Multivariate analyses of carcass traits for Angus cattle fitting reduced rank and factor analytic models

Genetic analysis of first lactation and lifetime performance traits in composite Vrindavani cattle: important considerations for higher milk production

The present study was aimed to evaluate the influence of non-genetic factors on several first lactation and lifetime performance traits and elucidate their genetic parameters in an organized Vrindavani cattle population. Data on eight first-lactation and thirteen lifetime traits were collected on 2400 cows with pedigree records that were reared during 33-year period (1989-2021). The first-lactation traits included age at first calving (AFC), total milk yield (FTMY), standard milk yield (FSMY305), peak yield (FPY), lactation length (FLL), dry period (FDP), service period (FSP) and calving interval (FCI). Whereas, the lifetime traits mainly included total lifetime milk yield (TLMY), total standard milk yield (TSMY), number of lactations completed (NL), total lactation length (TLL), herd life (HL), productive life (PL), average milk yield per day of herd life (TLMY/HL), average milk yield per day of productive life (TLMY/PL), average milk yield per day of productive life (TLMY/TLL). Other lifetime production traits included average service period (ASP), average dry period (ADP), average calving interval (ACI) and unproductive days (UD). The heritability estimates of first-lactation traits ranged between 0.026 and 0.228 and were found to be low for AFC (0.180 ± 0.042), FCI (0.191 ± 0.125), FSMY305 (0.145 ± 0.061), FTMY (0.165 ± 0.048), FDP (0.052 ± 0.049) and FSP (0.026 ± 0.033); however, FLL (0.229 ± 0.044) and FPY (0.202 ± 0.046) showed moderate heritability. Positive phenotypic correlation (p < 0.001) was revealed among FTMY, TLMY, TLL, HL and PL. The AFC produced a significant effect (p < 0.05) on several traits i,e, TLL, TLMY/HL, FSMY305, FPY, TLMY, HL and TLMY/PL. Lower AFC was associated with higher TLMY, TLL and TLMY/HL; while FSMY305, FPY, HL and TLMY/PL were higher in heifers that calved late in their life. The results revealed that AFC may be optimized with first lactation and lifetime traits for this population.

Genomic selection for genotype performance and stability using information on multiple traits and multiple environments

KEY MESSAGE

The inclusion of multiple traits and multiple environments within a partially separable factor analytic approach for genomic selection provides breeders with an informative framework to utilise genotype by environment by trait interaction for efficient selection. This paper develops a single-stage genomic selection (GS) approach which incorporates information on multiple traits and multiple environments within a partially separable factor analytic framework. The factor analytic linear mixed model is an effective method for analysing multi-environment trial (MET) datasets, but has not been extended to GS for multiple traits and multiple environments. The advantage of using all information is that breeders can utilise genotype by environment by trait interaction (GETI) to obtain more accurate predictions across correlated traits and environments. The partially separable factor analytic linear mixed model (SFA-LMM) developed in this paper is based on a three-way separable structure, which includes a factor analytic matrix between traits, a factor analytic matrix between environments and a genomic relationship matrix between genotypes. A diagonal matrix is then added to enable a different genotype by environment interaction (GEI) pattern for each trait and a different genotype by trait interaction (GTI) pattern for each environment. The results show that the SFA-LMM provides a better fit than separable approaches and a comparable fit to non-separable and partially separable approaches. The distinguishing feature of the SFA-LMM is that it will include fewer parameters than all other approaches as the number of genotypes, traits and environments increases. Lastly, a selection index is used to demonstrate simultaneous selection for overall performance and stability. This research represents an important continuation in the advancement of plant breeding analyses, particularly with the advent of high-throughput datasets involving a very large number of genotypes, traits and environments.

