Unequal allelic frequencies at the self‐incompatibility locus within local populations of Prunus avium L.: an effect of population structure?

In this paper, we investigated the genetic structure and distribution of allelic frequencies at the gametophytic self-incompatibility locus in three populations of Prunus avium L. In line with theoretical predictions under balancing selection, genetic structure at the self-incompatibility locus was almost three times lower than at seven unlinked microsatellites. Furthermore, we found that S-allele frequencies in wild cherry populations departed significantly from the expected isoplethic distribution towards which balancing selection is expected to drive allelic frequencies (i.e. identical frequency equal to the inverse of the number of alleles in the population). To assess whether this departure could be caused either by drift alone or by population structure, we used numerical simulations to compare our observations with allelic frequency distributions expected : (1) within a single deme from a subdivided population with various levels of differentiation; and (2) within a finite panmictic population with identical allelic diversity. We also investigated the effects of sample size and degree of population structure on tests of departure from isoplethic equilibrium. Overall, our results showed that the observed allele frequency distributions were consistent with a model of subdivided population with demes linked by moderate migration rate.     

TEMPORAL VARIATION IN ALLELE FREQUENCIES: TESTING THE RIGHT HYPOTHESIS

Although standard statistical tests (such as contingency chi-square or G tests) are not well suited to the analysis of temporal changes in allele frequencies, they continue to be used routinely in this context. Because the null hypothesis stipulated by the test is violated if samples are temporally spaced, the true probability of a significant test statistic will not equal the nominal α level, and conclusions drawn on the basis of such tests can be misleading. A generalized method, applicable to a wide variety of organisms and sampling schemes, is developed here to estimate the probability of a significant test statistic if the only forces acting on allele frequencies are stochastic ones (i.e., sampling error and genetic drift). Results from analyses and simulations indicate that the rate at which this probability increases with time is determined primarily by the ratio of sample size to effective population size. Because this ratio differs considerably among species, the seriousness of the error in using the standard test will also differ. Bias is particularly strong in cases in which a high percentage of the total population can be sampled (for example, endangered species). The model used here is also applicable to the analysis of parent-offspring data and to comparisons of replicate samples from the same generation. A generalized test of the hypothesis that observed changes in allele frequency can be satisfactorily explained by drift follows directly from the model, and simulation results indicate that the true α level of this adjusted test is close to the nominal one under most conditions.     

Development and Application of Genomic Control Methods for Genome-Wide Association Studies Using Non-Additive Models

Genome-wide association studies (GWAS) comprise a powerful tool for mapping genes of complex traits. However, an inflation of the test statistic can occur because of population substructure or cryptic relatedness, which could cause spurious associations. If information on a large number of genetic markers is available, adjusting the analysis results by using the method of genomic control (GC) is possible. GC was originally proposed to correct the Cochran-Armitage additive trend test. For non-additive models, correction has been shown to depend on allele frequencies. Therefore, usage of GC is limited to situations where allele frequencies of null markers and candidate markers are matched. In this work, we extended the capabilities of the GC method for non-additive models, which allows us to use null markers with arbitrary allele frequencies for GC. Analytical expressions for the inflation of a test statistic describing its dependency on allele frequency and several population parameters were obtained for recessive, dominant, and over-dominant models of inheritance. We proposed a method to estimate these required population parameters. Furthermore, we suggested a GC method based on approximation of the correction coefficient by a polynomial of allele frequency and described procedures to correct the genotypic (two degrees of freedom) test for cases when the model of inheritance is unknown. Statistical properties of the described methods were investigated using simulated and real data. We demonstrated that all considered methods were effective in controlling type 1 error in the presence of genetic substructure. The proposed GC methods can be applied to statistical tests for GWAS with various models of inheritance. All methods developed and tested in this work were implemented using R language as a part of the GenABEL package.     

