Genetic Variance-covariance Matrices Research Articles

Sampling distributions, biases, variances, and confidence intervals for genetic correlations.

Genetic correlations (rho ( g )) are frequently estimated from natural and experimental populations, yet many of the statistical properties of estimators of rho ( g ) are not known, and accurate methods have not been described for estimating the precision of estimates of rho ( g ). Our objective was to assess the statistical properties of multivariate analysis of variance (MANOVA), restricted maximum likelihood (REML), and maximum likelihood (ML) estimators of rho ( g ) by simulating bivariate normal samples for the one-way balanced linear model. We estimated probabilities of non-positive definite MANOVA estimates of genetic variance-covariance matrices and biases and variances of MANOVA, REML, and ML estimators of rho ( g ), and assessed the accuracy of parametric, jackknife, and bootstrap variance and confidence interval estimators for rho ( g ). MANOVA estimates of rho ( g ) were normally distributed. REML and ML estimates were normally distributed for rho ( g ) = 0.1, but skewed for rho ( g ) = 0.5 and 0.9. All of the estimators were biased. The MANOVA estimator was less biased than REML and ML estimators when heritability (H), the number of genotypes (n), and the number of replications (r) were low. The biases were otherwise nearly equal for different estimators and could not be reduced by jackknifing or bootstrapping. The variance of the MANOVA estimator was greater than the variance of the REML or ML estimator for most H, n, and r. Bootstrapping produced estimates of the variance of rho ( g ) close to the known variance, especially for REML and ML. The observed coverages of the REML and ML bootstrap interval estimators were consistently close to stated coverages, whereas the observed coverage of the MANOVA bootstrap interval estimator was unsatisfactory for some H, rho ( g ), n, and r. The other interval estimators produced unsatisfactory coverages. REML and ML bootstrap interval estimates were narrower than MANOVA bootstrap interval estimates for most H, rho ( g ), n, and r.

HOMOGENEITY OF THE GENETIC VARIANCE-COVARIANCE MATRIX FOR ANTIPREDATOR TRAITS IN TWO NATURAL POPULATIONS OF THE GARTER SNAKE THAMNOPHIS ORDINOIDES.

Quantitative genetic models of evolution rely on the genetic variance-covariance matrix to predict the phenotypic response to selection. Both prospective and retrospective studies of phenotypic evolution across generations rely on assumptions about the constancy of patterns of genetic covariance through time. In the absence of robust theoretical predictions about the stability of genetic covariances, this assumption must be tested with empirical comparisons of genetic parameters among populations and species. Genetic variance-covariance matrices were estimated for a suite of antipredator traits in two populations of the northwestern garter snake, Thamnophis ordinoides. The characters studied include color pattern and antipredator behaviors that interact to facilitate escape from predators. Significant heritabilities for all traits were detected in both populations. Genetic correlations and covariances were found among behaviors in both populations and between color pattern and behavior in one of the populations. Phenotypic means differed among populations, but pairwise comparisons revealed no heterogeneity of genetic parameters between the populations. The structure of the genetic variance-covariance matrix has apparently not changed significantly during the divergence of these two populations.

Bending of Parameter Estimates from Non-Orthogonal Data to Improve Genetic Gain Through Index Selection

The bending procedure for modification of the estimates of genetic and phenotypic variance-covariance matrices in balanced data (HAYES and HILL, 1981) has been extended to non-orthogonal data (unequal number of daughters per sire) and illustrated with the help of live data. Simulations conducted for different values of bending factor revealed that the expected genetic gain is improved significantly after modification of parameters.

Early selection in tree breeding: principles for applying index selection and inferring input parameters

Early selection is essential after the first generation of a tree-breeding programme. Two main issues that it raises are (i) the optimal age for selection for the next generation and (ii) how best to use early assessment data for the selection. Attention is given mainly to the latter issue. Early selection is considered as a special type of indirect selection that can embrace multiple traits. Where a sequence of early measurements has been made for single traits it is possible to treat the expressions of a trait at different dates as separate traits in a selection index. Constructing a least-squares selection index with this feature involves a number of questions. Outstanding among these is whether, and in what way, early measurements should be used as covariates for adjusting later ones. Such indices depend, however, on good estimates of phenotypic and genetic variance–covariance matrices. Potential aids for deriving and refining such matrices are reviewed, including the "Lambeth relationship" for age–age correlations and partial correlation analysis. However, analysis of the sensitivity of expected gains with respect to variations in assumed genetic parameters is strongly advocated before decisions are made on age of selection.

Design of multivariate selection experiments to estimate genetic parameters

Precise, unbiased estimates of genetic parameters, such as heritability and genetic covariance, are necessary to optimise breeding programs and to predict rates of change for various selection schemes. A classical method of estimation is to use high and low selection experiments. We consider two generation selection experiments when observations in the parental generation are only taken on one sex, resulting in half-sib family information. We consider cases of two standardised traits, with zero mean and unit variance, and assume that the traits are normally distributed. The genetic and phenotypic variance-covariance matrices for the traits are denoted by G and P respectively. The genetic variances and covariances of the standardised traits are then heritabilities and co-heritabilities.The construction of other designs is examined, assuming the phenotypic correlation of rp between traits is known. For simplicity, comparisons of designs are developed by considering the variance of the genetic variances when the traits are uncorrelated both phenotypically and genetically.

Quantitative Genetics in Natural Plant Populations: A Review of the Theory

The methods of quantitative genetics may be useful for quantifying constraints on the rate and direction of response to selection in natural plant populations. We review the assumptions necessary for accurate estimation and significance testing of quantitative-genetic parameters, and consider when such estimates may accurately predict a response to selection. Estimates of heritabilities and genetic correlations can provide good predictions of a long-term response to selection if (1) such estimates are reasonably accurate; (2) many genes contribute to genetic variances and covariances; (3) genetic variance-covariance matrices remain approximately constant through time (when major genes are important sources of variation, when selection is strong, or when populations are small, then rapid alterations in gene frequencies may occur, with consequent changes in the quantitative-genetic parameters); (4) genotype-environment interaction does not alter genetic parameters in new or unmeasured environments; and (5) populations are not inbred. The importance of these assumptions depends on the evolutionary scale of the questions we wish to ask. We discuss evolutionary interpretations of heritabilities and genetic correlations, emphasizing that the plasticity of quantitative-genetic parameters and the imprecision of their estimates are serious problems in the study of natural plant populations. It is important to separate the fundamental, long-term genetic constraints, which limit species evolution, from the transient genetic correlations that may be found in a particular population. It is unclear whether natural populations ever reach equilibrium conditions, in which the additive genetic variance for fitness is depleted. Under many circumstances it is difficult to deduce the past history of selection from the genetic architecture of metrical traits.

An Indirect Method of Forming Expressions for the Inverses of Genetic Variance-Covariance Matrices for Two Traits

Practical application of best linear unbiased prediction procedures to large data sets requires efficient use of computer time and storage. One way to increase effiency of computer use is to eliminate extraneous equations, i.e., equations for genetic predictions that are of no or little practical value. Rules are derived for obtaining, by inspection of the pedigree, a matrix representing the inverse of the genetic variance-covariance matrix for two traits in which all genetic values are not represented. The need to form the genetic variance-covariance matrix itself is eliminated. The condition in which the expression is the exact inverse is specified. An illustration is provided. Inbreeding is not considered.