Bayesian methods for examining Hardy-Weinberg equilibrium.
- Authors
- Wakefield, Jon
- Year
- 2010
- Journal
- Biometrics
- PMID
- 19459838
- DOI
- 10.1111/j.1541-0420.2009.01267.x
- PMCID
- PMC4535922
Testing for Hardy-Weinberg equilibrium is ubiquitous and has traditionally been carried out via frequentist approaches. However, the discreteness of the sample space means that uniformity of p-values under the null cannot be assumed, with enumeration of all possible counts, conditional on the minor allele count, offering a computationally expensive way of p-value calibration. In addition, the interpretation of the subsequent p-values, and choice of significance threshold depends critically on sample size, because equilibrium will always be rejected at conventional levels with large sample sizes. We argue for a Bayesian approach using both Bayes factors, and the examination of posterior distributions. We describe simple conjugate approaches, and methods based on importance sampling Monte Carlo. The former are convenient because they yield closed-form expressions for Bayes factors, which allow their application to a large number of single nucleotide polymorphisms (SNPs), in particular in genome-wide contexts. We also describe straightforward direct sampling methods for examining posterior distributions of parameters of interest. For large numbers of alleles at a locus we resort to Markov chain Monte Carlo. We discuss a number of possibilities for prior specification, and apply the suggested methods to a number of real datasets.
Prior distribution for a generic fixation index with (a) k = 4 alleles, (b) k = 9 alleles, given a Dirichlet prior with parameters 1, on the 10 (k = 4) and 45 (k = 9) allele frequencies.
Prior (top row) and posterior (bottom row) distributions based on 5000 samples, for the four group data and the single inbreeding coefficient f model. The MLE is indicated as the cross in (f). We examine p1 for illustration, and could just as easily have picked p2, p3, or p4.
Posterior summaries and MLEs for the fixation coefficients, in the four group example.
Bayes factor and p-value summaries for the GWAS for age-related macular degeneration data: (a) –log10 p-values from a χ2 test versus those from the exact test; (b) histogram of exact p-values (vertical axis truncated, the count at a p-value of 1 is indicated); (c) QQ-plot of observed versus expected –log10 p-values, assuming uniformity of p-values under the null; and (d) –log10 p-values against –log10 Bayes factors. The dashed lines corresponds to the Bonferroni threshold (p-value axes) and Bayes factor thresholds (Bayes factor axes).
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