Exact tests for association between alleles at arbitrary numbers of loci

Associations between allelic frequencies, within and between loci, can be tested for with an exact test. The probability of the set of multi-locus genotypes in a sample, conditional on the allelic counts, is calculated from multinomial theory under the hypothesis of no association. Alleles are then permuted and the conditional probability calculated for the permuted genotypic array. The proportion of arrays no more probable than the original sample provides the significance level for the test. An algorithm is provided for counting genotypes efficiently in the arrays, and the powers of the test presented for various kinds of association. The powers for the case when associations are generated by admixture of several populations suggest that exact tests are capable of detecting levels of association that would affect forensic calculations to a significant extent.     

Population structure and demographic inferences concerning the endangered onychophoran species Epiperipatus acacioi (Onychophora: Peripatidae)

Epiperipatus acacioi (Onychophora: Peripatidae) is an endemic species of the Atlantic rainforest in southeastern Brazil, with a restricted known distribution, found only in two nearby areas (Tripuí and Itacolomi). Mitochondrial gene COI sequences of 93 specimens collected across the known range of E. acacioi were used to assess the extant genetic diversity and patterns of genetic structure, as well as to infer the demographic history of this species. We found considerable variability within the populations, even though there has been recent environmental disturbance in these habitats. The samples from the two areas where this species is found showed significantly different COI sequences and constitute two distinct populations [exact test of sample differentiation (P = 0.0008) and pairwise F(ST) analyses (F(ST) = 0.214, P < 0.00001)]. However, there was little genetic differentiation among samples from different sampling sites within populations, suggesting that the potential for dispersal of E. acacioi greater than would have been expected, based on their cryptic behavior and reduced vagility. Mismatch analyses and neutrality tests revealed evidence of recent population expansion processes for both populations, possibly related to variations in the past distribution of this species.     

Exact tests of goodness of fit of log-linear models for rates

We propose Metropolis-Hastings sampling methods for estimating the exact conditional p-value for tests of goodness of fit of log-linear models for mortality rates and standardized mortality ratios. We focus on two-way tables, where the required conditional distribution is a multivariate noncentral hypergeometric distribution with known noncentrality parameter. Two examples are presented: a 2 x 3 table, where the exact results, obtained by enumeration, are available for comparison, and a 9 x 7 table, where Monte Carlo methods provide the only feasible approach for exact inference.     

Isolation by distance in the eastern oyster, Crassostrea virginica, in Chesapeake Bay

Intensive efforts are underway to restore depleted stocks of Crassostrea virginica in Chesapeake Bay. However, the extent of gene flow among local populations, an important force mediating the success of these endeavors, is poorly understood. Spatial and temporal population structures were examined in C. virginica from Chesapeake Bay using eight microsatellite loci. Deficits in heterozygosity relative to Hardy-Weinberg expectations were seen at all loci and were best explained by null alleles. Permutation tests indicated that heterozygote deficiency reduced power in tests of differentiation. Nonetheless, genotypic exact tests demonstrated significant levels of geographic differentiation overall, and a subtle pattern of isolation by distance (IBD) was observed. Comparisons between age classes failed to show differences in genotype frequencies, allelic richness, gene diversity, or differentiation as measured by F(ST), contrary to predictions made by the sweepstakes hypothesis. The IBD pattern could reflect an evolutionary equilibrium established because local gene flow predominates, or be influenced in either direction by recent anthropogenic activities. An evolutionary interpretation appears justified as more parsimonious, implying that local efforts to restore oyster populations will have local demographic payoffs, perhaps at the scale of tributaries or regional subestuaries within Chesapeake Bay.     

