Summary. 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.
Summary. Using simulation, the efficacy of penalized maximum likelihood estimation of genetic covariances when employing different strategies to determine the necessary tuning parameter is investigated. It is shown that errors in estimating the tuning factor from the data using cross-validation can reduce the percentage reduction in average loss at modest sample sizes from 70\% or more to 60\% or less. Mild penalization by limiting the change in likelihood is shown to perform well and to yield choices which are highly correlated with those based on the population parameters. Likelihood based selection of the tuning parameter is recommended as a simple and effective alternative to cross-validation.Meyer, K. (2011). "Performance of cross-validation and likelihood based strategies to select tuning factors for penalized estimation"
Summary. Problems inherent in multivariate, genetic analyses to estimate covariance components are discussed. New developments in methodology with the scope to yield `better' estimates are described, and their application is demonstrated for an analysis of carcass traits of beef cattle.Meyer, K. (2011). "Multivariate estimation of genetic parameters - Quo vadis?"
Summary. Parsimonious estimation of covariance matrices for multiple traits or repeated records is reviewed. Emphasis is placed on flexible models which do not require prior assumptions about the structure of covariance matrices, in particular parameterisations involving genetic principal components.Meyer, K. (2007). "Covariance structures for quantitative genetic analyses".
Summary. Fitting only the leading principal components allows genetic covariance matrices to be modelled parsimoniously, yielding reduced rank estimates. If principal components with non-zero variances are omitted from the model, genetic variation is moved into the covariance matrices for residuals or other random effects. The resulting bias in estimates of genetic eigen-values and -vectors is examined.Meyer, K. and Kirkpatrick, M. (2007). "A note on bias in reduced rank estimates of covariance matrices".
Summary. 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.Meyer, K. (2007). "Performance of REML algorithms in multivariate analyses fitting reduced rank and factor-analytic models".
Summary. Simulation and calculations based on the likelihood function are used to examine the sampling properties of maximum likelihood estimates of variance components due to sex-linked, maternal and dominance effects. Data are assumed to have been collected in an experiment designed specifically to provide a sufficient set of covariances between relatives for this task. Results show that variances due to sex-linked and dominance effects are inherently difficult to estimate. Moreover, the design suggested appears to be unsuitable to separate maternal, permanent environmental and dominance variances. The investigations shown are recommended as part of the design stage of quantitative genetic experiments.Meyer, K. (2007). "Scope for estimation of variances due to sex-linked, maternal and dominance effects in mixed model analyses".
Summary. Estimates of genetic principal components (PCs) have generally been obtained by first estimating a full rank covariance matrix and then determining its eigendecomposition. A better alternative is to estimate PCs directly, disregarding any PCs which do not explain a significant amount of variation. It is shown that this is readily implemented within the usual linear mixed model framework, requiring, in essence, only a simple reparameterisation, and that standard estimation techniques are applicable. Considering the important PCs only can yield highly parsimonious models, gives estimates of covariance matrices of reduced rank, and can reduce sampling variances. Moreover, computational requirements can be reduced dramatically, making routine higher-dimensional multivariate analyses of large data sets more feasible. PC analysis is illustrated considering a set of 14 carcass traits recorded on beef cattle.Meyer, K. (2006)."To have your steak and eat it : genetic principal component analysis of beef cattle data" Proceedings of the Eight World Congress on Genetic Applied to Livestock Production Communication No. 25-03. · Paper (8 pages, pdf) · Slides (4-up) (7 pages, pdf)
Summary. WOMBAT is a program for mixed model analysis and estimation of genetic parameters using restricted maximum likelihood. It is suitable for analyses of large data sets from animal breeding schemes, accommodating most models commonly fitted for such data. In addition to standard uni- and multivariate analyses, random regression models for different basis functions (including splines) are available. Reduced rank analyses fitting the leading principal components only are readily carried out. WOMBAT uses up-to-date techniques to order the mixed model equations, minimising computational requirements per likelihood evaluation, and a combination of average information and parameter expanded expectation maximisation algorithms to ensure fast and stable convergence. A Linux executable, manual and suite of worked examples can be downloaded from: http://didgeridoo.une.edu.au/km/wombat.phpMeyer, K. (2006)." ``WOMBAT'' - Digging deep for quantitative genetic analyses by restricted maximum likelihood" Proceedings of the Eight World Congress on Genetic Applied to Livestock Production Communication No. 27-14. · Paper (2 pages, pdf)
