Genetic evaluation of growth in nellore cattle by multiple-trait and random regression models

The objective of this study was to identify issues in genetic evaluation of beef cattle for growth by a random regression model (RRM). Genetic evaluation data included 2,946,847 records of up to nine sequential weights of 812,393 Nellore cattle measured at ages ranging from birth to 733 d. Models considered were a five-trait multiple-trait model (MTM) and a cubic RRM. The MTM included the effects of contemporary group, age of dam class, additive direct, additive maternal, and maternal permanent environment. Both additive effects were assumed correlated. The RRM included the same effects as MTM, with the addition of permanent and random error effects. The purpose of the random error effect, which was in addition to a residual effect with constant variance, was to model heterogeneous residual variances. All effects in RRM were modeled as cubic Legendre polynomials. Expected progeny differences (EPD) were obtained iteratively using a preconditioned conjugate gradient algorithm. Numerically accurate solutions with RRM were not obtained until the random regressions were orthogonalized. Computing requirements of RRM were reduced by more than 50%, without affecting the accuracy by removing regressions corresponding to very low eigen-values and by replacing the random error effects with weights. Afterward, the correlations between EPD from RRM and from MTM for EPD on selected weights were between 0.84 and 0.89. For sires with at least 50 progeny, these correlations increased to 0.92 to 0.97. Low correlations were caused by differences in parameters. The RRM applied to growth i s prone to numerical problems. Estimates of EPD with RRM may be more accurate than those with MTM only if accurate parameters are applied.     

Genetic evaluation of growth of Kenya Boran cattle using random regression models

Data consisting of 18 884 weight records collected from 1273 Boran cattle from birth to 24 months of age were used to estimate covariance functions and genetic parameters for growth of Boran cattle using random regression (RR) models under a situation of small herd size and inconsistent recording. The RR model fitted quadratic Legendre polynomials of age at recording for additive genetic and permanent environmental effects. Genetic variance increased from birth, reaching an asymptotic value at 455 days and was maximum at 525 days of age after which it gradually dropped. Permanent environmental variance increased throughout the trajectory. Estimates of temporary environmental variance were heterogeneous across ages. Direct heritability and permanent environmental variance as a proportion of phenotypic variance fluctuated greatly during the early ages but later stabilized at intermediate to later ages; the estimates ranged from 0.11 to 0.33 and from 0.18 to 0.83, respectively. Genetic correlation estimates were positive, ranging from 0.10 to unity. The estimates declined with increasing in lag between the age points. Phenotypic correlation pattern was erratic between early ages, negatively low (-0.02) between the extreme data points and moderate to highly positive (>0.50) between intermediate and later points, with prominent spikes along the diagonal. It is concluded that RR models have potential for modelling growth of Boran cattle, notwithstanding conditions of small herd sizes and inconsistent recording.     

Constructing covariance functions for random regression models for growth in Gelbvieh beef cattle

Genetic parameters for a random regression model of growth in Gelbvieh beef cattle were constructed using existing estimates. Information for variances along ages was provided by parameters used for routine Gelbvieh multiple-trait evaluation, and information on correlations among different ages was provided by random regression model estimates from literature studies involving Nellore cattle. Both sources of information were combined into multiple-trait estimates; corrected for continuity, smoothness, and general agreement with literature estimates; and extrapolated to 730 d. Covariance functions using standardized Legendre polynomials were fit for the following effects: additive genetic (direct and maternal), and animal and maternal permanent environment. Residual variances at different ages were fitted using linear splines with three knots. Fit was by least squares. The order of polynomials was varied from third to sixth. Increasing the fit beyond cubic provided small improvements in R2 and increased the number of small eigenvalues of covariance matrices, especially for the additive effect. Parameters for a random regression model in beef cattle can be constructed with negligible artifacts from literature estimates. Formulas can easily be modified for other types of polynomials and splines.     

Approximating prediction error covariances among additive genetic effects within animals in multiple-trait and random regression models

A method for approximating prediction error variances and covariances among estimates of individual animals’ genetic effects for multiple‐trait and random regression models is described. These approximations are used to calculate the prediction error variances of linear functions of the terms in the model. In the multiple‐trait case these are indexes of estimated breeding values, and for random regression models these are estimated breeding values at individual points on the longitudinal scale. Approximate reliabilities for terms in the model and linear functions thereof are compared with corresponding reliabilities obtained from the inverse of the coefficient matrix in the mixed model equations. Results show good agreement between approximate and ‘true’ values.     