The Exact Test for Cytonuclear Disequilibria

We extend the analysis of the statistical properties of cytonuclear disequilibria in two major ways. First, we develop the asymptotic sampling theory for the nonrandom associations between the alleles at a haploid cytoplasmic locus and the alleles and genotypes at a diploid nuclear locus, when there are an arbitrary number of alleles at each marker. This includes the derivation of the maximum likelihood estimators and their sampling variances for each disequilibrium measure, together with simple tests of the null hypothesis of no disequilibrium. In addition to these new asymptotic tests, we provide the first implementation of Fisher's exact test for the genotypic cytonuclear disequilibria and some approximations of the exact test. We also outline an exact test for allelic cytonuclear disequilibria in multiallelic systems. An exact test should be used for data sets when either the marginal frequencies are extreme or the sample size is small. The utility of this new sampling theory is illustrated through applications to recent nuclear-mtDNA and nuclear-cpDNA data sets. The results also apply to population surveys of nuclear loci in conjunction with markers in cytoplasmically inherited microorganisms.     

High genetic diversity in the remnant island population of hihi and the genetic consequences of re-introduction

The maintenance of genetic diversity is thought to be fundamental for the conservation of threatened species. It is therefore important to understand how genetic diversity is affected by the re-introduction of threatened species. We use establishment history and genetic data from the remnant and re-introduced populations of a New Zealand endemic bird, the hihi Notiomystis cincta, to understand genetic diversity loss and quantify the genetic effects of re-introduction. Our data do not support any recent bottleneck events in the remnant population. Furthermore, all genetic diversity measures indicate the remnant hihi population has retained high levels of genetic diversity relative to other New Zealand avifauna with similar histories of decline. Genetic diversity (N(A) , alleles per locus, allelic richness, F(IS) and H(S) ) did not significantly decrease in new hihi populations founded through re-introduction when compared to their source populations, except in the Kapiti Island population (allelic richness and H(S) ) which had very slow post-re-introduction population growth. The N(e) /N(c) ratio in the remnant population was high, but decreased in first-level re-introductions, which together with significant genetic differentiation between populations (F(ST) & Fisher's exact tests) suggest that extant populations are diverging as a result of founder effects and drift. Importantly, simulations of future allele loss predict that the number of alleles lost will be higher in populations with a slow population growth, fewer founding individuals and with nonrandom mating. Interestingly, this species has very high levels of extra-pair paternity which may reduce reproductive variance by allowing social and floater males to reproduce a life history trait that together with a large remnant population size may help maintain higher levels of genetic diversity than expected.     

Landscape structure, clonal propagation, and genetic diversity in Scandinavian populations of Arabidopsis lyrata (Brassicaceae)

Colonization history, landscape structure, and environmental conditions may influence patterns of neutral genetic variation because of their effects on gene flow and reproductive mode. We compared variation at microsatellite loci within and among 26 Arabidopsis lyrata populations in two disjunct areas of its distribution in northern Europe (Norway and Sweden). The two areas probably share a common colonization history but differ in size (Norwegian range markedly larger than Swedish range), landscape structure (mountains vs. coast), and habitat conditions likely to affect patterns of gene flow and opportunities for sexual reproduction. Within-population genetic diversity was not related to latitude but was higher in Sweden than in Norway. Population differentiation was stronger among Norwegian than among Swedish populations (F(ST) = 0.23 vs. F(ST) = 0.18). The frequency of clonal propagation (proportion of identical multilocus genotypes) increased with decreasing population size, was higher in Norwegian than in Swedish populations, but was not related to altitude or substrate. Differences in genetic structure are discussed in relation to population characteristics and range size in the two areas. The results demonstrate that the possibility of clonal propagation should be considered when developing strategies for sampling and analyzing data in ecological and genetic studies of this emerging model species.     

