A Bayesian hierarchical model for allele frequencies

Allelic frequency estimation in presence of uncertain priors

In this paper, we assume that allele frequencies are random variables and follow certain statistical distributions. However, specifying an appropriate informative prior distribution with specific hyperparameters seems to be a major issue. Assuming that prior information varies over some classes of priors, we develop the concept of robust Bayes estimation into the context of allele frequency estimation. We first assume that the region of interest is a single locus and the prior information is represented in terms of a class of Beta distributions, and present explicit forms of the resulting Bayes and robust Bayes estimators. We then extend our results to biallelic k-loci and multi-allelic k-loci cases within the region of interest. We perform a simulation study to measure performance of the proposed robust Bayes estimators against some Bayes estimators associated with specific hyperparameters. The simulations reflect satisfactory performance of the proposed robust Bayes estimators when there is no evidence implying the actual prior distribution.

Estimating Allele Frequencies

Methods of estimating allele frequencies from data on unrelated and related individuals are described in this chapter. For samples of unrelated individuals with simple codominant markers, the natural estimators of allele frequencies can be used. For genetic data on related individuals, maximum likelihood estimation (MLE) can be applied to compute allele frequencies. Factors that influence allele frequencies in populations are also explained.

Estimating allele frequencies

Methods of estimating allele frequencies from data on unrelated and related individuals are described in this chapter. For samples of unrelated individuals with simple codominant markers, the natural estimator of allele frequencies can be used. For genetic data on related individuals, maximum likelihood estimation (MLE) can be applied to compute allele frequencies. Factors that influence allele frequencies in populations are also explained.

Evaluating differences in linkage disequilibrium between populations

We propose two methods to evaluate the statistical significance of differences in linkage disequilibrium (LD) between populations, where LD is measured by the standardised parameter D'. The first method is based on bootstrapping individuals within populations in order to test LD differences for each pair of loci. Using this approach we propose a solution to the problem of testing multiple locus-pairs by means of a single test for the number of pairs that exhibit significant LD differences among populations. The second method provides the Bayesian posterior probability that one population has greater LD than the other for each locus pair. Both methods can handle genotypes with unknown phase, and are demonstrated using two data sets. For the purpose of demonstration, we apply the methods to two different sets of data from humans. First, we explore the issue of LD differences between reproductively isolated populations using a new data set of twelve Xq25 microsatellites, typed in four European populations. Second, we examine evidence for LD differences between Alzheimer cases and controls from the Icelandic population using 19 single nucleotide polymorphisms (SNPs) from a 97 kb region flanking the Apolipoprotein E (APOE) gene on chromosome 19.

A genome-scan method to identify selected loci appropriate for both dominant and codominant markers: a Bayesian perspective

Identifying loci under natural selection from genomic surveys is of great interest in different research areas. Commonly used methods to separate neutral effects from adaptive effects are based on locus-specific population differentiation coefficients to identify outliers. Here we extend such an approach to estimate directly the probability that each locus is subject to selection using a Bayesian method. We also extend it to allow the use of dominant markers like AFLPs. It has been shown that this model is robust to complex demographic scenarios for neutral genetic differentiation. Here we show that the inclusion of isolated populations that underwent a strong bottleneck can lead to a high rate of false positives. Nevertheless, we demonstrate that it is possible to avoid them by carefully choosing the populations that should be included in the analysis. We analyze two previously published data sets: a human data set of codominant markers and a Littorina saxatilis data set of dominant markers. We also perform a detailed sensitivity study to compare the power of the method using amplified fragment length polymorphism (AFLP), SNP, and microsatellite markers. The method has been implemented in a new software available at our website (http://www-leca.ujf-grenoble.fr/logiciels.htm).

IMPROVING POPULATION-SPECIFIC ALLELE FREQUENCY ESTIMATES BY ADAPTING SUPPLEMENTAL DATA: AN EMPIRICAL BAYES APPROACH

Estimation of the allele frequency at genetic markers is a key ingredient in biological and biomedical research, such as studies of human genetic variation or of the genetic etiology of heritable traits. As genetic data becomes increasingly available, investigators face a dilemma: when should data from other studies and population subgroups be pooled with the primary data? Pooling additional samples will generally reduce the variance of the frequency estimates; however, used inappropriately, pooled estimates can be severely biased due to population stratification. Because of this potential bias, most investigators avoid pooling, even for samples with the same ethnic background and residing on the same continent. Here, we propose an empirical Bayes approach for estimating allele frequencies of single nucleotide polymorphisms. This procedure adaptively incorporates genotypes from related samples, so that more similar samples have a greater influence on the estimates. In every example we have considered, our estimator achieves a mean squared error (MSE) that is smaller than either pooling or not, and sometimes substantially improves over both extremes. The bias introduced is small, as is shown by a simulation study that is carefully matched to a real data example. Our method is particularly useful when small groups of individuals are genotyped at a large number of markers, a situation we are likely to encounter in a genome-wide association study.