Putative Panmixia in Restricted Populations of Trypanosoma cruzi Isolated from Wild Triatoma infestans in Bolivia

Trypanosoma cruzi, the causative agent of Chagas disease, is subdivided into six discrete typing units (DTUs; TcI–TcVI) of which TcI is ubiquitous and genetically highly variable. While clonality is the dominant mode of propagation, recombinant events play a significant evolutive role. Recently, foci of wild Triatoma infestans have been described in Bolivia, mainly infected by TcI. Hence, for the first time, we evaluated the level of genetic exchange within TcI natural potentially panmictic populations (single DTU, host, area and sampling time). Seventy-nine TcI stocks from wild T. infestans, belonging to six populations were characterized at eight microsatellite loci. For each population, Hardy-Weinberg equilibrium (HWE), linkage disequilibrium (LD), and presence of repeated multilocus genotypes (MLG) were analyzed by using a total of seven statistics, to test the null hypothesis of panmixia (H₀). For three populations, none of the seven statistics allowed to rejecting H₀; for another one the low size did not allow us to conclude, and for the two others the tests have given contradictory results. Interestingly, apparent panmixia was only observed in very restricted areas, and was not observed when grouping populations distant of only two kilometers or more. Nevertheless it is worth stressing that for the statistic tests of "HWE", in order to minimize the type I error (i. e. incorrect rejection of a true H₀), we used the Bonferroni correction (BC) known to considerably increase the type II error ( i. e. failure to reject a false H₀). For the other tests (LD and MLG), we did not use BC and the risk of type II error in these cases was acceptable. Thus, these results should be considered as a good indicator of the existence of panmixia in wild environment but this must be confirmed on larger samples to reduce the risk of type II error.     

metasim 1.0: an individual‐based environment for simulating population genetics of complex population dynamics

metasim provides a flexible environment in which to perform individual‐based population genetic simulations. A wide range of landscape‐level dynamics, population structures, and within‐population demographies can be represented using the framework implemented in this software. In addition, temporal variation in all demographic characteristics can be simulated, both deterministically and stochastically. Such simulations can be used to produce null distributions of genotypes under realistic conditions. These genotypic data can then be used by a variety of analytical programs to develop null expectations of any population genetic statistic estimated from genotypic data.     

The estimation of Wright's fixation index from genotypic frequencies

For populations in which the genotypic frequencies are estimated, it is customary to test the deviation of the estimates from those predicted by the Hardy-Weinberg Law using the x² goodness-of-fit statistic. Maximum likelihood estimation ofWright's F statistic and its variance furnishes a similar test. Following the derivation of the variance of , this note reviews the relation between the two tests. The F statistic possesses the advantages of full sufficiency and an easier formulation of the power function. Its bias is negligible when sample size exceeds 20.     

Simple allelic-phenotype diversity and differentiation statistics for allopolyploids

The analysis of genetic diversity within and between populations is a routine task in the study of diploid organisms. However, population genetic studies of polyploid organisms have been hampered by difficulties associated with scoring and interpreting molecular data. This occurs because the presence of multiple alleles at each locus often precludes the measurement of genotype or allele frequencies. In allopolyploids, the problem is compounded because genetically distinct isoloci frequently share alleles. As a result, analysis of genetic diversity patterns in allopolyploids has tended to rely on the interpretation of phenotype frequencies, which loses information available from allele composition. Here, we propose the use of a simple allelic-phenotype diversity statistic (H') that measures diversity as the average number of alleles by which pairs of individuals differ. This statistic can be extended to a population differentiation measure (F'ST), which is analogous to FST. We illustrate the behaviour of these statistics using coalescent computer simulations that show that F'ST behaves in a qualitatively similar way to FST, thus providing a useful way to quantify population differentiation in allopolyploid species.     

Population genetics of the fissiparous holothurian Stichopus chloronotus (Aspidochirotida) on the Great Barrier Reef, Australia

 Population genetic structure was studied in one nearshore and two offshore populations of Stichopus chloronotus, a common holothurian species on Indo-Pacific coral reefs. Genetic variation at five polymorphic loci was examined using allozyme electrophoresis. The nearshore population consisted almost exclusively of male individuals, and more males than females were found in all populations studied. Deviations of heterozygosity from that predicted under Hardy-Weinberg equilibrium indicated that asexual reproduction occurred in all populations. Estimates of the level of asexual reproduction using the ratios of the number of sexually produced individuals to sample size, observed genotypic diversity to expected genotypic diversity, and number of genotypes to sample size confirmed that this reproductive mode was more important at the nearshore reef compared to the two offshore reefs. There were large differences in genotypic frequencies between males and females. F-statistics on clonal genotypic frequencies were not statistically significant between populations for neither females or males, suggesting high dispersal of larvae between reefs. A higher mortality of females during larval or early post-settlement stages, or reduced dispersal capability of female larvae are the most likely reasons for biased sex ratios. Accepted: 23 November 1998     