Summary. Features of algorithms to locate the maximum of the likelihood function in restricted maximum likelihood (REML) estimation are briefly reviewed. Differences between average information (AI) and expectation maximisation (EM) algorithms, in particular the 'parameter expanded' variant of EM (PX-EM), are highlighted. Convergence rates of AI, EM and PX-EM algorithms are contrasted for several 'difficult' practical examples of analyses of beef cattle data, involving numerous traits or multiple random effects, and thus many parameters to be estimated. Results suggest that more reliable, and often faster convergence of REML analyses can be achieved by combining algorithms : Exploit the stability and good performance of the PX-EM algorithm in the first few iterates, then switch to the AI algorithm with rapid convergence close to the maximum of the likelihood function.Meyer, K. (2006)."PX x AI : algorithmics for better convergence in restricted maximum likelihood estimation" Proceedings of the Eight World Congress on Genetic Applied to Livestock Production Communication No. 24-15. · Paper (4 pages, pdf) · Slides (4-up) (4 pages, pdf)
Summary. Computational requirements for sparse matrix factorisation or inversion are highly dependent on the `fill-in' created. This can be reduced by judicious re-ordering of equations. It is shown that use of newer ordering strategies, with corresponding computer code available in the public domain, can reduce the time required for ordering and computational requirements of analyses dramatically.Meyer, K. (2005). "Ordering strategies to reduce computational requirements in variance component estimation" . In Proceedings Association for Advancement of Animal Breeding and Genetics 16: 282-285. · Paper (4 pages, pdf)
Keywords: Mixed models, variance components estimation, sparse matrix factorisation, ordering
Summary. A simulation study investigating relative errors and sampling variances of reduced rank estimates of genetic covariance functions from random regression analyses estimating the leading principal components only, is presented. The example considered pertains to covariance functions for growth of beef cattle. It is demonstrated that the leading principal components are estimated most accurately, and that reduced rank estimates yield estimates of covariance functions with similar errors than full rank estimates. Furthermore, it is shown that substantial repartitioning between genetic and permanent environmental covariances can occur if either is modelled with too few principal components. Results emphasize the need for a judicious choice among the possible combinations of rank of fit for different covariance functions.Meyer, K. (2005). "Sampling behaviour of reduced rank estimates of genetic covariance functions". In Proceedings Association for Advancement of Animal Breeding and Genetics 16 : 286-289. · Paper (4 pages, pdf)
Keywords: Reduced rank, sampling variances, bias, random regression, covariance function
Summary. Multivariate restricted maximum likelihood analyses for a large data set comprising eight traits were carried out, estimating the leading 3, 4, 5 and 6 genetic principal components only. Traits were eye muscle area, percentage intra-muscular fat, and fat depth at the 12/13th rib and P8 sites, treating records on bulls and heifers or steers as different traits. The resulting, reduced rank estimates of genetic covariance matrices for analyses fitting 5 or 6 principal components agreed closely with an estimate from pooled, bivariate analysis. It is shown that reduced rank estimation can result in substantial reduction in computational requirements, compared to standard analyses fitting unstructured covariance matrices, and thus facilitate higher-dimensional multivariate analyses.Meyer, K. (2005). "Reduced rank estimates of the genetic covariance matrix for live ultra-sound scan traits". In Proceedings Association for Advancement of Animal Breeding and Genetics 16 : 56-59. · Paper (4 pages, pdf)
Keywords : Genetic parameters, beef cattle, scan traits, reduced rank, principal components
Summary. Estimates of variance components and genetic parameters for growth of Angus cattle from birth to 820 days of age were obtained from a random regression analysis. Trajectories were modelled through quadratic B-spline functions with 7 random regression coefficients for direct genetic and permanent environmental effects, and 5 coefficients for the corresponding maternal effects. Data comprised a large set of field records, selecting only animals with at least 4 weights recorded. Results showed smooth estimates of variances, increasing with weight and age, with good interpolation for early ages with few records and little end-of-range problems at the highest ages. On the whole, estimates of genetic parameters were consistent with literature results. However, estimates of heritabilities for weaning weight were higher for direct and lower for maternal effects than usually found, indicating a different partitioning between direct and maternal variances in random regression analyses than in standard multivariate analyses.Meyer, K. (2005). "Estimates of covariance functions for growth of Angus cattle from random regression analyses fitting B-spline functions". In Proceedings Association for Advancement of Animal Breeding and Genetics 16 :52-55. · Paper (4 pages, pdf)
Keywords : Genetic parameters, beef cattle, growth, random regression, B-splines