Empirical Bayes inference of pairwise F(ST) and its distribution in the genome

Populations often have very complex hierarchical structure. Therefore, it is crucial in genetic monitoring and conservation biology to have a reliable estimate of the pattern of population subdivision. F(ST)'s for pairs of sampled localities or subpopulations are crucial statistics for the exploratory analysis of population structures, such as cluster analysis and multidimensional scaling. However, the estimation of F(ST) is not precise enough to reliably estimate the population structure and the extent of heterogeneity. This article proposes an empirical Bayes procedure to estimate locus-specific pairwise F(ST)'s. The posterior mean of the pairwise F(ST) can be interpreted as a shrinkage estimator, which reduces the variance of conventional estimators largely at the expense of a small bias. The global F(ST) of a population generally varies among loci in the genome. Our maximum-likelihood estimates of global F(ST)'s can be used as sufficient statistics to estimate the distribution of F(ST) in the genome. We demonstrate the efficacy and robustness of our model by simulation and by an analysis of the microsatellite allele frequencies of the Pacific herring. The heterogeneity of the global F(ST) in the genome is discussed on the basis of the estimated distribution of the global F(ST) for the herring and examples of human single nucleotide polymorphisms (SNPs).

Genetic structure of chimpanzee populations

Little is known about the history and population structure of our closest living relatives, the chimpanzees, in part because of an extremely poor fossil record. To address this, we report the largest genetic study of the chimpanzees to date, examining 310 microsatellites in 84 common chimpanzees and bonobos. We infer three common chimpanzee populations, which correspond to the previously defined labels of "western," "central," and "eastern," and find little evidence of gene flow between them. There is tentative evidence for structure within western chimpanzees, but we do not detect distinct additional populations. The data also provide historical insights, demonstrating that the western chimpanzee population diverged first, and that the eastern and central populations are more closely related in time.

An Empirical Bayes Method for Updating Inferences in Analysis of Quantitative Trait Loci Using Information From Related Genome Scans

Individual genome scans for quantitative trait loci (QTL) mapping often suffer from low statistical power and imprecise estimates of QTL location and effect. This lack of precision yields large confidence intervals for QTL location, which are problematic for subsequent fine mapping and positional cloning. In prioritizing areas for follow-up after an initial genome scan and in evaluating the credibility of apparent linkage signals, investigators typically examine the results of other genome scans of the same phenotype and informally update their beliefs about which linkage signals in their scan most merit confidence and follow-up via a subjective–intuitive integration approach. A method that acknowledges the wisdom of this general paradigm but formally borrows information from other scans to increase confidence in objectivity would be a benefit. We developed an empirical Bayes analytic method to integrate information from multiple genome scans. The linkage statistic obtained from a single genome scan study is updated by incorporating statistics from other genome scans as prior information. This technique does not require that all studies have an identical marker map or a common estimated QTL effect. The updated linkage statistic can then be used for the estimation of QTL location and effect. We evaluate the performance of our method by using extensive simulations based on actual marker spacing and allele frequencies from available data. Results indicate that the empirical Bayes method can account for between-study heterogeneity, estimate the QTL location and effect more precisely, and provide narrower confidence intervals than results from any single individual study. We also compared the empirical Bayes method with a method originally developed for meta-analysis (a closely related but distinct purpose). In the face of marked heterogeneity among studies, the empirical Bayes method outperforms the comparator.

Empirical bayes method for incorporating data from multiple genome scans

Individual genome scans tend to have low power and can produce markedly biased estimates of QTL effects. Further, the confidence interval for their location is often prohibitively large for subsequent fine mapping and positional cloning. Given that a large number of genome scans have been conducted, not to mention the large number of variables and subsets tested, it is difficult to confidently rule out type 1 error as an explanation for significant effects even when there is apparent replication in a separate data set. We adapted Empirical Bayes (EB) methods [1] to analyze data from multiple genome scans simultaneously and alleviate each of these problems while still allowing for different QTL population effects across studies. We investigated the effects of using the EB method to include data from background studies to update the results of a single study of interest via simulation and demonstrated that it has a stable confidence level over a wide range of parameters defining the background studies and increased the power to detect linkage, even when some of the background studies were null or had QTL effect at other markers. This EB method for incorporating data from multiple studies into genome scan analyses seems promising.

