Components of Variance Underlying Fitness in a Natural Population of the Great Tit, 'Parus major'
Traits that are closely associated with fitness tend to have lower heritabilities (h²) values than those that are not. This has commonly been interpreted as evidence that natural selection tends to deplete genetic variation more rapidly for traits more closely associated with fitness (a corollary of Fisher’s Fundamental Theorem), but Price and Schluter (1991) suggested the pattern might be due to higher residual variance in traits more closely related to fitness. The relationship between eleven different traits for females, and eight traits for males and overall fitness (lifetime recruitment) was quantified for great tits ('Parus major') studied in their natural environment of Wytham Wood, England, using data collected over 38 years. Heritabilities and the coefficients of additive genetic and residual variance (CVA and CVR respectively) were estimated using an "animal model". In both males and females a trait’s correlation (r) with fitness was negatively related to its h2, but positively related to its CVR. CVA was not related to the traits correlation with fitness in either sex . This is the third study using directly measured fitness in a wild population in a natural environment to show the important role of residual variation in determining the pattern of lower heritabilities for traits more closely related to fitness, as predicted by Price & Schluter (1991).
Penalized maximum likelihood estimates of genetic covariance matrices with shrinkage towards phenotypic dispersion
A simulation study examining the effects of 'regularization' on estimates of genetic covariance matrices for small samples is presented. This is achieved by penalizing the likelihood, and three types of penalties are examined. It is shown that regularized estimation can substantially enhance the accuracy of estimates of genetic parameters. Penalties shrinking estimates of genetic covariances or correlations towards their phenotypic counterparts acted somewhat differently to those aimed reducing the spread of sample eigenvalues. While improvements of estimates were found to be comparable overall, shrinkage of genetic towards phenotypic correlations resulted in least bias.
Cheverud revisited: Scope for joint modelling of genetic and environmental covariance matrices
Multivariate estimation fitting a common structure to estimates of genetic and environmental covariance matrices is examined in a simple simulation study. It is shown that such parsimonious estimation can considerably reduce sampling variation. However, if the assumption of similarity in structure does not hold at least approximately, bias in estimates of the genetic covariance matrix can be substantial. For small samples and more than a few traits, structured estimation is likely to reduce mean square error even if bias is quite large. Hence such models should be used cautiously.
WOMBAT: A tool for mixed model analyses in quantitative genetics by restricted maximum likelihood (REML)
WOMBAT is a software package for quantitative genetic analyses of continuous traits, fitting a linear, mixed model; estimates of covariance components and the resulting genetic parameters are obtained by restricted maximum likelihood. A wide range of models, comprising numerous traits, multiple fixed and random effects, selected genetic covariance structures, random regression models and reduced rank estimation are accommodated. WOMBAT employs up-to-date numerical and computational methods. Together with the use of efficient compilers, this generates fast executable programs, suitable for large scale analyses. Use of WOMBAT is illustrated for a bivariate analysis. The package consists of the executable program, available for LINUX and WINDOWS environments, manual and a set of worked example, and can be downloaded free of charge from http://agbu.une.edu.au/~kmeyer/wombat.html
Performance of REML algorithms in multivariate analyses fitting reduced rank and factor-analytic models
Convergence behaviour of restricted maximum likelihood algorithms in multivariate analyses imposing a factor-analytic structure on covariance matrices is examined. Results indicate that estimation for such models can entail a more difficult maximisation problem than 'unstructured' estimation. On the other hand, if only factors explaining negligible variation are omitted, convergence can be faster as parameters at the boundaries of the parameter space have been eliminated. The 'parameter expanded' expectation maximisation algorithm tends to require many more iterates than the 'average information' algorithm, but is useful, in particular when combined with the latter.
Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices
Eigenvalues and eigenvectors of covariance matrices are important statistics for multivariate problems in many applications, including quantitative genetics. Estimates of these quantities are subject to different types of bias. This article reviews and extends the existing theory on these biases, considering a balanced one-way classification and restricted maximum-likelihood estimation. Biases are due to the spread of sample roots and arise from ignoring selected principal components when imposing constraints on the parameter space, to ensure positive semidefinite estimates or to estimate covariance matrices of chosen, reduced rank. In addition, it is shown that reduced-rank estimators that consider only the leading eigenvalues and -vectors of the 'between-group' covariance matrix may be biased due to selecting the wrong subset of principal components. In a genetic context, with groups representing families, this bias is inverse proportional to the degree of genetic relationship among family members, but is independent of sample size. Theoretical results are supplemented by a simulation study, demonstrating close agreement between predicted and observed bias for large samples. It is emphasized that the rank of the genetic covariance matrix should be chosen sufficiently large to accommodate all important genetic principal components, even though, paradoxically, this may require including a number of components with negligible eigenvalues. A strategy for rank selection in practical analyses is outlined.