Analyses of Nuclear ldhA Gene and mtDNA Control Region Sequences of Atlantic Northern Bluefin Tuna Populations

There has been considerable debate about whether the Atlantic northern bluefin tuna exist as a single panmictic unit. We have addressed this issue by examining both mitochondrial DNA control region nucleotide sequences and nuclear gene ldhA allele frequencies in replicate size or year class samples of northern bluefin tuna from the Mediterranean Sea and the northwestern Atlantic Ocean. Pairwise comparisons of multiple year class samples from the 2 regions provided no evidence for population subdivision. Similarly, analyses of molecular variance of both mitochondrial and ldhA data revealed no significant differences among or between samples from the 2 regions. These results demonstrate the importance of analyzing multiple year classes and large sample sizes to obtain accurate estimates when using allele frequencies to characterize a population. It is important to note that the absence of genetic evidence for population substructure does not unilaterally constitute evidence of a single panmictic population, as genetic differentiation can be prevented by large population sizes and by migration.     

Effect of population stratification on case-control association studies. I. Elevation in false positive rates and comparison to confounding risk ratios (a simulation study)

OBJECTIVES: This is the first of two articles discussing the effect of population stratification on the type I error rate (i.e., false positive rate). This paper focuses on the confounding risk ratio (CRR). It is accepted that population stratification (PS) can produce false positive results in case-control genetic association. However, which values of population parameters lead to an increase in type I error rate is unknown. Some believe PS does not represent a serious concern, whereas others believe that PS may contribute to contradictory findings in genetic association. We used computer simulations to estimate the effect of PS on type I error rate over a wide range of disease frequencies and marker allele frequencies, and we compared the observed type I error rate to the magnitude of the confounding risk ratio. METHODS: We simulated two populations and mixed them to produce a combined population, specifying 160 different combinations of input parameters (disease prevalences and marker allele frequencies in the two populations). From the combined populations, we selected 5000 case-control datasets, each with either 50, 100, or 300 cases and controls, and determined the type I error rate. In all simulations, the marker allele and disease were independent (i.e., no association). RESULTS: The type I error rate is not substantially affected by changes in the disease prevalence per se. We found that the CRR provides a relatively poor indicator of the magnitude of the increase in type I error rate. We also derived a simple mathematical quantity, Delta, that is highly correlated with the type I error rate. In the companion article (part II, in this issue), we extend this work to multiple subpopulations and unequal sampling proportions. CONCLUSION: Based on these results, realistic combinations of disease prevalences and marker allele frequencies can substantially increase the probability of finding false evidence of marker disease associations. Furthermore, the CRR does not indicate when this will occur.     

A Model for Analysis of Population Structure

Arguments have been presented for the appropriateness of a multinomial Dirichlet distribution for describing single-locus genotypic frequencies in a subdivided population. This distribution is defined as a function of allele frequency, the average (over the entire population) inbreeding coefficient and the correlation between genotypes within a subdivision. Alternative parameterizations and their genetic interpretations are given.-We then show how information from a sample drawn from this subdivided population, in the absence of pedigrees, can be combined with the multinomial Dirichlet model to form a likelihood function. This likelihood function is then used as the basis for estimation and testing hypotheses concerning the genetic parameters of the model. Comparisons of this approach to the alternative procedure of Cockerham (1969) and (1973) are made using human data obtained from Tecumseh, Michigan and Monte Carlo simulations.-Finally, implications of these results to statistical inference and to mutation rates are presented.     

