B. S. Weir

ESTIMATING F‐STATISTICS FOR THE ANALYSIS OF POPULATION STRUCTURE

This journal frequently contains papers that report values of F-statistics estimated from genetic data collected from several populations. These parameters, FST, FIT, and FIS, were introduced by Wright (1951), and offer a convenient means of summarizing population structure. While there is some disagreement about the interpretation of the quantities, there is considerably more disagreement on the method of evaluating them. Different authors make different assumptions about sample sizes or numbers of populations and handle the difficulties of multiple alleles and unequal sample sizes in different ways. Wright himself, for example, did not consider the effects of finite sample size. The purpose of this discussion is to offer some unity to various estimation formulae and to point out that correlations of genes in structured populations, with which F-statistics are concerned, are expressed very conveniently with a set of parameters treated by Cockerham (1 969, 1973). We start with the parameters and construct appropriate estimators for them, rather than beginning the discussion with various data functions. The extension of Cockerhams work to multiple alleles and loci will be made explicit, and the use of jackknife procedures for estimating variances will be advocated. All of this may be regarded as an extension of a recent treatment of estimating the coancestry coefficient to serve as a mea-

Plant population genetics, breeding, and genetic resources

A resource for students and research workers in population genetics, molecular evolution, evolutionary biology, ecological genetics, forestry and crop improvement.

Detecting Marker-Disease Association by Testing for Hardy-Weinberg Disequilibrium at a Marker Locus

We review and extend a recent suggestion that fine-scale localization of a disease-susceptibility locus for a complex disease be done on the basis of deviations from Hardy-Weinberg equilibrium among affected individuals. This deviation is driven by linkage disequilibrium between disease and marker loci in the whole population and requires a heterogeneous genetic basis for the disease. A finding of marker-locus Hardy-Weinberg disequilibrium therefore implies disease heterogeneity and marker-disease linkage disequilibrium. Although a lack of departure of Hardy-Weinberg disequilibrium at marker loci implies that disease susceptibilityweighted linkage disequilibria are zero, given disease heterogeneity, it does not follow that the usual measures of linkage disequilibrium are zero. For disease-susceptibility loci with more than two alleles, therefore, care is needed in the drawing of inferences from marker Hardy-Weinberg disequilibria.

Estimation of gene flow from F-statistics

We present theory clarifying the general behavior of FST‐based and GST‐based estimators of gene flow, and confirm these predictions with simulations. In particular, we use the correlation of genes within groups within populations to define an estimator. The theoretical value of the correlation doe not depend on the number of groups in a population, and properties of the estimated correlation do not depend on the number of groups sampled or the number of individuals sampled per group. This invariance is in contrast to properties of GST. For a complete census of a population, bias and variance considerations would suggest the use of the GST‐based estimator of gene flow, but lack of knowledge of population size or group number in practice suggests preference be given to the correlation‐based estimator. We acknowledge that these estimators require that several conditions of a population‐genetic model be met, since they do not make use of direct observations on the flow of genes. Our results differ from some of those based on simulation in a series of recent papers by M. Slatkin.

Tests for Linkage and Association in Nuclear Families

The transmission/disequilibrium test (TDT) originally was introduced to test for linkage between a genetic marker and a disease-susceptibility locus, in the presence of association. Recently, the TDT has been used to test for association in the presence of linkage. The motivation for this is that linkage analysis typically identifies large candidate regions, and further refinement is necessary before a search for the disease gene is begun, on the molecular level. Evidence of association and linkage may indicate which markers in the region are closest to a disease locus. As a test of linkage, transmissions from heterozygous parents to all of their affected children can be included in the TDT; however, the TDT is a valid chi2 test of association only if transmissions to unrelated affected children are used in the analysis. If the sample contains independent nuclear families with multiple affected children, then one procedure that has been used to test for association is to select randomly a single affected child from each sibship and to apply the TDT to those data. As an alternative, we propose two statistics that use data from all of the affected children. The statistics give valid chi2 tests of the null hypothesis of no association or no linkage and generally are more powerful than the TDT with a single, randomly chosen, affected child from each family.

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.

A systematic statistical linear modeling approach to oligonucleotide array experiments.

We outline and describe steps for a statistically rigorous approach to analyzing probe-level Affymetrix GeneChip data. The approach employs classical linear mixed models and operates on a gene-by-gene basis. Forgoing any attempts at gene presence or absence calls, the method simultaneously considers the data across all chips in an experiment. Primary output includes precise estimates of fold change (some as low as 1.1), their statistical significance, and measures of array and probe variability. The method can accommodate complex experiments involving many kinds of treatments and can test for their effects at the probe level. Furthermore, mismatch probe data can be incorporated in different ways or ignored altogether. Data from an ionizing radiation experiment on human cell lines illustrate the key concepts.

Variances and covariances of squared linkage disequilibria in finite populations

Analysis of linkage disequilibrium D among restriction sites or bases in DNA sequences, arising from mutations in finite populations, depends on a knowledge of the variance-covariance structure of measures such as D2 between different pairs of sites. This requires evaluation of the eighth moments of gene frequencies among two, three, and four loci, and the necessary methodology is derived here and results are computed. While primary emphasis is placed on disequilibrium arising from mutation or gene conversion, the methodology also allows for the joint effects of only drift and recombination. Numerical results confirm that squared linkage disequilibria can have high variances and covariances.

Covariances of relatives stemming from a population undergoing mixed self and random mating

We consider covariances of all parent and first-generation relatives from outcrossing or self-fertilization in a parent population that is in equilibrium with respect to these processes. The results, which are for any number of alleles and loci with additive and dominance effects, are phrased in terms of six quadratic genetic components whose coefficients are given by descent measures for equilibrium populations. Because of the variation in the inbreeding coefficients for this system of mating, the expressions include joint contributions of loci to the variances and covariances of relatives. By inclusion of the full complement of relatives, all quadratic components can be estimated. The findings of Ghai (1982, Biometrics 38, 87-92) for compound functions of the covariances with two alleles at a single locus are analyzed in terms of the more general model.

Bayesian statistics in genetics: a guide for the uninitiated

Statistical analyses are used in many fields of genetic research. Most geneticists are taught classical statistics, which includes hypothesis testing, estimation and the construction of confidence intervals; this framework has proved more than satisfactory in many ways. What does a Bayesian framework have to offer geneticists? Its utility lies in offering a more direct approach to some questions and the incorporation of prior information. It can also provide a more straightforward interpretation of results. The utility of a Bayesian perspective, especially for complex problems, is becoming increasingly clear to the statistics community; geneticists are also finding this framework useful and are increasingly utilizing the power of this approach.