Reduced rank analysis of morphometric and functional traits in Campolina horses

Multitrait models can increase the accuracy of breeding value prediction and reduce bias due to selection by using traits measured before and after it has occurred. However, as the number of traits grows, a similar trend is expected for the number of parameters to be estimated, which directly affects the computing power and the amount of data required. The aim of the present study was to apply reduced rank (principal components model-PCM) and factor analytical models (FAM), to estimate (co)variance components for nineteen traits, jointly evaluated in a single analysis in Campolina horses. A total of 18 morphometric traits (MT) and one gait visual score (GtS), along with genealogical records of 48,806 horses, were analysed under a restricted maximum likelihood framework. Nine PCM, nine FAM and one standard multitrait model (MTM) were fitted to the data and compared to find the best suitable model. Based on Bayesian information criterion, the best model was the FAM option, considering five common factors (FAM5). After performing an intraclass analysis, none of MT were genetically negatively correlated, whereas GtS was negatively related to all MT, except for the genetic correlations among GtS and BLL, and between GtS and BLLBL (0.01 and 0.10 respectively). From all MT, two traits were derived computing ratios involving other traits, those had negative correlations with others MT, but all favourable for selection. Similar patterns were observed between the genetic parameters obtained from MTM and FAM5 respectively. The heritability estimates ranged from 0.09 (head width) to 0.47 (height at withers). Our results indicated that FAM was efficient to reduce the multitrait analysis dimensionality, and therefore, traits can be combined based on the first three eigenvectors from the additive genetic (co)variance matrix. In addition, there was sufficient genetic variation for selection, benefiting its potential implementation in a breeding program.

MegaLMM: Mega-scale linear mixed models for genomic predictions with thousands of traits

Large-scale phenotype data can enhance the power of genomic prediction in plant and animal breeding, as well as human genetics. However, the statistical foundation of multi-trait genomic prediction is based on the multivariate linear mixed effect model, a tool notorious for its fragility when applied to more than a handful of traits. We present MegaLMM, a statistical framework and associated software package for mixed model analyses of a virtually unlimited number of traits. Using three examples with real plant data, we show that MegaLMM can leverage thousands of traits at once to significantly improve genetic value prediction accuracy.

Selection indexes using principal component analysis for reproductive, beef and milk traits in Simmental cattle

Selection indexes in dual-purpose cattle should include beef, milk and reproductive traits. The principal component analysis is a multivariate technique that allows researchers to explore relationships between explanatory variables and traits of interest. The objective of this study was to construct selection indexes for tropical dual-purpose Simmental cattle based on principal components. The evaluated traits were weight at 8 months of age; age at first calving; cumulative first-lactation milk yield at 60, 150, 210 and 305 days; and first calving interval. The selection indexes were estimated as the sum of the products of the estimated breeding values for the seven traits times their respective eigenvectors for the first three principal components. The three selection indexes from principal components analysis generated favourable expected genetic progress for all the traits. However, a selection index with a high expected genetic progress for all traits could not be obtained. The principal component analysis allows breeders to have a selection index that simultaneously improves milk, beef and reproductive traits in dual-purpose Simmental cattle. Because a selection index yielding high expected genetic progress for all traits could not be achieved, the decision to use a specific selection index will depend on the specific conditions of the market, the local needs and the farmer preference.

Genetic principal components for reproductive and productive traits in Holstein cows reared under tropical conditions

The objective of this study was to compare the standard multi-trait model and five reduced-rank models fitted to the first principal components and genetic parameter estimates in order to determine the most appropriate method to model the covariance structure of reproductive and productive traits in Brazilian Holstein cows. Individual records of the following traits from 5217 cows were analyzed: 305-day milk yield (MY305), peak yield, milk yield per day of calving interval, days from calving to first estrus, days from calving to last service (CLS), calving interval (CI), and gestation length. Schwarz's Bayesian information criterion was used to compare the different models. The results indicated that four principal components were necessary to model the genetic (co)variance structure, reducing the number of parameters to be estimated. Analysis of genetic and phenotypic correlations showed that milk production-related traits were strongly correlated with each other (ranging from 0.74 to 0.99), while the correlation of these traits with the reproductive traits was weak (ranging from - 0.14 to 0.27). Heritability estimates for the traits ranged from 0.03 to 0.18. The reproductive traits CLS and CI and the production trait MY305 should be included as selection criteria in dairy cattle breeding programs because they are correlated with the first two principal components, retaining 91% of the genetic variability of the data.

Genetic parameters and principal components analysis of breeding value for birth and weaning weight in Egyptian buffalo

OBJECTIVE

The objectives of the current study were to study the main environmental factors affecting birth weight (BW) and weaning weight (WW), estimate variance components, genetic parameters and genetic trend and to evaluate the variability and relationships among breeding value of BW and WW using principal components analysis (PCA).

