C. Clark Cockerham

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-

VARIANCE OF GENE FREQUENCIES

Inbreeding, gene frequency variance, and their corresponding effective population numbers are now commonplace terms in population genetics. The concepts and much of the theory are classical (Wright, 1921, 1931; Fisher, 1930). More recent refinements and extensions of the theory by Crow and associates (Crow, 1954; Crow and Morton, 1955; Kimura and Crow, 1963a, b) have been primarily concerned with distinguishing between the inbreeding effect on heterozygosity and on the variance of gene frequencies which are so intimately connected in finite populations. The purpose of the present paper is to relate the two in a way which the author thinks is meaningful and easy to grasp. Further, correlational measures are made compatible with probability measures of identity by descent and a simple basis is provided for the analysis of the variance of gene frequencies in experimental or natural populations. The procedure is to work with the variance of a linear function and to incorporate the role that the inbreeding and coancestry of individuals play in this variance. First, let us develop this role. We let aij index the jth allele in the ith individual and introduce a measure of frequency xij defined by

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.

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.

THE GOODNESS-OF-FIT TEST FOR DETECTING NATURAL SELECTION IN RANDOM MATING POPULATIONS

Suppose, now, that a sample is taken from such a population and that the three genotypes are distinguishable. There will be three observed frequencies, nl, ns and ns, but only two independent observations since there is the linear restriction The issue has recently been raised in the pages of this journal and elsewhere as to the possibility of judging the relative fitnesses of genotypes from the frequencies of these genotypes in populations. Wallace (1958) has provided several numerical examples where severe selection pressures do not result in appreciable deviations of the observed zygotic frequencies from expectation under Hardy-Weinberg equilibrium. Novitski and Dempster (1958) attempted by means of a digital computer to estimate the adaptive values of genotypes in Drosophila melanogaster using only the observed genotypic frequencies. They found that the computer produced an infinite variety of best fit values for the fitnesses. We are indebted to them for first calling our attention to this problem and to Dr. Bruce Wallace for urging us to publish our findings. If the frequency of an allele B in a population is p and that of its alternate allele b ia q = 1 p, then following random mating but before any natural selection has occurred the zygotes will be in the relative frequencies

Quadratic analyses of reciprocal crosses.

Three different models, a two-way factorial model for familiarity, an orthogonalizing transform of this model to a diallel model, and a bio model more representative of the biological situation, are interrelated in terms of their components of variance and covariance. It is clarified that there are five components that can be reckoned with in the analysis of reciprocal crosses, including distinct maternal and paternal variances. Estimation of the components and tests of hypotheses concerning them are outlined for two types of mating designs with reciprocals. One deisgn involves a factorial mating design between two distinct sets of parents or parental lines and the other a diallel of all crosses from a single set of parents or parental lines. Both designs provide the same types of information and similar tests of hypotheses. At least some parts of the analyses corresponding to the factorial model are required to separate the maternal and paternal variances. A least squares partitioning of the sums of squares according to the diallel model, but with expectations expressed in terms of the bio model, provides most of the tests of hypotheses of interest. Worked examples are given.

Digenic descent measures for finite populations

The development of a set of two-locus descent measures is reviewed. The three digenic measures, inbreeding coefficient and parental and recombinant descent coefficients, are considered in detail. The derivations of these three in pedigrees, fixed mating systems, and random mating in monoecious or dioecious populations are given. General expressions for digenic frequencies and disequilibria functions at any time are found by applying the three digenic descent measures to two types of initial populations. The final or equilibrium status of the population is also given. As the inbreeding coefficient is the same as the recombinant descent coefficient in the case of complete linkage, avoidance or promotion of early inbreeding has similar effects on the two coefficients. Estimable components of linkage disequilibrium and other measures of association within and among populations are elaborated.

Frequency-Dependent Selection in Randomly Mating Populations

A general model which accommodates different fitness values of an individual in association with each other type of individual is studied. The model in conjunction with population composition, in which individuals are assumed to be dispersed at random, determines the relative fitness values of individuals. Expressions for the change in gene frequency and for the mean are derived for one locus with two alleles in a random mating population. Equilibrium and max/min mean compositions generally require solving third-degree equations, but explicit expressions are found for models with general levels of dominance and some of their variants. Protected polymorphisms, genetic loads, inbreeding depressions, and the cost of a gene substitution are also considered. Many additional features of the statics and dynamics of populations are introduced with the effects of population composition on fitnesses. A single globally stable equilibrium can exist without any dominance. There may be multiple equilibria, up to three, with various stability characteristics, some of which are protected polymorphisms. There need not be any correspondence between compositions for max/min means and those for equilibria. Inbreeding depressions may be curvilinear, and even stable equilibrium populations may show negative inbreeding depressions. Very different models may give the same genetic load. The number of genetic deaths required for a gene substitution may be as few as twice the population size. One appeal of the model is that it encompasses several classes of models, including those with constant fitness values, and thus provides a basis for determining the appropriate model. After considering alternative experiments and measures of discrimination, we conclude that only survival values of the genotypes for several population compositions would be sufficient to discriminate between certain models.

Random and fixed effects in plant genetics.

SummaryA general model for any type of genetic entry is developed which takes into account both the factorial model of gene effects and the ancestral sources, whether inbred lines or outbred varieties, of the genes.Utilizing the model, various genetic designs of fixed entries are explored for the estimation of genetic effects and the testing of genetic hypotheses. These designs consisted of generation means — parents, crosses, various types of backcrosses, and so on — stemming from one or more pairs of parents, and of hybrid combinations from factorial mating designs. Limitations, from the standpoint of genetic effects that can be estimated and genetic hypotheses that can be tested, are developed in considerable detail.When entries from the factorial mating designs are considered to be random, attention is focused on the estimation of genetic variances, rather than effects, and on the concomitant changes in the tests of genetic hypotheses. While there is considerable improvement over fixed entries in the number of types of genetic variances that can be estimated, and of genetic hypotheses that can be tested, they are still very limited in contrast to what would be most desirable.

Descent measures for two loci with some applications.

Abstract For any four genes, two at each of two loci, in a population, a 15 component descent measure has been introduced. These components are the probabilities of the 15 possible arrangements on a set of initial gametes of those genes of which the four of interest are copies. Since identity by descent of genes is equivalent to their being copies of a single gene on an initial gamete, descent measures have inbreeding coefficients as special cases. The individual descent measure, defined for four genes on two uniting gametes can be evaluated for any pedigree by means of an algorithm developed here. If initial gametic frequencies are specified, descent measures allow genotypic frequencies and disequilibria functions at one and two loci to be found. The procedures are illustrated for selfing and for sib mating. Several applications of the descent measures are discussed.