Statistical properties of Teng and Risch's sibship type tests for detecting an association between disease and a candidate allele

Risch and Teng [Genome Res 1998;8:1273-1288] and Teng and Risch [Genome Res 1999;9:234-241] proposed a class of transmission/disequilibrium test-like statistical tests based on the difference between the estimated allele frequencies in the affected and control populations. They evaluated the power of a variety of family-based and nonfamily-based designs for detecting an association between a candidate allele and disease. Because they were concerned with diseases with low penetrances, their power calculations assumed that unaffected individuals can be treated as a random sample from the population. They predicted that this assumption rendered their sample size calculations slightly conservative. We generalize their partial ascertainment conditioning by including the status of the unaffected sibs in the calculations of the distribution and power of the statistic used to compare the allele frequency in affected offspring to the estimated frequency in the parents, based on sibships with genotyped affected and unaffected sibs. Sample size formulas for our full ascertainment methods are presented. The sample sizes for our procedure are compared to those of Teng and Risch. The numerical results and simulations indicate that the simplifying assumption used in Teng and Risch can produce both conservative and anticonservative results. The magnitude of the difference between the sample sizes needed by their partial ascertainment approximation and the full ascertainment is small in the circumstances they focused on but can be appreciable in others, especially when the baseline penetrances are moderate. Two other statistics, using different estimators for the variance of the basic statistic comparing the allele frequencies in the affected and unaffected sibs are introduced. One of them incorporates an estimate of the null variance obtained from an auxiliary sample and appears to noticeably decrease the sample sizes required to achieve a prespecified power.     

Gene identity and genetic differentiation of populations in the finite island model

A formula for the variance of gene identity (homozygosity) was derived for the case of neutral mutations using diffusion approximations for the changes of gene frequencies in a subdivided population. It is shown that when gene flow is extremely small, the variance of gene identity for the entire population at equilibrium is smaller than that of the panmictic population with the same mean gene identity. On the other hand, although a large amount of gene flow makes a subdivided population equivalent to a panmictic population, there is an intermediate range of gene flow in which population subdivision can increase the variance. This increase results from the increased variance between colonies. In such a case, each colony has a predominant allele, but the predominant type may differ from colony to colony. The formula for obtaining the variance allows us to study such statistics as the coefficient of gene differentiation and the correlation of heterozygosity. Computer simulations were conducted to study the distribution of gene identity as well as to check the validity of the analytical formulas. Effects of selection were also studied by simulations.     

Genetic structure of a population sample of apomictic dandelions

In Northern Europe, dandelion populations consist solely of triploid or higher polyploid apomicts. Without a regular sexual cycle or lateral gene transmission, a clonal structure is expected for Taraxacum apomicts, although this was not found by compatibility analysis. In this study, we investigate whether this observation could be suported by performing independent tests based on data from hypervariable microsatellite markers as well as more conservative data based on allozymes and matrilinear cpDNA markers. In addition, population genetic methods were used to test departure from panmictic expectations, which is expected for clonal populations. Results indicated that many data sets, again, did not agree with expectations from clonal evolution because only small groups of genotypes exhibit no marker incompatibility. Population genetic analysis revealed that virtually all genotypes, but not individuals, agreed with random segregation and genotypic equilibria. Exceptions were genotypes with rare allozyme alleles or nearly identical microsatellite genotypes. Consequently, a population sample of apomictic dandelions essentially harbours genotypes that resulted from segregation and/or recombination and only a few genotypes that may have differentiated by somatic mutations.     

Homogeneity test of rate ratios in stratified matched-pair studies

This paper investigates homogeneity test of rate ratios in stratified matched-pair studies on the basis of asymptotic and bootstrap-resampling methods. Based on the efficient score approach, we develop a simple and computationally tractable score test statistic. Several other homogeneity test statistics are also proposed on the basis of the weighted least-squares estimate and logarithmic transformation. Sample size formulae are derived to guarantee a pre-specified power for the proposed tests at the pre-given significance level. Empirical results confirm that (i) the modified score statistic based on the bootstrap-resampling method performs better in the sense that its empirical type I error rate is much closer to the pre-specified nominal level than those of other tests and its power is greater than those of other tests, and is hence recommended, whilst the statistics based on the weighted least-squares estimate and logarithmic transformation are slightly conservative under some of the considered settings; (ii) the derived sample size formulae are rather accurate in the sense that their empirical powers obtained from the estimated sample sizes are very close to the pre-specified nominal powers. A real example is used to illustrate the proposed methodologies.     