METHODS

A total of 16,370 records were collected from 8,271 buffalo calves. Genetic parameters and breeding values were estimated using a bivariate animal model which includes direct, maternal and permanent maternal effects. These estimates were standardized and used in PCA.

RESULTS

The direct heritability estimates were 0.06 and 0.41 for BW and WW, respectively whereas direct maternal heritability values were 0.03 and 0.14, respectively. Proportions of variance due to permanent environmental effects of dam were 0.455 and 0.280 for BW and WW respectively. The genetic correlation between BW and WWs was weak approaching zero, but the maternal correlation was 0.26. The first two principal components (PC1 and PC2) were estimated utilizing the standardized breeding values according to Kaiser method. The total variance explained by the first two PCs was 71.17% in which 45.91% and 25.25% were explained by PC1 and PC2, respectively. The direct breeding values of BW were related to PC2 but those of WW and maternal breeding values of BW and WWs were associated with PC1.

CONCLUSION

The results of genetic parameters and PCA indicate that BW and WWs were not genetically correlated and improving growth traits of Egyptian buffaloes could be achieved using WW without any adverse effect by BW.

Maternal body composition in seedstock herds. 3. Multivariate analysis using factor analytic models and cluster analysis

Considerable information exists on genetic relationships of body composition and carcass quality of young and finished beef cattle. However, there is a dearth of information on genetic relationships of cow body composition over time and, also, relationships with young-animal body-composition measures. The aim of the present study is to understand genetic relationships among various cow body-composition traits of Angus cows over time, from yearling to weaning of a second calf at ~3.5 years. To determine genetic correlations among various composition traits over time, a multi-trait–multi-time analysis is required. For the Maternal Productivity Project, this necessitates modelling of five traits (namely weight and ultrasound measure for loin eye muscle area (EMA), rib fat, P8 rump fat and intramuscular fat) by five time combinations (recordings at yearling then pre-calving and weaning in first and second parity). The approach was based on including all 25 trait-by-time combinations in an analysis using factor analytic models to approximate the genetic covariance matrix. Various models for the residual covariance structure were investigated. The analyses yielded correlations that could be compared with those of past studies reported in the literature and, also, to a set of bivariate analyses. Clustering of the genetic multi-trait–multi-time correlation structure resulted in a separation of traits (weight and EMA, and the fat traits) and also of time effects into early (heifer = before first lactation) and late (cow = post-first lactation) measurements.

Genetic Evaluation of Dual-Purpose Buffaloes (Bubalus bubalis) in Colombia Using Principal Component Analysis

Genealogy and productive information of 48621 dual-purpose buffaloes born in Colombia between years 1996 and 2014 was used. The following traits were assessed using one-trait models: milk yield at 270 days (MY270), age at first calving (AFC), weaning weight (WW), and weights at the following ages: first year (W12), 18 months (W18), and 2 years (W24). Direct additive genetic and residual random effects were included in all the traits. Maternal permanent environmental and maternal additive genetic effects were included for WW and W12. The fixed effects were: contemporary group (for all traits), sex (for WW, W12, W18, and W24), parity (for WW, W12, and MY270). Age was included as covariate for WW, W12, W18 and W24. Principal component analysis (PCA) was conducted using the genetic values of 133 breeding males whose breeding-value reliability was higher than 50% for all the traits in order to define the number of principal components (PC) which would explain most of the variation. The highest heritabilities were for W18 and MY270, and the lowest for AFC; with 0.53, 0.23, and 0.17, respectively. The first three PCs represented 66% of the total variance. Correlation of the first PC with meat production traits was higher than 0.73, and it was -0.38 with AFC. Correlations of the second PC with maternal genetic component traits for WW and W12 were above 0.75. The third PC had 0.84 correlation with MY270. PCA is an alternative approach for analyzing traits in dual-purpose buffaloes and reduces the dimension of the traits.