Summary. Estimating variances due sire X herd effects requires a large number of sires to be represented in multiple herds, otherwise too much variation between animals is `picked up' as variance due to sire X herd effects, and heritability estimates are biased downwards. Such bias can be avoided by using only records for progeny of multiple-herd use sires to estimate the additive genetic variance. Estimates of variance components and genetic parameters from analyses of weights of Hereford cattle are presented for herds with high, medium and low proportions of sires used across multiple herds. Results from analyses estimating separate genetic variances for progeny of single and multiple herd sires, estimating a joint genetic variance for all animals, and ignoring sire X herd effects are compared.Meyer, K. (2003). "Estimates of variances due to sire x herd effects for weights of Hereford cattle." In Proceedings Association for Advancement of Animal Breeding and Genetics 15 :131-134. · Paper (4 pages, pdf)
Keywords : Sire X herd effects, variance components, weight, beef cattle
Summary. Estimates of covariance functions for weights of Angus cattle from birth to 3000 days of age were obtained using Bayesian analysis. Data consisted of records in 69 herds with at least 50 mature cow weights, and records in 6 additional herds with 60% or more animals having at least four weights, 551,259 records on 197,915 animals in total. The model of analysis fitted contemporary groups and cubic regressions on orthogonal polynomials of age nested within sex, birth type, dam age class and lactation status as fixed effects. Random effects fitted were cubic and quartic regressions on orthogonal polynomials of age for animals' direct genetic and permanent environmental effects, and quadratic regressions, restricted to 0 to 600 days of age, for maternal genetic and environmental effects. Measurement error variances were modelled through a step function with 32 classes, yielding 69 covariance components to be estimated.Meyer, K. (2003). "First estimates of covariance functions for lifetime growth of Angus cattle." In Proceedings Association for Advancement of Animal Breeding and Genetics 15 : 395-398. · Paper (4 pages, pdf)
Keywords : Beef cattle, growth, covariance function, genetic parameters
Summary. Restricted maximum likelihood estimates of genetic correlations between between live ultrasound scan measurements and days to calving were obtained from bivariate analyses. Scan traits considered were fat depth at the 12/13-th rib, P8 fat depth, percentage intramuscular fat and eye muscle area, treating records for heifers or steers and bulls as separate traits. Analyses were carried out including all days to calving records, and considering the subset of cows only which had a 'complete sequence' of records, beginning with a first mating record. Heritability estimates for days to calving were low, about 3% with a repeatability of 18%. Estimates of genetic correlations were low to moderate, and consistently negative for fat depth measurements, i.e. animals with a higher genetic potential for fat deposition tended to have better reproductive performance.Meyer, K. and Johnston, D.J. (2003). "Estimates of genetic correlations between live ultrasound scan traits and days to calving in Hereford cattle." In Proceedings Association for Advancement of Animal Breeding and Genetics 15 : 387-390. Paper (4 pages, pdf)
Keywords : Genetic parameters, beef cattle, scan traits, reproductive performance
Summary. Approximate prediction error covariances among estimates of random regression coefficients for direct genetic effects were obtained for two beef cattle data sets using an extension of the method of Graser et al. (1997). From these, approximate accuracies of breeding value estimates for birth, 200 day, 400 day and 600 day weight were calculated. Corresponding 'exact' values were determined from the inverse of the coefficient matrix in the mixed model equations (MME), estimating the diagonal blocks of the inverse pertaining to random regression coefficients using Gibbs sampling. Approximate and 'exact' values were contrasted with empirical accuracies, obtained as correlations between true and estimated genetic values in a simulation study. Results showed good agreement between approximate and 'exact' values.Meyer, K. and Tier, B. (2003). "Approximate and 'exact' accuracies of breeding value estimates for growth of beef cattle from random regression analyses." In Proceedings Association for Advancement of Animal Breeding and Genetics 15 : 391-394. · Paper (4 pages, pdf)
Keywords : Random Regression, genetic evaluation, accuracy, beef cattle
Summary. Weights of cows in the Wokalup selection experiment, recorded on a monthly basis, were analysed using a random regression model. Records from 19 to 84 months of age were considered. Analyses fitted a phenotypic animal effect only, i.e. did not attempt to separate genetic and permanent environmental effects due to the animal, and ignored relationships between animals. Curves fitted included orthogonal (Legendre) polynomials of age, up to an order of fit of 20, segmented quadratic polynomials and Fourier series. Data were characterised by cyclic, seasonal variation not only in means but also in variances which was not removed by transformation to logarithmic scale. A large number of parameters was required to model this variation in variances adequately.Meyer, K. (1999). Modelling phenotypic variation in monthly weights of Australian beef cows using a random regression model. Proceedings of the 13th Conference of the Association for Advancement of Animal Breeding and Genetics, Mandurah, WA, July 4-7, 1999.