Assessing the goodness of fit of logistic regression models in large samples: A modification of the Hosmer-Lemeshow test

Evaluating the goodness of fit of logistic regression models is crucial to ensure the accuracy of the estimated probabilities. Unfortunately, such evaluation is problematic in large samples. Because the power of traditional goodness of fit tests increases with the sample size, practically irrelevant discrepancies between estimated and true probabilities are increasingly likely to cause the rejection of the hypothesis of perfect fit in larger and larger samples. This phenomenon has been widely documented for popular goodness of fit tests, such as the Hosmer-Lemeshow test. To address this limitation, we propose a modification of the Hosmer-Lemeshow approach. By standardizing the noncentrality parameter that characterizes the alternative distribution of the Hosmer-Lemeshow statistic, we introduce a parameter that measures the goodness of fit of a model but does not depend on the sample size. We provide the methodology to estimate this parameter and construct confidence intervals for it. Finally, we propose a formal statistical test to rigorously assess whether the fit of a model, albeit not perfect, is acceptable for practical purposes. The proposed method is compared in a simulation study with a competing modification of the Hosmer-Lemeshow test, based on repeated subsampling. We provide a step-by-step illustration of our method using a model for postneonatal mortality developed in a large cohort of more than 300 000 observations.     

Neutrality tests of highly polymorphic restriction-fragment-length polymorphisms.

The allele frequency data of Baird et al. were tested using Ewens-Watterson sampling theory for goodness of fit to the infinite-alleles model of neutral evolution. Although probes of both the HRAS-1 and D14S1 loci identify highly diverse restriction-fragment-length polymorphisms, the observed values of gene identity (F) and the common allele frequency (C) are not significantly different from the neutral expectation. Allele frequency distributions show a tendency toward a deficit in diversity for HRAS-1 and a slight excess diversity for D14S1. The direction of these departures is consistent with potential selective effects of the Harvey-ras oncogene and hitchhiking of the D14S1 locus to closely linked immunoglobulin genes. Direct chi 2-tests of goodness of fit of the observed and expected allele frequency distributions reveal significant departures in the caucasoid and Hispanic HRAS-1 distributions but not in any of the other tests.     

Molecular and quantitative genetic divergence among populations of house mice with known evolutionary histories

Evolutionary biologists have long been interested in the processes influencing population differentiation, but separating the effects of neutral and adaptive evolution has been an obstacle for studies of population subdivision. A recently developed method allows tests of whether disruptive (ie, spatially variable) or stabilizing (ie, spatially uniform) selection is influencing phenotypic differentiation among subpopulations. This method, referred to as the F(ST) vs Q(ST) comparison, separates the total additive genetic variance into within- and among-population components and evaluates this level of differentiation against a neutral hypothesis. Thus, levels of neutral molecular (F(ST)) and quantitative genetic (Q(ST)) divergence are compared to evaluate the effects of selection and genetic drift on phenotypic differentiation. Although the utility of such comparisons appears great, its accuracy has not yet been evaluated in populations with known evolutionary histories. In this study, F(ST) vs Q(ST) comparisons were evaluated using laboratory populations of house mice with known evolutionary histories. In this model system, the F(ST) vs Q(ST) comparisons between the selection groups should reveal quantitative trait differentiation consistent with disruptive selection, while the F(ST) vs Q(ST) comparisons among lines within the selection groups should suggest quantitative trait differentiation in agreement with drift. We find that F(ST) vs Q(ST) comparisons generally produce the correct evolutionary inference at each level in the population hierarchy. Additionally, we demonstrate that when strong selection is applied between populations Q(ST) increases relative to Q(ST) among populations diverging by drift. Finally, we show that the statistical properties of Q(ST), a variance component ratio, need further investigation.     

Comparisons between QST and FST—how wrong have we been?