Genetic parameters for sensory traits in longissimus muscle and their associations with tenderness, marbling score, and intramuscular fat in Angus cattle

The objective of this study was to estimate heritabilities for sensory traits and genetic correlations among sensory traits and with marbling score (MS), Warner-Bratzler shear force (WBSF), and intramuscular fat content (IMFC). Samples of LM from 2,285 Angus cattle were obtained and fabricated into steaks for laboratory analysis and 1,720 steaks were analyzed by a trained sensory panel. Restricted maximum likelihood procedures were used to obtain estimates of variance and covariance components under a multitrait animal model. Estimates of heritability for MS, IMFC, WBSF, tenderness, juiciness, and connective tissue traits were 0.67, 0.38, 0.19, 0.18, 0.06, and 0.25, respectively. The genetic correlations of MS with tenderness, juiciness, and connective tissue were estimated to be 0.57 ± 0.14, 1.00 ± 0.17, and 0.49 ± 0.13, all positive and strong. Estimated genetic correlations of IMFC with tenderness, juiciness, and connective tissue were 0.56 ± 0.16, 1.00 ± 0.21, and 0.50 ± 0.15, respectively. The genetic correlations of WBSF with tenderness, juiciness, and connective tissue were all favorable and estimated to be -0.99 ± 0.08, -0.33 ± 0.30 and -0.99 ± 0.07, respectively. Strong and positive genetic correlations were estimated between tenderness and juiciness (0.54 ± 0.28) and between connective tissue and juiciness (0.58 ± 0.26). In general, genetic correlations were large and favorable, which indicated that strong relationships exist and similar gene and gene networks may control MS, IMFC, and juiciness or WBSF, panel tenderness, and connective tissue. The results from this study confirm that MS currently used in selection breeding programs has positive genetic correlations with tenderness and juiciness and, therefore, is an effective indicator trait for the improvement of tenderness and juiciness in beef. This study also indicated that a more objective measure, particularly WBSF, a trait not easy to improve through phenotypic selection, is an excellent candidate trait for genomic selection aimed at improving eating satisfaction.

Use of robust multivariate linear mixed models for estimation of genetic parameters for carcass traits in beef cattle

Assumptions of normality of residuals for carcass evaluation may make inferences vulnerable to the presence of outliers, but heavy-tail densities are viable alternatives to normal distributions and provide robustness against unusual or outlying observations when used to model the densities of residual effects. We compare estimates of genetic parameters by fitting multivariate Normal (MN) or heavy-tail distributions (multivariate Student's t and multivariate Slash, MSt and MS) for residuals in data of hot carcass weight (HCW), longissimus muscle area (REA) and 12th to 13th rib fat (FAT) traits in beef cattle using 2475 records from 2007 to 2008 from a large commercial operation in Nebraska. Model comparisons using deviance information criteria (DIC) favoured MSt over MS and MN models, respectively. The posterior means (and 95% posterior probability intervals, PPI) of v for the MSt and MS models were 5.89 ± 0.90 (4.35, 7.86) and 2.04 ± 0.18 (1.70, 2.41), respectively. Smaller values of posterior densities of v for MSt and MS models confirm that the assumption of normally distributed residuals is not adequate for the analysis of the data set. Posterior mean (PM) and posterior median (PD) estimates of direct genetic variances were variable with MSt having the highest mean value followed by MS and MN, respectively. Posterior inferences on genetic variance were, however, comparable among the models for FAT. Posterior inference on additive heritabilities for HCW, REA and FAT using MN, MSt and MS models indicated similar and moderate heritability comparable with the literature. Posterior means of genetic correlations for carcass traits were variable but positive except for between REA and FAT, which showed an antagonistic relationship. We have demonstrated that genetic evaluation and selection strategies will be sensitive to the assumed model for residuals.

Genetic relationships between live animal scan traits and carcass traits of Australian Angus bulls and heifers

Genetic parameters of four ultrasound live-scan traits and five carcass traits of Australian Angus cattle were examined with regard to sex and age of the scanned individuals. Live-scans were subdivided according to whether the observation was obtained from a bull or a heifer. In addition, two age subset (‘young’ and ‘old’) within sex were formed by k-means clustering around two centres within sex according to the age at scanning. REML estimates for heritabilities, genetic, residual and phenotypic correlations for each trait and trait combination were derived from a series of uni-, bi- and tri-variate analysis. Statistically significant age effects could be found for heritablities of scan intra-muscular fat content in heifers and scan fat depth at P8 site and scan rib fat depth in bulls, and for genetic correlations between the scan traits fat depth at P8 site, rib fat depth and eye muscle area. However, differences in heritablities between age sets within sex did not exceed 0.05, and genetic correlations between scan traits of ‘young’ and ‘old’ animals were at least 0.9. Differences between genetic correlations of abattoir carcass traits and ‘young’ and ‘old’ live-scan traits, respectively, were not significant due to high standard errors but up to 0.44. The larger of these differences were found for combinations of scan-traits and non-target carcass traits and not for combination of scan-traits and their actual carcass target traits. Thus, although some results suggest an age effect on the genetic parameters of scan traits, the extent of this effect is of limited impact on breeding value accuracy and genetic gain of scan traits. Furthermore, a possible age effect on correlations to economically important carcass traits need to be underpinned by more carcass traits observations in order to get unambiguous results allowing to draw consequences of scanning younger individuals for accuracy of breeding values and genetic gain in carcass traits.

