Additive Genetic Covariance Matrices Research Articles

Exploiting genetic variability in the trajectory of lactation yield and somatic cell score with each progressing parity

The inclusion of reproductive performance in dairy cow breeding schemes has resulted in a cumulative improvement in genetic merit for reproductive performance; this improvement should manifest in longer productive lives through a reduced requirement for involuntary culling. Nonetheless, the average length of dairy cow productive life has not changed in most populations, suggesting that risk factors for culling, especially in older cows, are possibly more associated with lower yield or high somatic cell score (SCS) than compromised reproductive performance. The objective of the present study was to understand the dynamics of lactation yields and SCS in dairy cows across parities and, in doing so, quantify the potential to alter this trajectory through breeding. After edits, 3,470,520 305-d milk, fat, and protein yields, as well as milk fat and protein percentage and somatic cell count records from 1,162,473 dairy cows were available for analysis. Random regression animal models were used to identify the parity in which individual cows reached their maximum lactation yields, and highest average milk composition and SCS; also estimated from these models were the (co)variance components for yield, composition, and SCS per parity across parities. Estimated breeding values for all traits per parity were calculated for cows reaching ≥fifth parity. Of the cows included in the analyses, 91.0%, 92.2%, and 83.4% reached maximum milk, fat, and protein yield in fifth parity, respectively. Conversely, 95.9% of cows reached their highest average fat percentage in first parity and 62.9% of cows reached their highest average protein percentage in third parity. In contrast to both milk yield and composition traits, 98.4% of cows reached their highest average SCS in eighth parity. Individual parity estimates of heritability for milk yield traits, milk composition, and SCS ranged from 0.28 to 0.44, 0.47 to 0.69, and 0.13 to 0.23, respectively. The strength of the genetic correlations per trait among parities was inversely related to the interval between the parities compared; the weakest genetic correlation was 0.67 (standard error = 0.02) between milk yield in parities 1 and 8. Eigenvalues and eigenfunctions of the additive genetic covariance matrices for all investigated traits revealed potential to alter the trajectory of parity profiles for milk yield, milk composition, and SCS. This was further demonstrated when evaluating the trajectories of animal estimated breeding values per parity.

Genomic mating in outbred species: predicting cross usefulness with additive and total genetic covariance matrices.

Diverse crops are both outbred and clonally propagated. Breeders typically use truncation selection of parents and invest significant time, land, and money evaluating the progeny of crosses to find exceptional genotypes. We developed and tested genomic mate selection criteria suitable for organisms of arbitrary homozygosity level where the full-sibling progeny are of direct interest as future parents and/or cultivars. We extended cross variance and covariance variance prediction to include dominance effects and predicted the multivariate selection index genetic variance of crosses based on haplotypes of proposed parents, marker effects, and recombination frequencies. We combined the predicted mean and variance into usefulness criteria for parent and variety development. We present an empirical study of cassava (Manihot esculenta), a staple tropical root crop. We assessed the potential to predict the multivariate genetic distribution (means, variances, and trait covariances) of 462 cassava families in terms of additive and total value using cross-validation. Most variance (89%) and covariance (70%) prediction accuracy estimates were greater than zero. The usefulness of crosses was accurately predicted with good correspondence between the predicted and the actual mean performance of family members breeders selected for advancement as new parents and candidate varieties. We also used a directional dominance model to quantify significant inbreeding depression for most traits. We predicted 47,083 possible crosses of 306 parents and contrasted them to those previously tested to show how mate selection can reveal the new potential within the germplasm. We enable breeders to consider the potential of crosses to produce future parents (progeny with top breeding values) and varieties (progeny with top own performance).

The Skull Integration Pattern and Internal Constraints in Myotis myotis–Myotis blythii Species Group (Vespertilionidae, Chiroptera) Might be Shaped by Natural Selection During Evolution Along the Genetic Line of Least Resistance