Principal Component and Factor Analytic Models In International Sire Evaluation
Various studies have addressed the challenge of variance component estimation for multiple-trait across country evaluation (MACE) and attempted to ease the burden of the estimation process. Several of these have focused on using the decomposition of the genetic covariance matrices into the pertaining matrices of eigenvalues and -vectors, namely principal component (PC) and factor analytic (FA) approaches (e.g., Leclerc et al., 2005; Mäntysaari, 2004). For highly correlated traits, some eigenvalues have only a very small effect on the genetic variation. This is utilized by ignoring the PCs with negligible effects. For the PC approach this results in dimension reduction. The FA model also includes trait specific variances. This results in a full rank (co)variance (VCV) matrix unless some of the latter are zero. Leclerc et al. (2005) studied both PC and FA approaches for a sub-set of well-linked base countries, performing dimension reduction for this sub-set and estimating the contribution of the remaining countries to these PCs or factors. Mäntysaari (2004) introduced a bottom-up PC approach: this begins with a sub-set of countries, adding in the remaining countries sequentially. By examining in each step whether or not the new country increases the rank of the genetic VCV matrix, it only fits PCs with non-negligible eigenvalues and thus avoids over-parameterized models. Direct estimation of the important genetic principal components only has been proposed by Kirkpatrick and Meyer (2004). However, this requires the appropriate rank to be known or to be estimated. Similarly, we can estimate a VCV matrix imposing a FA structure directly. The bottom-up approach has recently been tested for variance component estimation for MACE with promising results (Tyrisevä et al., 2009). Both direct PC and FA approaches have been applied to beef cattle data sets, and have demonstrated their potential to be used for large, multi-trait data sets (e.g., Meyer, 2007a). The objectives of this study are to assess the impact of alternative parameterizations (PC and FA) for the estimation of variance components on practical predictions of breeding values with MACE.
Pooling Estimates of Covariance Components Using a Penalized Maximum Likelihood Approach
Estimates of large genetic covariance matrices are commonly obtained by pooling results from a series of analyses of small subsets of traits. Procedures available to pool the part-estimates differ in their efficacy in accounting for unequal accuracies of estimates and sampling correlations, and ensuring that pooled matrices are within the parameter space. We propose a maximum likelihood (ML) approach to combine estimates, treating sets from individual part-analyses as matrices of mean squares and cross-products from independent families. This facilitates simultaneous pooling of estimates for all sources of variation considered, readily allows for weighted estimation or a given structure of the pooled matrices, and provides a framework for regularized estimation by penalizing the likelihood. A simulation study is presented, comparing the quality of combined estimates for several procedures, including truncation or shrinkage of either canonical or individual matrix eigen-values, iterative summation of expanded part matrices, and the ML approach, considering a range of penalties. Shrinking eigen-values of individual matrices towards their mean reduced losses in the pooled estimates, but substantially increased proportional losses in their phenotypic counterparts and thus yielded estimates differing most from corresponding full multivariate analyses of all traits. Assuming a simple pseudo-pedigree structure when combining estimates for all random effects simultaneously using ML allowed sampling correlations between estimates of different components from the same part-analysis to be approximated sufficiently to yield pooled matrices closest to full multivariate results, with little change in phenotypic components. Imposing a mild penalty to shrink matrices for random effects towards their sum proved highly advantageous, markedly reducing losses in estimates and more than compensating for the reduction in efficiency of using the data inherent in analyses by parts. Penalized ML provides a flexible alternative to current methods for pooling estimates from part-analyses with good sampling properties, and should be adopted more widely.
The effect of imprinted genes on carcass traits in Australian Angus and Hereford cattle
Imprinted loci are those where the level of expression of an allele depends upon the allele's parent of origin. Imprinting is a widespread phenomenon and parent-of-origin effects have been reported for many qualitative and quantitative traits, in particular carcass traits. The effect of parent-of-origin effects on three quantitative traits - eye muscle area and fat depth at the P8 and 12/13th rib sites - measured on Angus and Hereford heifers and bull calves was examined. Parent-of-origin effects accounted for 12-45% of the total genetic variation for these traits.
Analysing quantitative parent-of-origin effects with examples from ultrasonic measures of body composition in Australian beef cattle
Parent-of-origin effects arise when an individual's genes are modified during gametogenesis. Commonly known as imprinting, affected genes may be completely, or partially, suppressed. Individual loci in mice, human and sheep are known to be imprinted, and the quantitative effects of imprinted loci have been found for many carcass traits in cattle and pigs. Differentiating between five types of loci - direct additive loci and partially and completely imprinted loci by sires and dams - is not possible as their effects are confounded such that only three of seven parameters can be estimated. An analysis of Australian Hereford and Angus heifers and bulls for four ultrasonic measures of body composition - eye muscle area, rib fat, rump fat and intramuscular fat per cent - found parent-of-origin effects for both parents in most trait-gender data sets and that they were an average of 28% of the total genetic variance. No parent-of-origin effects were found for Hereford bull intramuscular fat per cent and the maternal parent-of-origin effects were not significant for Angus Heifer eye muscle area.