Principal component and factor analytic models in international sire evaluation

BACKGROUND

Interbull is a non-profit organization that provides internationally comparable breeding values for globalized dairy cattle breeding programmes. Due to different trait definitions and models for genetic evaluation between countries, each biological trait is treated as a different trait in each of the participating countries. This yields a genetic covariance matrix of dimension equal to the number of countries which typically involves high genetic correlations between countries. This gives rise to several problems such as over-parameterized models and increased sampling variances, if genetic (co)variance matrices are considered to be unstructured.

METHODS

Principal component (PC) and factor analytic (FA) models allow highly parsimonious representations of the (co)variance matrix compared to the standard multi-trait model and have, therefore, attracted considerable interest for their potential to ease the burden of the estimation process for multiple-trait across country evaluation (MACE). This study evaluated the utility of PC and FA models to estimate variance components and to predict breeding values for MACE for protein yield. This was tested using a dataset comprising Holstein bull evaluations obtained in 2007 from 25 countries.

RESULTS

In total, 19 principal components or nine factors were needed to explain the genetic variation in the test dataset. Estimates of the genetic parameters under the optimal fit were almost identical for the two approaches. Furthermore, the results were in a good agreement with those obtained from the full rank model and with those provided by Interbull. The estimation time was shortest for models fitting the optimal number of parameters and prolonged when under- or over-parameterized models were applied. Correlations between estimated breeding values (EBV) from the PC19 and PC25 were unity. With few exceptions, correlations between EBV obtained using FA and PC approaches under the optimal fit were ≥ 0.99. For both approaches, EBV correlations decreased when the optimal model and models fitting too few parameters were compared.

CONCLUSIONS

Genetic parameters from the PC and FA approaches were very similar when the optimal number of principal components or factors was fitted. Over-fitting increased estimation time and standard errors of the estimates but did not affect the estimates of genetic correlations or the predictions of breeding values, whereas fitting too few parameters affected bull rankings in different countries.

Principal component approach in variance component estimation for international sire evaluation

Background

The dairy cattle breeding industry is a highly globalized business, which needs internationally comparable and reliable breeding values of sires. The international Bull Evaluation Service, Interbull, was established in 1983 to respond to this need. Currently, Interbull performs multiple-trait across country evaluations (MACE) for several traits and breeds in dairy cattle and provides international breeding values to its member countries. Estimating parameters for MACE is challenging since the structure of datasets and conventional use of multiple-trait models easily result in over-parameterized genetic covariance matrices. The number of parameters to be estimated can be reduced by taking into account only the leading principal components of the traits considered. For MACE, this is readily implemented in a random regression model.

Methods

This article compares two principal component approaches to estimate variance components for MACE using real datasets. The methods tested were a REML approach that directly estimates the genetic principal components (direct PC) and the so-called bottom-up REML approach (bottom-up PC), in which traits are sequentially added to the analysis and the statistically significant genetic principal components are retained. Furthermore, this article evaluates the utility of the bottom-up PC approach to determine the appropriate rank of the (co)variance matrix.

Results

Our study demonstrates the usefulness of both approaches and shows that they can be applied to large multi-country models considering all concerned countries simultaneously. These strategies can thus replace the current practice of estimating the covariance components required through a series of analyses involving selected subsets of traits. Our results support the importance of using the appropriate rank in the genetic (co)variance matrix. Using too low a rank resulted in biased parameter estimates, whereas too high a rank did not result in bias, but increased standard errors of the estimates and notably the computing time.

Conclusions

In terms of estimation's accuracy, both principal component approaches performed equally well and permitted the use of more parsimonious models through random regression MACE. The advantage of the bottom-up PC approach is that it does not need any previous knowledge on the rank. However, with a predetermined rank, the direct PC approach needs less computing time than the bottom-up PC.