Evolutionary dynamics of covariation patterns in craniometric traits was studied for bats from the Myotis myotis–Myotis blythii species group (Mammalia, Chiroptera, Vespertilionidae) namely M.m.myotis, M.b.oxygnathus, M.b.omari, M.b.blythii, and M.b.altaicus. These species evolved towards increasing the size. One more species, M.dasycneme, which is phylogenetically, morphologically, and ecologically rather distant from these OTUs, also was studied for comparison with them. A set of 30 craniometric traits was studied for each of the OTUs using the quantitative genetics approaches. Phenotypic covariance matrices were used as proxies for additive genetic covariance matrices. The analysis has shown that multivariate divergence of the studied Myotis OTUs was governed by selection rather than by random drift and that these OTUs evolved in the direction close to the line of least evolutionary resistance. As overall size increases, the skull integration, respondability, and evolvability increase, while flexibility slightly decreases. Such a pattern can possibly be explained as a consequence of adaptation of M.blythii and especially M.myotis to prey on considerably large hard-shelled insects. A high cohesion of skull structures is needed for the effective functioning of their jaw apparatus. For Myotis bats with other foraging strategies (such as M. dasycneme that catches insects mainly using the claws of hind feet) this cohesion seems to be of lesser importance.

Multi-Trait analysis of growth traits: fitting reduced rank models using principal components for Simmental beef cattle

ABSTRACT: The aim of this research was to evaluate the dimensional reduction of additive direct genetic covariance matrices in genetic evaluations of growth traits (range 100-730 days) in Simmental cattle using principal components, as well as to estimate (co)variance components and genetic parameters. Principal component analyses were conducted for five different models-one full and four reduced-rank models. Models were compared using Akaike information (AIC) and Bayesian information (BIC) criteria. Variance components and genetic parameters were estimated by restricted maximum likelihood (REML). The AIC and BIC values were similar among models. This indicated that parsimonious models could be used in genetic evaluations in Simmental cattle. The first principal component explained more than 96% of total variance in both models. Heritability estimates were higher for advanced ages and varied from 0.05 (100 days) to 0.30 (730 days). Genetic correlation estimates were similar in both models regardless of magnitude and number of principal components. The first principal component was sufficient to explain almost all genetic variance. Furthermore, genetic parameter similarities and lower computational requirements allowed for parsimonious models in genetic evaluations of growth traits in Simmental cattle.

Dominance genetic variance for traits under directional selection in Drosophila serrata.

In contrast to our growing understanding of patterns of additive genetic variance in single- and multi-trait combinations, the relative contribution of nonadditive genetic variance, particularly dominance variance, to multivariate phenotypes is largely unknown. While mechanisms for the evolution of dominance genetic variance have been, and to some degree remain, subject to debate, the pervasiveness of dominance is widely recognized and may play a key role in several evolutionary processes. Theoretical and empirical evidence suggests that the contribution of dominance variance to phenotypic variance may increase with the correlation between a trait and fitness; however, direct tests of this hypothesis are few. Using a multigenerational breeding design in an unmanipulated population of Drosophila serrata, we estimated additive and dominance genetic covariance matrices for multivariate wing-shape phenotypes, together with a comprehensive measure of fitness, to determine whether there is an association between directional selection and dominance variance. Fitness, a trait unequivocally under directional selection, had no detectable additive genetic variance, but significant dominance genetic variance contributing 32% of the phenotypic variance. For single and multivariate morphological traits, however, no relationship was observed between trait-fitness correlations and dominance variance. A similar proportion of additive and dominance variance was found to contribute to phenotypic variance for single traits, and double the amount of additive compared to dominance variance was found for the multivariate trait combination under directional selection. These data suggest that for many fitness components a positive association between directional selection and dominance genetic variance may not be expected.

Impact of variation in length of individual testing periods on estimation of (co)variance components of a random regression model for feed intake of growing pigs

A simulation study was conducted to assess the influence of differences in the length of individual testing periods on estimates of (co)variance components of a random regression model for daily feed intake of growing pigs performance tested between 30 and 100 kg live weight. A quadratic polynomial in days on test with fixed regressions for sex, random regressions for additive genetic and permanent environmental effects and a constant residual variance was used for a bivariate simulation of feed intake and daily gain. (Co)variance components were estimated for feed intake only by means of a Bayesian analysis using Gibbs sampling and restricted maximum likelihood (REML). A single trait random regression model analogous to the one used for data simulation was used to analyse two versions of the data: full data sets with 18 weekly means of feed intake per animal and reduced data sets with the individual length of testing periods determined when tested animals reached 100 kg live weight. Only one significant difference between estimates from full and reduced data (REML estimate of genetic covariance between linear and quadratic regression parameters) and two significant differences from expected values (Gibbs estimates of permanent environmental variance of quadratic regression parameters) occurred. These differences are believed to be negligible, as the number lies within the expected range of type I error when testing at the 5% level. The course of test day variances calculated from estimates of additive genetic and permanent environmental covariance matrices also supports the conclusion that no bias in estimates of (co)variance components occurs due to the individual length of testing periods of performance‐tested growing pigs. A lower number of records per tested animal only results in more variation among estimates of (co)variance components from reduced compared with full data sets. Compared with the full data, the effective sample size of Gibbs samples from the reduced data decreased to 18% for residual variance and increased up to five times for other (co)variances. The data structure seems to influence the mixing of Gibbs chains.