Studies on multiple trait and random regression models for genetic evaluation of beef cattle for growth

A simulation study examined issues important for genetic evaluation of growth in beef cattle by random regression models with cubic Legendre polynomials (RRML) and linear splines with three knots (RRMS) compared with multiple-trait models (MTM). Parameters for RRML were obtained by conversion from covariance functions. Parameters for MTM and RRMS were extracted from RRML at 1, 205, and 365 d; parameters for RRMS were the same as MTM for all effects except the permanent environment and the residual. Four data sets were generated assuming RRML included records at 1, 205, and 365 d; at 1, 160 to 250, and 320 to 410 d; at 1, 100, 205, 300, and 365 d; and at 1, 55 to 145, 160 to 250, 275 to 325, and 320 to 410 d. Accuracies were computed as correlations between the true (simulated) and predicted breeding values. With the first data set, excellent agreement in accuracy was obtained for all models. With the second data set, the accuracy of MTM dropped by up to 1.5% compared with the first data set, but accuracy was unchanged for both RRML and RRMS. With the third (fourth) data set, accuracies of RRML were up to 2.4% (2.5%) higher than with the first (second) data set. Small differences in accuracy between RRML and RRMS were found with the third and fourth data sets, which were traced to inflated correlations especially between 1 and 205 d in RRMS; inflation could be decreased by adding one extra knot at 100 d to RRMS. Diagonalization of random coefficients was crucial for RRML but not for RRMS, resulting in approximately six (two) times faster convergence with RRML (RRMS). Reduction of dimensionality in RRML associated with small eigenvalues caused a less accurate evaluation for birth weight. Genetic evaluation of growth by RRM requires careful implementation. The RRMS is simpler to implement than the RRML.     

Covariance functions and random regression models for cow weight in beef cattle

Data from the first four cycles of the Germplasm Evaluation program at the U.S. Meat Animal Research Center were used to evaluate weights of Angus, Hereford, and F1 cows produced by crosses of 22 sire and 2 dam (Angus and Hereford) breeds. Four weights per year were available for cows from 2 through 8 yr of age (AY) with age in months (AM). Weights (n = 61,798) were analyzed with REML using covariance function-random regression models (CF-RRM), with regression on orthogonal (Legendre) polynomials of AM. Models included fixed regression on AM and effects of cow line, age in years, season of measurement, and their interactions; year of birth; and pregnancy-lactation codes. Random parts of the models fitted RRM coefficients for additive (a) and permanent environmental (c) effects. Estimates of CF were used to estimate covariances among all ages. Temporary environmental effects were modeled to account for heterogeneity of variance by AY. Quadratic fixed regression was sufficient to model population trajectory and was fitted in all analyses. Other models varied order of fit and rank of coefficients for a and c. A parsimonious model included linear and quartic regression coefficients for a and c, respectively. A reduced cubic order sufficed for c. Estimates of all variances increased with age. Estimates for older ages disagreed with estimates using traditional bivariate models. Plots of covariances for c were smooth for intermediate, but erratic for extreme ages. Heritability estimates ranged from 0.38 (36 mo) to 0.78 (94 mo), with fluctuations especially for extreme ages. Estimates of genetic correlations were high for most pairs of ages, with the lowest estimate (0.70) between extreme ages (19 and 103 mo). Results suggest that although cow weights do not fit a repeatability model with constant variances as well as CF-RRM, a repeatability model might be an acceptable approximation for prediction of additive genetic effects.     

Genetic evaluation of dairy cattle for conformation traits using random regression models

Random regression models were applied to eight conformation traits (i.e. stature, rump angle, thurl width, rear leg set, rear udder width, rear udder height, udder depth, and fore udder attachment) of Holstein cows from the northeastern United States. Covariates for fixed and random regressions included age and age‐squared for six of the traits, and two additional covariates were included for rear udder width and rear udder height. Other effects in the model were herd—year‐classifier and months in milk. Fixed covariates were nested within year of birth of the cow. Variance components were estimated using Bayesian theory and Gibbs sampling procedure. Estimated breeding values from the random regression models were compared to two single trait models. The first model utilized only the first classification record of the cow in first lactation, and the second model utilized all classifications of the cow in a simple repeatability model. Additive genetic merit for conformation traits changed with the age of the animal. Some traits were affected by age more than others. The single trait, single record model and the simple repeatability model were not appropriate in predicting breeding values at mature ages for rear udder width and rear udder height.     