Genetic evaluation of Angus cattle for carcass marbling using ultrasound and genomic indicators

The objectives were to estimate genetic parameters needed to elucidate the relationships of a molecular breeding value (MBV) for marbling, intramuscular fat (IMF) of yearling bulls measured with ultrasound, and marbling score (MRB) of slaughtered steers, and to assess the utility of MBV and IMF in predicting the breeding value for MRB. Records for MRB (n = 38,296) and IMF (n = 6,594) were from the American Angus Association database used for national cattle evaluation. A total of 1,006 records of MBV were used in this study. (Co)variance components were estimated with ASREML, fitting an animal model with fixed contemporary groups for MRB and IMF similar to those used in the Angus national genetic evaluation. The overall mean was the only fixed effect included in the model for MBV. Heritability estimates for carcass measures were 0.48 +/- 0.03, 0.31 +/- 0.03, and 0.98 +/- 0.05 for MRB, IMF, and MBV, respectively. Genetic correlations of IMF and MBV with MRB were 0.56 +/- 0.09 and 0.38 +/- 0.10, respectively. The genetic correlation between IMF and MBV was 0.80 +/- 0.22. These results indicate the MBV evaluated may yield a greater genetic advance of approximately 20% when used as an indicator trait for genetic prediction of MRB compared with IMF. However, neither of these indicators alone provides sufficient information to produce highly accurate prediction of breeding value for the economically relevant trait MRB. Given that the goal is a highly accurate prediction of true breeding value for MRB, results of this work point to the need to 1) continue progeny testing, and 2) continue increasing the genetic correlation between the MBV and MRB.

Factor-analytic models for genotype x environment type problems and structured covariance matrices

Background

Analysis of data on genotypes with different expression in different environments is a classic problem in quantitative genetics. A review of models for data with genotype × environment interactions and related problems is given, linking early, analysis of variance based formulations to their modern, mixed model counterparts.

Results

It is shown that models developed for the analysis of multi-environment trials in plant breeding are directly applicable in animal breeding. In particular, the 'additive main effect, multiplicative interaction' models accommodate heterogeneity of variance and are characterised by a factor-analytic covariance structure. While this can be implemented in mixed models by imposing such structure on the genetic covariance matrix in a standard, multi-trait model, an equivalent model is obtained by fitting the common and specific factors genetic separately. Properties of the mixed model equations for alternative implementations of factor-analytic models are discussed, and extensions to structured modelling of covariance matrices for multi-trait, multi-environment scenarios are described.

Conclusion

Factor analytic models provide a natural framework for modelling genotype × environment interaction type problems. Mixed model analyses fitting such models are likely to see increasing use due to the parsimonious description of covariance structures available, the scope for direct interpretation of factors as well as computational advantages.

Genetics of meat quality and carcass traits and the impact of tenderstretching in two tropical beef genotypes

Meat quality and carcass traits were measured for 2180 feedlot finished Brahman (BRAH) and Tropical Composite (TCOMP) steers to investigate genetic and non-genetic influences on shear force, and other meat quality traits. Genetic and phenotypic correlations were estimated between carcass and meat quality traits, and with live animal measurements collected in steers from weaning to feedlot exit, and their heifer half-sibs up to their first mating, which were managed in Australia’s tropical or subtropical environments. Left sides of carcasses were tenderstretched (hung by the aitch-bone) while right sides were conventionally hung (by the Achilles tendon). Tenderstretching reduced mean shear force by 1.04 kg, and phenotypic variance by 77% of that observed in conventionally hung sides. Genotype differences existed for carcass traits, with TCOMP carcasses significantly heavier, fatter, with greater eye muscle area, and lower retail beef yield than BRAH. TCOMP had lower shear force, and higher percent intramuscular fat. Meat quality and carcass traits were moderately heritable, with estimates for shear force and compression of 0.33 and 0.19 for BRAH and 0.32 and 0.20 for TCOMP respectively. In both genotypes, estimates of heritability for carcass traits (carcass weight, P8 and rib fat depths, eye muscle area and retail beef yield) were consistently moderate to high (0.21 to 0.56). Shear force and compression were genetically correlated with percent intramuscular fat (r g = –0.26 and –0.57, respectively), and meat colour (r g = –0.41 and –0.68, respectively). For TCOMP, lower shear force was genetically related to decreased carcass P8 fat depth (r g = 0.51). For BRAH steers and heifers measured at pasture, fatness traits and growth rates were genetically correlated with shear force, although the magnitude of these relationships varied with time of measurement. Net feed intake was significantly genetically correlated with carcass rib fat depth (r g = 0.49), eye muscle area (r g = –0.42) and retail beef yield (r g = –0.61). These results demonstrate that selection to improve production and carcass traits can impact meat quality traits in tropically adapted cattle, and that genotype specific evaluations will be necessary to accommodate different genetic relationships between meat quality, carcass and live animal traits.