Multivariate segregation analysis using the mixed model

Most major genes involved in the etiology of complex diseases are likely to have pleiotropic effects on a number of intervening quantitative traits. Methods of segregation analysis that incorporate the additional information from such multiple traits will exhibit greater power for detecting the effects of major genes and allow explicit tests of major locus pleiotropy hypotheses. In this study, we present a new method for multivariate segregation analysis that utilizes a multivariate generalization of Hasstedt's [1982] technique for calculating approximate mixed model likelihoods on pedigrees. The method is based on a simplification of the multivariate conditional likelihood via a transformation that simultaneously orthogonalizes the residual additive genetic and environmental covariance matrices. This transformation allows the multivariate conditional likelihood to be factored into the product of independent univariate conditional likelihoods. Resulting computations are relatively fast, making it feasible to analyze multiple traits in extended pedigrees. We demonstrate our method with a bivariate analysis of high-density lipoprotein cholesterol (HDL-C) and apolipoprotein AI (apo AI) serum levels in 585 pedigreed baboons.

A Note on Multiple Solutions in Multivariate Restricted Maximum Likelihood Covariance Component Estimation

Restricted maximum likelihood has gained wide acceptance for estimating variance components by animal breeders. Converged solutions from maximization algorithms are usually considered REML solutions. However, neither the expectation maximization nor the derivative-free downhill simplex algorithm guarantees convergence to the global maximum. To investigate the occurrence of multiple solutions, residual and additive genetic covariance matrices for two traits were estimated by the derivative-free approach using the downhill simplex method and an EM algorithm for a multivariate animal model. Expectation maximization yielded three different converged solutions for the data set used, while downhill simplex identified two of the three expectation maximization solutions. Multiple solutions from both expectation maximization and downhill simplex suggest the existence of local maxima. Visualization of the log-likelihood surface however, did not support this view. Instead, points of inflection in some of the dimensions could be the cause for multiple solutions. The results prove that multiple solutions can exist, casting doubt on the merit of algorithms—like expectation maximization and downhill simplex—that do not guarantee global maximization. Downhill simplex was superior to expectation maximization. In terms of central processing unit time it was faster by a factor of 22 misidentifying only one solution instead of 2 as expectation maximization did.

Selection index: economic weights for maximum simultaneous genetic gain

Selection indices that maximize the correlation between an individual organism's index score and its breeding value frequently require a priori known "economic" weights before the optimum phenotypic weights can be estimated. The long generation intervals and economic uncertainty that surround forest tree breeding can make the choice of weights arbitrary. In this paper an algorithm is introduced for finding "economic" weights that will ensure maximum simultaneous progress in all index traits. At the outset the traits are assumed to be of equal preference. The solutions are functions of the eigenvalues and eigenvectors of a quadratic form of the additive genetic and phenotypic covariance matrices. Examples of applications in tree breeding emphasize the practical aspects of the method.

EFFECT OF AN EXPERIMENTAL BOTTLENECK ON MORPHOLOGICAL INTEGRATION IN THE HOUSEFLY

Three measures of multivariate integration were derived from both additive genetic covariance and correlation matrices estimated from parent-offspring covariances to investigate the effect of bottlenecks of different sizes on genetic integration of morphological traits in the housefly, Musca domestica L. Bottleneck lines were initiated with one, four, or 16 pairs of flies sampled from a natural outbred (control) population. Bottlenecks of intermediate size significantly increased the average genetic correlation among traits, resulting in nearly isomorphic variation among all traits in these lines. Single-pair bottlenecks significantly disrupted the trait interrelationships, and the suites of traits identified by principal components of the additive genetic correlation and covariance matrices for the control population were no longer evident in these bottleneck lines. The alteration of the genetic relationships among traits as a result of a bottleneck suggests that nonadditive components of genetic variation affecting these traits were present in the control line. We discuss the implications of nonadditive gene action, particularly epistasis, for speciation via bottlenecks.