Two methods for parameter estimation using multiple-trait models and beef cattle field data

Two methods are presented for estimating variances and covariances from beef cattle field data using multiple-trait sire models. Both methods require that the first trait have no missing records and that the contemporary groups for the second trait be subsets of the contemporary groups for the first trait; however, the second trait may have missing records. One method uses pseudo expectations involving quadratics composed of the solutions and the right-hand sides of the mixed model equations. The other method is an extension of Henderson's Simple Method to the multiple trait case. Neither of these methods requires any inversions of large matrices in the computation of the parameters; therefore, both methods can handle very large sets of data. Four simulated data sets were generated to evaluate the methods. In general, both methods estimated genetic correlations and heritabilities that were close to the Restricted Maximum Likelihood estimates and the true data set values, even when selection within contemporary groups was practiced. The estimates of residual correlations by both methods, however, were biased by selection. These two methods can be useful in estimating variances and covariances from multiple-trait models in large populations that have undergone a minimal amount of selection within contemporary groups.     

Application of robust procedures for estimation of breeding values in multiple-trait random regression test-day model

Robust procedures for estimation of breeding values were applied to multiple‐trait random regression test‐day (TD) model to reduce the influence of outliers on inferences. Robust estimation methods consisted of correcting selected observations (defined as outliers) in the process of solving mixed‐model equations in such a way that ‘new’ observations gave residuals (actual observation minus predicted) within k residual standard deviations for a given day in milk in 305‐day lactation. Data were 980 503 TD records on 63 346 Canadian Jersey cows. Milk, fat, protein and somatic cell score in the first three lactations were analysed jointly in the model that included fixed herd‐TD effect and regressions within region–age–season of calving, and regressions with random coefficients for animal genetic and permanent environmental effects. All regressions were orthogonal polynomials of order 4. Robust procedures for k = 1.5, 2.0, 2.5, 2.75 and 3.0 were contrasted with the regular best linear unbiased prediction (BLUP) method in terms of numbers and distributions of outliers, and estimated breeding values (EBV) of animals. Distributions of outliers were similar across traits and lactations. Early days in milk (from 5 to 15) were associated with larger frequency of outliers compared with the remaining part of lactation. Several, computationally simple, robust methods (for k > 2.0) reduced the influence of outlier observations in the model and improved the overall model performance. Differences in rankings of animals from robust evaluations were small compared with the regular BLUP method. No clear associations between changes in EBV (rankings) of top animals from different methods and the occurrence of outliers were detected.     

Genetic evaluation of growth in Barki sheep using random regression models

The objective of the current study was to estimate covariance components of growth at different ages from birth to yearling in Barki lambs. A total of 16,496 records for body weights at birth (W0), 3 (W3), 6 (W6), 9 (W9), and 12 (12) months of age for Barki lambs were available. Two statistical approaches were used; multi-trait (MT) and random regression (RR) animal models assuming two random effects only, additive genetic effect (σ²_a) and permanent environmental effect (σ²_pe) of the animal. Regarding the RR model, Legendre polynomials (LP) of different orders for the random parts were compared in order to evaluate the most appropriate model. Bayesian information and Akaike information criteria suggested that the optimal RR model included the third order for fixed effect of lamb age and σ²_pe, and fourth order of LP for σ²_a (LP₃₄₃). Estimates of direct heritability (h²_a) from LP₃₄₃ showed an ascending pattern, as it was 0.06 ± 0.03 for birth weight and reached to the peak at 9 months (0.42 ± 0.02). Thereafter, it declined again at the end of trajectory (12 months of age; 0.27 ± 0.03). The MT model showed a fluctuated pattern and lower estimates of h²_a (0.19 ± 0.03, 0.11 ± 0.02, 0.12 ± 0.02, 0.11 ± 0.03, and 0.16 ± 0.04 for W0, W3, W6, W9, and W12, respectively). Considerably, similar ascending patterns of the ratio of σ²_pe to phenotypic variance were reported from both RR (from 3 to 50%) and MT models (from 5 to 20%). Of interest, the RR model showed higher predicting ability of the breeding values compared with the MT model, which is an indicator for the suitability of RR models for analyzing the consecutive growth traits in sheep. Results suggested that the Barki sheep has a potential for genetic selection based on weight at different ages with selection likely to be more efficient at 9 months of age.