Keywords: Random regression, growth curve, mature weights, beef cattle.
Summary. Records for scanned eye muscle area, P8 fat depth, fat depth at 12/13th rib and scanning weight for Brahman and Santa Gertrudis cattle were analysed fitting age at scanning as a linear covariable within sex in the model of analysis, pre-adjusting records for age, and pre-adjusting records for weight at scanning. Heritability estimates were moderate, higher at older ages, and tended to be higher when adjusting for weight than for age. High correlations between the two fat depth measures were unaffected by the method of adjustment, while genetic correlations between eye muscle area and weight were reduced from moderate when adjusting for age to zero when adjusting for weight. Keywords : Scan records, Bos indicus, genetic parametersMeyer, K. (1999). Estimates of genetic parameters for scan measurement in Australian Brahmans and Santa Gertrudis adjusting records for age versus adjusting for weight at scanning. Proceedings of the 13th Conference of the Association for Advancement of Animal Breeding and Genetics, Mandurah, WA, July 4-7, 1999. · Paper (4 pages, doc) · Slides
Summary. Pelvic measurements on about 7500 Angus cattle, consisting of pelvic height, pelvic width, pelvic area and hip height records obtained as part of the MRC funded validation project were correlated to calving ease scores for about 20,000 calvings. Direct-maternal genetic correlation estimates for calving ease were moderately antagonistic (-0.5). Estimates of direct correlations between calving ease and pelvic measurements taken between 300 and 700 days of age ranged from 0.21 to 0.65, while corresponding direct-maternal genetic correlations were -0.27 to -0.51. Keywords : Genetic parameters, calving ease, pelvic measurementsMeyer, K. and Graser, H.-U. (1999). Estimates of genetic correlations between pelvic measurements and calving ease for Australian Angus. Proceedings of the 13th Conference of the Association for Advancement of Animal Breeding and Genetics, Mandurah, WA, July 4-7, 1999. · Paper (4 pages, doc) · Slides
Summary. Age adjusted field scanning records for eye muscle area, P8 fat depth, fat depth at 12/13th rib and scanning weight for Australian Angus, Hereford, Polled Hereford and Santa Gertrudis were analysed treating measurements for heifers/steers and bulls as different traits. Estimates of variances, heritabilities and genetic correlations are given. Average genetic correlations between sexes (across breeds) were 0.92, 0.69, 0.77 and 0.93 for the four traits, respectively. Results have been implemented into BREEDPLAN (V4.1) using scan records of heifers and bulls as different traits.Meyer, K. and Graser, H.-U. (1999). Estimates of parameters for scan records of Australian beef cattle treating records on males and females as different traits. Proceedings of the 13th Conference of the Association for Advancement of Animal Breeding and Genetics, Mandurah, WA, July 4-7, 1999. · Paper (Word document, 4 pages) · Slides
Keywords: Scan records, genetic parameters, correlation between sexes
Summary. Covariance functions are the `infinite-dimensional' equivalents to covariance matrices for longitudinal data, i.e. many, `repeated' records per individual taken over a period of time. Their properties are reviewed and illustrated with a numerical example. Restricted Maximum Likelihood estimation of genetic and phenotypic covariance functions fitting an animal model is described.Meyer, K. (1997). "Estimation of genetic and phenotypic covariance functions for longitudinal data". Proceedings of the 12th Conference of the Australian Association for Animal Breeding and Genetics, Dubbo, NSW. · Paper (4 pages, pdf).