QUANTITATIVE GENETICS OF MORPHOLOGICAL DIFFERENTIATION IN PEROMYSCUS. I. TESTS OF THE HOMOGENEITY OF GENETIC COVARIANCE STRUCTURE AMONG SPECIES AND SUBSPECIES.

Phenotypic and additive genetic covariance matrices were estimated for 15 morphometric characters in three species and subspecies of Peromyscus. Univariate and multivariate ANOVAs indicate these groups are highly diverged in all characters, P. leucopus having the largest body size, P. maniculatus bairdii the smallest, and P. maniculatus nebrascensis being intermediate. Comparing the structure of P and G within each taxon revealed significant similarities in all three cases. This proportionality was strong enough to justify using P in the place of G to analyze evolutionary processes using quantitative genetic models when G can not be estimated, as in fossil material. However, the similarity between genetic and phenotypic covariance structures is sufficiently low that estimates of the genetic parameters should be used when possible. The additive genetic covariance matrices were compared to examine the assumption that they remain constant during evolution, an assumption which underlies many applications of quantitative-genetic models. While matrix permutation tests indicated statistically significant proportionality between the genetic covariance structures of the two P. maniculatus subspecies, there is no evidence of significant genetic structural similarity between species. This result suggests that the assumption of constant genetic covariance structure may be valid only within species. (It does not, however, necessarily imply a causal relationship between speciation and heterogeneity of genetic covariance structures.) The low matrix correlation for the two P. maniculatus subspecies' genetic covariance matrices indicates G may not be functionally constant, even within species. The lack of similarity observed here may be due partly to sampling variation.

Exact first- and second-order moments of estimates of components of covariance

SUMMARY This paper develops formulae for the first two moments of estimates of covariance for the general multivariate 'one-way' and 'two-way' models. The results are used to obtain the large sample dispersion matrix for estimated coefficients of two types of genetic selection indexes. These dispersion matrices provide the necessary extension of known results in balanced models to the unbalanced case. The problems of estimation associated with variance and covariance component analysis with unbalanced data have been of major concern to the statistical geneticist. This is primarily because most of the selection procedures applied to livestock require a knowledge of genetic variances and covariances which usually have to be estimated from the analyses of hierarchical models. Invariably, there is a marked lack of balance in the data thus rendering standard formulae for the variances of the estimates inapplicable. Serious consideration to these problems has been given by Henderson (1953), Searle (1956) and Hartley & Rao (1967). Searle gave particular attention to the one-way analysis of variance and covariance and used matrix methods to calculate the moments of the various estimators. Other work in this area concerns analysis of variance models of varying complexity; see Searle (1958, 1961), Mahamunulu (1963) and Blischke (1966). It is the purpose of the present paper to extend and complement existing results. With the advent of high speed computers, matrix operations can be handled with great speed and hence formulae for expectations and covariances of sums of squares and products can be left in a general computable form. Thus, explicit algebraic evaluation of each case is, in most cases, not only time-consuming but unnecessary. We consider here the general one-way and two-way analysis of covariance model with fixed and random effects. The number of variables included in the analysis is assumed to be arbitrary and this seems to lead to somewhat involved notation and algebra. However, general results are required in order to solve a number of practical problems. We give two examples from statistical genetics. In the theory of animal breeding interest centres around certain phenotypic and genetic parameters. Suppose that k characters of a particular breed of animal are relevant from the point of view of a selection programme. Then we let P and G be the phenotypic and additive genetic covariance matrices for the k characters and we consider two types of selection index which are based on these matrices.

Selection for an Optimum Growth Curve

Growth and development can be regarded as a stochastic process in continuous time. Moreover, in some situations of primary production, certain growth patterns may be more economical, or otherwise more desirable, than others. In this paper an attempt is made to develop techniques which could be used to exert some selection pressure for optimal growth curves. A discrete solution to this problem is suggested which should be relatively easy to apply in practice. However, the complete discussion of the situation requires the introduction of certain integral equations to replace the usual matrix equations of the classical theory. The conventional phenotypic and additive genetic covariance matrices give way to continuous kernels and, as expected, it is found that the continuous and the discrete theories are similar. The usual case of unrestricted selection is also developed for the continuous time model and the selection of several characters towards respective optimal curves simultaneously is also treated. Two different numerical procedures for solving the integral equations are proposed.