On inferring presence of an individual in a mixture: a Bayesian approach

Abstract

Homer and others (2008. Resolving individuals contributing trace amounts of DNA to highly complex mixtures using high-density SNP genotyping microarrays. PLoS Genetics 4, e1000167) recently showed that, given allele frequency data for a large number of single nucleotide polymorphisms in a sample together with corresponding population "reference" frequencies, by typing an individual's DNA sample at the same set of loci it can be inferred whether or not the individual was a member of the sample. This observation has been responsible for precautionary removal of large amounts of summary data from public access. This and further work on the problem has followed a frequentist approach. This paper sets out a Bayesian analysis of this problem which clarifies the role of the reference frequencies and allows incorporation of prior probabilities of the individual's membership in the sample.

Fig. 1.

A simulation study. For each of the partial prior knowledge cases shown in Table 1, 1000 simulations were run for genotype (binomial) data under each of H₁ (In sample) and H₀ (Not in). The box plots show the distribution of values of the Gaussian approximation to log₁₀(Bayes factor).

Fig. 2.

Gaussian approximation log Bayes factors versus exact values. This figure shows the results of 20 simulations under each of H₀ and H₁ for binomial data with N = 100,K = 200 and P = 3450.

Fig. 3.

Effect of correlation between variables. This figure shows the results of 100 simulations under each of H₀ and H₁ for binomial data with N = 100,K = 200 and P = 3450 when the variables are intercorrelated according to an AR(1) model with lag 1 correlations 0.5. (a) No correction for correlation. (b) Correction using estimated inverse correlation matrix from a sample of size 200. (c) Correction using a sample of size 500. (d) Correction using a sample of size 1000.

Fig. 4.

A real example using chromosome 20 data drawn from the Wellcome Trust Case Control Consortium (N = 145, P = 4743 and K = 1455). The ordinate in (b) is calculated under the assumption of a nonrepresentative reference sample (F_ST = 0.003), while that in (c) ignores linkage disequilibrium. Otherwise Bayes factors are calculated on the assumption that the reference sample is representative and allowing for linkage disequilibrium between SNPs.