Best linear unbiased allele-frequency estimation in complex pedigrees

Many types of genetic analyses depend on estimates of allele frequencies. We consider the problem of allele-frequency estimation based on data from related individuals. The motivation for this work is data collected on the Hutterites, an isolated founder population, so we focus particularly on the case in which the relationships among the sampled individuals are specified by a large, complex pedigree for which maximum likelihood estimation is impractical. For this case, we propose to use the best linear unbiased estimator (BLUE) of allele frequency. We derive this estimator, which is equivalent to the quasi-likelihood estimator for this problem, and we describe an efficient algorithm for computing the estimate and its variance. We show that our estimator has certain desirable small-sample properties in common with the maximum likelihood estimator (MLE) for this problem. We treat both the case when parental origin of each allele is known and when it is unknown. The results are extended to prediction of allele frequency in some set of individuals S based on genotype data collected on a set of individuals R. We compare the mean-squared error of the BLUE, the commonly used naive estimator (sample frequency) and the MLE when the latter is feasible to calculate. The results indicate that although the MLE performs the best of the three, the BLUE is close in performance to the MLE and is substantially easier to calculate, making it particularly useful for large complex pedigrees in which MLE calculation is impractical or infeasible. We apply our method to allele-frequency estimation in a Hutterite data set.

Design and analysis of admixture mapping studies

Admixture between populations originating on different continents can be exploited to detect disease susceptibility loci at which risk alleles are distributed differentially between these populations. We first examine the statistical power and mapping resolution of this approach in the limiting situation in which gamete admixture and locus ancestry are measured without uncertainty. We show that, for a rare disease, the most efficient design is to study affected individuals only. In a typical African American population (two-way admixture proportions 0.8/0.2, ancestry crossover rate 2 per 100 cM), a study of 800 affected individuals has 90% power to detect at P values <10(-5) a locus that generates a risk ratio of 2 between populations, with an expected mapping resolution (size of 95% confidence region for the position of the locus) of 4 cM. In practice, to infer locus ancestry from marker data requires Bayesian computationally intensive methods, as implemented in the program ADMIXMAP. Affected-only study designs require strong prior information on the frequencies of each allele given locus ancestry. We show how data from unadmixed and admixed populations can be combined to estimate these ancestry-specific allele frequencies within the admixed population under study, allowing for variation between allele frequencies in unadmixed and admixed populations. Using simulated data based on the genetic structure of the African American population, we show that 60% of information can be extracted in a test for linkage using markers with an ancestry information content of 36% at 3-cM spacing. As in classic linkage studies, the most efficient strategy is to use markers at a moderate density for an initial genome search and then to saturate regions of putative linkage with additional markers, to extract nearly all information about locus ancestry.

Methods for high-density admixture mapping of disease genes

Admixture mapping (also known as "mapping by admixture linkage disequilibrium," or MALD) has been proposed as an efficient approach to localizing disease-causing variants that differ in frequency (because of either drift or selection) between two historically separated populations. Near a disease gene, patient populations descended from the recent mixing of two or more ethnic groups should have an increased probability of inheriting the alleles derived from the ethnic group that carries more disease-susceptibility alleles. The central attraction of admixture mapping is that, since gene flow has occurred recently in modern populations (e.g., in African and Hispanic Americans in the past 20 generations), it is expected that admixture-generated linkage disequilibrium should extend for many centimorgans. High-resolution marker sets are now becoming available to test this approach, but progress will require (a). computational methods to infer ancestral origin at each point in the genome and (b). empirical characterization of the general properties of linkage disequilibrium due to admixture. Here we describe statistical methods to estimate the ancestral origin of a locus on the basis of the composite genotypes of linked markers, and we show that this approach accurately estimates states of ancestral origin along the genome. We apply this approach to show that strong admixture linkage disequilibrium extends, on average, for 17 cM in African Americans. Finally, we present power calculations under varying models of disease risk, sample size, and proportions of ancestry. Studying approximately 2500 markers in approximately 2500 patients should provide power to detect many regions contributing to common disease. A particularly important result is that the power of an admixture mapping study to detect a locus will be nearly the same for a wide range of mixture scenarios: the mixture proportion should be 10%-90% from both ancestral populations.

Significant linkage on chromosome 10p in families with bulimia nervosa

Bulimia nervosa (BN) is strongly familial, and additive genetic effects appear to contribute substantially to the observed familiality. In turn, behavioral components of BN, such as self-induced vomiting, are reliably measured and heritable. To identify regions of the genome harboring genetic variants conferring susceptibility to BN, we conducted a linkage analysis of multiplex families with eating disorders that were identified through a proband with BN. Linkage analysis of the entire sample of 308 families yielded a double peak, with the highest nonparametric multipoint maximum LOD score (MLS), of 2.92, on chromosome 10. Given the high heritability of self-induced vomiting and the reliability with which it can be measured, we performed linkage analysis in a subset (n=133) of families in which at least two affected relatives reported a symptom pattern that included self-induced vomiting. The highest MLS (3.39) observed was on chromosome 10, between markers D10S1430 and D10S1423. These results provide evidence of the presence of a susceptibility locus for BN on chromosome 10p. Using simulations, we demonstrate that both of these scores, 2.92 and 3.39, meet the widely accepted criterion for genomewide significance. Another region on 14q meets the criterion for genomewide suggestive linkage, with MLSs of 1.97 (full sample) and 1.75 (subset) at 62 centimorgans from p-ter.