The comparison between quantitative genetic divergence (Q(ST) ) and neutral genetic divergence (F(ST) ) among populations has become the standard test for historical signatures of selection on quantitative traits. However, when the mutation rate of neutral markers is relatively high in comparison with gene flow, estimates of F(ST) will decrease, resulting in upwardly biased comparisons of Q(ST) vs. F(ST) . Reviewing empirical studies, the difference between Q(ST) and F(ST) is positively related to marker heterozygosity. After refuting alternative explanations for this pattern, we conclude that marker mutation rate indeed has had a biasing effect on published Q(ST) -F(ST) comparisons. Hence, it is no longer clear that populations have commonly diverged in response to divergent selection. We present and discuss potential solutions to this bias. Comparing Q(ST) with recent indices of neutral divergence that statistically correct for marker heterozygosity (Hedrick's G'st and Jost's D) is not advised, because these indices are not theoretically equivalent to Q(ST) . One valid solution is to estimate F(ST) from neutral markers with mutation rates comparable to those of the loci underlying quantitative traits (e.g. SNPs). Q(ST) can also be compared to Φ(ST) (Phi(ST) ) of amova, as long as the genetic distance among allelic variants used to estimate Φ(ST) reflects evolutionary history: in that case, neutral divergence is independent of mutation rate. In contrast to their common usage in comparisons of Q(ST) and F(ST) , microsatellites typically have high mutation rates and do not evolve according to a simple evolutionary model, so are best avoided in Q(ST) -F(ST) comparisons.     

Genetic basis of adaptation in Arabidopsis thaliana: local adaptation at the seed dormancy QTL DOG1

Local adaptation provides an opportunity to study the genetic basis of adaptation and investigate the allelic architecture of adaptive genes. We study delay of germination 1 (DOG1), a gene controlling natural variation in seed dormancy in Arabidopsis thaliana and investigate evolution of dormancy in 41 populations distributed in four regions separated by natural barriers. Using F(ST) and Q(ST) comparisons, we compare variation at DOG1 with neutral markers and quantitative variation in seed dormancy. Patterns of genetic differentiation among populations suggest that the gene DOG1 contributes to local adaptation. Although Q(ST) for seed dormancy is not different from F(ST) for neutral markers, a correlation with variation in summer precipitation supports that seed dormancy is adaptive. We characterize dormancy variation in several F(2) -populations and show that a series of functionally distinct alleles segregate at the DOG1 locus. Theoretical models have shown that the number and effect of alleles segregatin at quantitative trait loci (QTL) have important consequences for adaptation. Our results provide support to models postulating a large number of alleles at quantitative trait loci involved in adaptation.     

An Investigation on the Allelic Chi‐Square Test Used in Genetic Association Studies

Case‐control studies are primary study designs used in genetic association studies. Sasieni (Biometrics 1997, 53, 1253–1261) pointed out that the allelic chi‐square test used in genetic association studies is invalid when Hardy‐Weinberg equilibrium (HWE) is violated in a combined population. It is important to know how much type I error rate is deviated from the nominal level under violated HWE. We examine bounds of type I error rate of the allelic chi‐square test. We also investigate power of the goodness‐of‐fit test for HWE which can be used as a guideline for selecting an appropriate test between the allelic chi‐square test and the modified allelic chi‐square test, the latter of which was proposed for cases of violated HWE. In small samples, power is not large enough to detect the Wright's inbreeding model of small values of inbreeding coefficient. Therefore, when the null hypothesis of HWE is barely accepted, the modified test should be considered as an alternative method. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Population structure, genetic variation and morphological diversity in indigenous sheep of Ethiopia

We investigated genetic and morphological diversity and population structure of 14 traditional sheep populations originating from four ecological zones in Ethiopia (sub-alpine, wet highland, sub-humid lowland and arid lowland). All animals (n = 672) were genotyped for 17 microsatellite markers and scored for 12 morphological characters. The sheep were initially classified as fat-tailed (11 populations), thin-tailed (one population) and fat-rumped sheep (two populations). These classifications are thought to correspond to three consecutive introduction events of sheep from the Near-East into East Africa. For the 14 populations, allelic richness ranged from 5.87 to 7.51 and expected heterozygosity (H(E)) from 0.66 to 0.75. Genetic differentiations (F(ST) values) between all pairs of populations, except between sub-alpine populations, were significantly different from zero (P < 0.001). Cluster analysis of morphological characters and a dendrogram constructed from genetic distances were broadly consistent with the classification into fat-tailed, thin-tailed and fat-rumped sheep. Bayesian cluster analysis using microsatellite markers indicated that there has been further genetic differentiation after the initial introduction of sheep into Ethiopia. Investigation of factors associated with genetic variation showed that an isolation-by-distance model, independently of other factors, explained most of the observed genetic variation. We also obtained a strong indication of adaptive divergence in morphological characters, patterns of morphological variation being highly associated with ecology even when the effect of neutral genetic divergence (F(ST)) was parcelled out in partial Mantel tests. Using a combination of F(ST) values, Bayesian clustering analysis and morphological divergence, we propose a classification of Ethiopian sheep into six breed groups and nine breeds.     