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.

Patterns of quantitative genetic variation in multiple dimensions

A fundamental question for both evolutionary biologists and breeders is the extent to which genetic correlations limit the ability of populations to respond to selection. Here I view this topic from three perspectives. First, I propose several nondimensional statistics to quantify the genetic variation present in a suite of traits and to describe the extent to which correlations limit their selection response. A review of five data sets suggests that the total variation differs substantially between populations. In all cases analyzed, however, the "effective number of dimensions" is less than two: more than half of the total genetic variation is explained by a single combination of traits. Second, I consider how patterns of variation affect the average evolutionary response to selection in a random direction. When genetic variation lies in a small number of dimensions but there are a large number of traits under selection, then the average selection response will be reduced substantially from its potential maximum. Third, I discuss how a low genetic correlation between male fitness and female fitness limits the ability of populations to adapt. Data from two recent studies of natural populations suggest this correlation can diminish or even erase any genetic benefit to mate choice. Together these results suggest that adaptation (in natural populations) and genetic improvement (in domesticated populations) may often be as much constrained by patterns of genetic correlation as by the overall amount of genetic variation.

National cattle evaluation system for combined analysis of carcass characteristics and indicator traits recorded by using ultrasound in Angus cattle

The objectives were to 1) evaluate genetic relationships of sex-specific indicators of carcass merit obtained by using ultrasound with carcass traits of steers; 2) estimate genetic parameters needed to implement combined analyses of carcass and indicator traits to produce unified national cattle evaluations for LM area, subcutaneous fat depth (SQF), and marbling (MRB), with the ultimate goal of publishing only EPD for the carcass traits; and 3) compare resulting evaluations with previous ones. Four data sets were extracted from the records of the American Angus Association from 33,857 bulls, 33,737 heifers, and 1,805 steers that had measures of intramuscular fat content (IMF), LM area (uLMA), and SQF derived from interpretation of ultrasonic imagery, and BW recorded at the time of scanning. Also used were 38,296 records from steers with MRB, fat depth at the 12th to 13th rib interface (FD), carcass weight, and carcass LM area (cLMA) recorded on slaughter. (Co)variance components were estimated with ASREML by using the same models as used for national cattle evaluations by the American Angus Association. Heritability estimates for carcass measures were 0.45 +/- 0.03, 0.34 +/- 0.02, 0.40 +/- 0.02, and 0.33 +/- 0.02 for MRB, FD, carcass weight, and cLMA, respectively. Genetic correlations of carcass measures from steers with ultrasonic measures from bulls and heifers indicated sex-specific relationships for IMF (0.66 +/- 0.05 vs. 0.52 +/- 0.06) and uLMA (0.63 +/- 0.06 vs. 0.78 +/- 0.05), but not for BW at scanning (0.46 +/- 0.07 vs. 0.40 +/- 0.07) or SQF (0.53 +/- 0.06 vs. 0.55 +/- 0.06). For each trait, estimates of genetic correlations between bulls and heifers measured by using ultrasound were greater than 0.8. Prototype national cattle evaluations were conducted by using the estimated genetic parameters, resulting in some reranking of sires relative to previous analyses. Rank correlations of high-impact sires were 0.91 and 0.84 for the joint analysis of MRB and IMF with previous separate analyses of MRB and IMF, respectively. Corresponding results for FD and SQF were 0.90 and 0.90, and for cLMA and uLMA were 0.79 and 0.89. The unified national cattle evaluation for carcass traits using measurements from slaughtered animals and ultrasonic imagery of seed stock in a combined analysis appropriately weights information from these sources and provides breeders estimates of genetic merit consistent with traits in their breeding objectives on which to base selection decisions.

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).