to be truly associated and $$p$$-values are required to be small due to multiple testing problems.

#### References

Stephens and Balding