Genetic and phenotypic population divergence on a microgeographic scale in brown trout

Salmonid populations of many rivers are rapidly declining. One possible explanation is that habitat fragmentation increases genetic drift and reduces the populations' potential to adapt to changing environmental conditions. We measured the genetic and eco-morphological diversity of brown trout (Salmo trutta) in a Swiss stream system, using multivariate statistics and Bayesian clustering. We found large genetic and phenotypic variation within only 40 km of stream length. Eighty-eight percent of all pairwise F(ST) comparisons and 50% of the population comparisons in body shape were significant. High success rates of population assignment tests confirmed the distinctiveness of populations in both genotype and phenotype. Spatial analysis revealed that divergence increased with waterway distance, the number of weirs, and stretches of poor habitat between sampling locations, but effects of isolation-by-distance and habitat fragmentation could not be fully disentangled. Stocking intensity varied between streams but did not appear to erode genetic diversity within populations. A lack of association between phenotypic and genetic divergence points to a role of local adaptation or phenotypically plastic responses to habitat heterogeneity. Indeed, body shape could be largely explained by topographic stream slope, and variation in overall phenotype matched the flow regimes of the respective habitats.     

The effects of dominance, regular inbreeding and sampling design on Q(ST), an estimator of population differentiation for quantitative traits

To test whether quantitative traits are under directional or homogenizing selection, it is common practice to compare population differentiation estimates at molecular markers (F(ST)) and quantitative traits (Q(ST)). If the trait is neutral and its determinism is additive, then theory predicts that Q(ST) = F(ST), while Q(ST) > F(ST) is predicted under directional selection for different local optima, and Q(ST) < F(ST) is predicted under homogenizing selection. However, nonadditive effects can alter these predictions. Here, we investigate the influence of dominance on the relation between Q(ST) and F(ST) for neutral traits. Using analytical results and computer simulations, we show that dominance generally deflates Q(ST) relative to F(ST). Under inbreeding, the effect of dominance vanishes, and we show that for selfing species, a better estimate of Q(ST) is obtained from selfed families than from half-sib families. We also compare several sampling designs and find that it is always best to sample many populations (>20) with few families (five) rather than few populations with many families. Provided that estimates of Q(ST) are derived from individuals originating from many populations, we conclude that the pattern Q(ST) > F(ST), and hence the inference of directional selection for different local optima, is robust to the effect of nonadditive gene actions.     

Testing goodness of fit of a uniform truncation model

Several goodness-of-fit tests of a lifetime distribution have been suggested in the literature; many take into account censoring and/or truncation of event times. In some contexts, a goodness-of-fit test for the truncation distribution is of interest. In particular, better estimates of the lifetime distribution can be obtained when knowledge of the truncation law is exploited. In cross-sectional sampling, for example, there are theoretical justifications for the assumption of a uniform truncation distribution, and several studies have used it to improve the efficiency of their survival estimates. The duality of lifetime and truncation in the absence of censoring enables methods for testing goodness of fit of the lifetime distribution to be used for testing goodness of fit of the truncation distribution. However, under random censoring, this duality does not hold and different tests are required. In this article, we introduce several goodness-of-fit tests for the truncation distribution and investigate their performance in the presence of censored event times using simulation. We demonstrate the use of our tests on two data sets.     