Keywords : Covariance functions, longitudinal data, genetic parameters, REML
Summary. Restricted maximum likelihood estimates of genetic parameters for weaning weight were obtained fitting a regression on maternal phenotype to account for direct-maternal environmental covariances. For Herefords there was a substantial negative regression on dam's phenotype (up to -0.2), accompanied by small, negative estimates of the direct-maternal genetic covariance. For Angus and Limousin, the direct-maternal genetic covariance was clearly more important than its environmental counterpart, i.e. an estimate of the direct-maternal genetic correlation of about -0.5 could not be attributed to a negative environmental relationship not taken into account. Fitting a sire X herd-year interaction as an additional random effect reduced estimates of the direct-maternal genetic covariance for these breeds, resulting in corresponding correlation estimates of -0.3 to -0.2.Meyer. K. (1997). "Weaning weight revisited : Estimates of genetic paraneters fitting a regression on maternal phenotype". Proceedings of the 12th Conference of the Australian Association for Animal Breeding and Genetics, Dubbo, NSW. · Paper (4 pages, pdf).
Keywords: Beef Cattle, weaning weight, maternal effects, genetic parameters
Summary. Genetic and residual covariance functions and temporary environmental variances were estimated for January weights of cows in the Wokalup selection experiment, recorded at ages from 2 to 10 years. Various subsets of the data and orders of fit (k) from 2 to a were considered. For Wokalups, a quadratic function (k=3) described the data adequately while a cubic coefficient (k=4 ) was required for Herefords. Results indicate that a minimum of 3 `traits' is necessary to model the growth curve of beef cattle.Meyer, K. (1997). "Estimates of covariance functions for mature weight of beef cows in the Wokalup selection experiment". Proceedings of the 12th Conference of the Australian Association for Animal Breeding and Genetics, Dubbo, NSW. · Paper (4 pages, pdf).
Keywords : Beef cattle, genetic parameters, mature weight, covariance function
Estimates of genetic parameters for cannon bone length at birth and hip height at weaning and their correlations with growth traits were obtained for the two herds in the Wokalup selection experiment. Wokalups were heavier, had longer cannon bones and were taller at weaning than Polled Herefords, and exhibited more genetic and phenotypic variation. There were small genetic and permanent environmental effects, mainly attributable to maternal effects on weight at recording. Genetic correlations with growth traits were moderate to high, on the whole higher for Wokalups than Polled Herefords, and of similar magnitude to those between weights.Meyer, K. (1995). "Estimates of genetic parameters for cannon bone length in beef cattle". Proceedings of the 11th Conference of the Australian Association for Animal Breeding and Genetics, Adelaide, 242-245. · Paper (4 pages, pdf).
Genetic parameters and adjustment factors for birth, weaning, yearling and final weight were estimated for the New Zealand Angus population, fitting an animal model including maternal genetic and permanent environmental effects as additional random effects. Overall, pooled covariance matrices agreed well with those for Australian Angus, though heritability estimates for birth weight were somewhat lower than in Australian Angus. \BP\ estimates of breeding values and their accuracies were obtained for each population separately. Correlations between estimates for sires with accurate proofs in both countries agreed with their expectations, giving no indications of genotype X environment interactions.Meyer, K. and Garrick, D.J. (1995). "Scope for a joint genetic evaluation of New Zealand and Australian Angus cattle". Proceedings of the 11th Conference of the Australian Association for Animal Breeding and Genetics, Adelaide, pp. 250-253. · Paper (4 pages, pdf).
Weights of Polled Hereford and `Wokalup' cows in the Wokalup selection experiment were obtained monthly, with up to 11 years of records per cow available. Mature weights were analysed treating them as repeated records per animal and by fitting a Gompertz growth curve for each animal. Estimates of heritabilities for mature weight were 0.3 to 0.4 for Herefords and 0.5 to 0.6 for Wokalups, and 0.3 for rate of maturation in both breeds. Genetic correlations with cannon bone length at birth were moderate to high, with earlier maturing animals tending to have shorter cannon bones. Implications on selection for mature size are discussed.Meyer, K. and Carrick, M.J. (1995). "Estimates of genetic parameters for mature weight for beef cows in the Wokalup selection experiment". Proceedings of the 11th Conference of the Australian Association for Animal Breeding and Genetics, Adelaide, 246-249