PHYSIOLOGICAL AND BEHAVIORAL ADAPTATION TO VARYING ENVIRONMENTS: A MATHEMATICAL MODEL

We develop a mathematical model to explore the evolution of habitat selection and physiological adaptation in a heterogeneous environment. The model assumes the following conditions: 1) a panmictic population of infinite size; 2) prereproductive individuals mobile enough to move between patches; 3) alleles at one locus code for absence or presence of adaptation to detrimental patches; 4) alleles at a second locus code for absence or presence of behavior(s) that cause avoidance of the detrimental patches; 5) additive effects of alleles controlling physiology and behavior; 6) frequency-independent fitness. Results of the model indicate that nontrivial, polymorphic equilibria do not exist. The pattern of genotypic fitnesses and the initial allelic frequencies can influence whether the population adapts by physiological or behavioral mechanisms, or by both. Linkage between the two loci can alter the outcome of evolution, given specified genotypic fitness values and initial allelic frequencies.     

CONTRIBUTIONS OF SEXUAL AND ASEXUAL REPRODUCTION TO POPULATION STRUCTURE IN THE CLONAL SOFT CORAL,ALCYONIUM RUDYI

Numerous studies of population structure in sessile clonal marine invertebrates have demonstrated low genotypic diversity and nonequilibrium genotype frequencies within local populations that are monopolized by relatively few, highly replicated genets. All of the species studied to date produce planktonic sexual propagules capable of dispersing long distances; despite local genotypic disequilibria, populations are often panmictic over large geographic areas. The population structure paradigm these species represent may not be typical of the majority of clonal invertebrate groups, however, which are believed to produce highly philopatric sexual propagules. I used allozyme variation to examine the population structure of the temperate soft coral, Alcyonium rudyi, a typical clonal species whose sexually produced larvae and asexually produced ramets both have very low dispersal capabilities. Like other clonal plants and invertebrates, the local population dynamics of A. rudyi are dominated by asexual reproduction, and recruitment of new sexually produced genets occurs infrequently. As expected from its philopatric larval stage, estimates of genetic differentiation among populations of A. rudyi were highly significant at all spatial scales examined (mean θ = 0.300 among 20 populations spanning a 1100-km range), suggesting that genetic exchange seldom occurs among populations separated by as little as a few hundred meters. Mapping of multilocus allozyme genotypes within a dense aggregation of A. rudyi ramets confirmed that dispersal of asexual propagules is also very limited: members of the same genet usually remain within < 50 cm of one another on the same rock surface. Unlike most previously studied clonal invertebrates, populations of A. rudyi do not appear to be dominated by a few widespread genets: estimates of genotypic diversity (G_o) within 20 geographically distinct populations did not differ from expectations for outcrossing, sexual populations. Despite theoretical suggestions that philopatric dispersal combined with typically small effective population sizes should promote inbreeding in clonal species, inbreeding does not appear to contribute significantly to the population structure of A. rudyi. Genet genotype frequencies conformed to Hardy-Weinberg expectations in all populations, and inbreeding coefficients (f) were close to zero. In general, the population structure of A. rudyi did not differ significantly from that observed among outcrossing sexual species with philopatric larval dispersal. Age estimates suggest, however, that genets of A. rudyi live for many decades. Genet longevity may promote high genotypic diversity within A. rudyi populations and may be the most important evolutionary consequence of clonal reproduction in this species and the many others that share its dispersal characteristics.     

Sample size calculations for noninferiority trials with Poisson distributed count data

Clinical trials with Poisson distributed count data as the primary outcome are common in various medical areas such as relapse counts in multiple sclerosis trials or the number of attacks in trials for the treatment of migraine. In this article, we present approximate sample size formulae for testing noninferiority using asymptotic tests which are based on restricted or unrestricted maximum likelihood estimators of the Poisson rates. The Poisson outcomes are allowed to be observed for unequal follow‐up schemes, and both the situations that the noninferiority margin is expressed in terms of the difference and the ratio are considered. The exact type I error rates and powers of these tests are evaluated and the accuracy of the approximate sample size formulae is examined. The test statistic using the restricted maximum likelihood estimators (for the difference test problem) and the test statistic that is based on the logarithmic transformation and employs the maximum likelihood estimators (for the ratio test problem) show favorable type I error control and can be recommended for practical application. The approximate sample size formulae show high accuracy even for small sample sizes and provide power values identical or close to the aspired ones. The methods are illustrated by a clinical trial example from anesthesia.