Phylogeography of the montane caddisfly Drusus discolor: evidence for multiple refugia and periglacial survival

We studied the genetic population structure and phylogeography of the montane caddisfly Drusus discolor across its entire range in central and southern Europe. The species is restricted to mountain regions and exhibits an insular distribution across the major mountain ranges. Mitochondrial sequence data (COI) of 254 individuals from the entire species range is analysed to reveal population genetic structure. The data show little molecular variation within populations and regions, but distinct genetic differentiation between mountain ranges. Most populations are significantly differentiated based on F(ST) and exact tests of population differentiation and most haplotypes are unique to a single mountain range. Phylogenetic analyses reveal deep divergence between geographically isolated lineages. Combined, these results suggest that past fragmentation is the prominent process structuring the populations across Europe. We use tests of selective neutrality and mismatch distributions, to study the demographic population history of regions with haplotype overlap. The high level of genetic differentiation between mountain ranges and estimates of demographic history provide evidence for the existence of multiple glacial refugia, including several in central Europe. The study shows that these aquatic organisms reacted differently to Pleistocene cooling than many terrestrial species. They persisted in numerous refugia over multiple glacial cycles, allowing many local endemic clades to form.     

Global analysis of regional differences in craniometric diversity and population substructure.

Estimates of genetic diversity in major geographic regions are frequently made by pooling all individuals into regional aggregates. This method can potentially bias results if there are differences in population substructure within regions, since increased variation among local populations could inflate regional diversity. A preferred method of estimating regional diversity is to compute the mean diversity within local populations. Both methods are applied to a global sample of craniometric data consisting of 57 measurements taken on 1734 crania from 18 local populations in six geographic regions: sub-Saharan Africa, Europe, East Asia, Australasia, Polynesia, and the Americas. Each region is represented by three local populations. Both methods for estimating regional diversity show sub-Saharan Africa to have the highest levels of phenotypic variation, consistent with many genetic studies. Polynesia and the Americas both show high levels of regional diversity when regional aggregates are used, but the lowest mean local population diversity. Regional estimates of F(ST) made using quantitative genetic methods show that both Polynesia and the Americas also have the highest levels of differentiation among local populations, which inflates regional diversity. Regional differences in F(ST) are directly related to the geographic dispersion of samples within each region; higher F(ST) values occur when the local populations are geographically dispersed. These results show that geographic sampling can affect results, and suggest caution in making inferences regarding regional diversity when population substructure is ignored.     

Population genetic structure of wild and hatchery black rockfish Sebastes inermis in Korea, assessed using cross-species microsatellite markers

The population structure of the black rockfish, Sebastes inermis (Sebastidae), was estimated using 10 microsatellite loci developed for S. schlegeli on samples of 174 individuals collected from three wild and three hatchery populations in Korea. Reduced genetic variation was detected in hatchery strains [overall number of alleles (N(A)) = 8.07; allelic richness (A(R)) = 7.37; observed heterozygosity (H(O)) = 0.641] compared with the wild samples (overall N(A) = 8.43; A(R) = 7.83; H(O) = 0.670), but the difference was not significant. Genetic differentiation among the populations was significant (overall F(ST) = 0.0237, P < 0.05). Pairwise F(ST) tests, neighbor-joining tree, and principal component analyses showed significant genetic heterogeneity among the hatchery strains and between wild and hatchery strains, but not among the wild populations, indicating high levels of gene flow along the southern coast of Korea, even though the black rockfish is a benthic, non-migratory marine species. Genetic differentiation among the hatchery strains could reflect genetic drift due to intensive breeding practices. Thus, in the interests of optimal resource management, genetic variation should be monitored and inbreeding controlled within stocks in commercial breeding programs. Information on genetic population structure based on cross-species microsatellite markers can aid in the proper management of S. inermis populations.