     

Autoregressive and random regression test-day models for multiple lactations in genetic evaluation of Brazilian Holstein cattle

Autoregressive (AR) and random regression (RR) models were fitted to test-day records from the first three lactations of Brazilian Holstein cattle with the objective of comparing their efficiency for national genetic evaluations. The data comprised 4,142,740 records of milk yield (MY) and somatic cell score (SCS) from 274,335 cows belonging to 2,322 herds. Although heritabilities were similar between models and traits, additive genetic variance estimates using AR were 7.0 (MY) and 22.2% (SCS) higher than those obtained from RR model. On the other hand, residual variances were lower in both traits when estimated through AR model. The rank correlation between EBV obtained from AR and RR models was 0.96 and 0.94 (MY) and 0.97 and 0.95 (SCS), respectively, for bulls (with 10 or more daughters) and cows. Estimated annual genetic gains for bulls (cows) obtained using AR were 46.11 (49.50) kg for MY and −0.019 (−0.025) score for SCS; whereas using RR these values were 47.70 (55.56) kg and −0.022 (−0.028) score. Akaike information criterion was lower for AR in both traits. Although AR model is more parsimonious, RR model assumes genetic correlations different from the unity within and across lactations. Thus, when these correlations are relatively high, these models tend to yield to similar predictions; otherwise, they will differ more and RR model would be theoretically sounder.     

Genetic correlations between body condition scores and fertility in dairy cattle using bivariate random regression models

Genetic correlations between body condition score (BCS) and fertility traits in dairy cattle were estimated using bivariate random regression models. BCS was recorded by the Swiss Holstein Association on 22,075 lactating heifers (primiparous cows) from 856 sires. Fertility data during first lactation were extracted for 40,736 cows. The fertility traits were days to first service (DFS), days between first and last insemination (DFLI), calving interval (CI), number of services per conception (NSPC) and conception rate to first insemination (CRFI). A bivariate model was used to estimate genetic correlations between BCS as a longitudinal trait by random regression components, and daughter's fertility at the sire level as a single lactation measurement. Heritability of BCS was 0.17, and heritabilities for fertility traits were low (0.01-0.08). Genetic correlations between BCS and fertility over the lactation varied from: -0.45 to -0.14 for DFS; -0.75 to 0.03 for DFLI; from -0.59 to -0.02 for CI; from -0.47 to 0.33 for NSPC and from 0.08 to 0.82 for CRFI. These results show (genetic) interactions between fat reserves and reproduction along the lactation trajectory of modern dairy cows, which can be useful in genetic selection as well as in management. Maximum genetic gain in fertility from indirect selection on BCS should be based on measurements taken in mid lactation when the genetic variance for BCS is largest, and the genetic correlations between BCS and fertility is strongest.     

Evaluation of methods for computing approximate accuracies of predicted breeding values in maternal random regression models for growth traits in beef cattle

The objective of this study was to determine the suitability of 2 methods for computing approximate accuracies of predicted breeding values, in which accuracy was defined as the squared correlation between the predicted and true breeding value, when modeling growth traits in beef cattle using random regression (RR) models. The first method (Strabel et al., S-M-B) was designed for use with multitrait models; thus, its use with RR models requires the clustering of measurements into different traits. The second method (Tier and Meyer, T-M) was more general, because it accounted for random coefficients other than zeros and ones and thus it could be used directly when fitting RR models. To investigate the performance of both methods, their results were compared with the true accuracies using a balanced simulated data set. The largest difference between approximate and true average accuracies for direct effects was observed at 205 d when S-M-B was used (4.6% males and 8.8% females). With regard to maternal effects, the largest differences in average accuracies were observed at 205 d in males when S-M-B was used (31.8%) and at the same age in females but when using T-M (33.3%). In general, bias increased for direct effect accuracies in males at the tails of the accuracy range, but for females and for maternal effect accuracies in both sexes, bias increased as accuracy increased. When a population was simulated to create large numbers of progeny for base females that did not have individual records, much greater errors were observed in the regression of approximate values on the true ones. When both approximate methods were compared using a real beef cattle data set, a good agreement was observed, particularly for direct effect accuracies in sires [i.e., at 205 d, the regressions were 0.98 (direct) and 0.95 (maternal) with r(2) over 0.99]. The largest discrepancies for sires between the methods were observed at 205 d for direct (2.7%) and maternal (16.3%) effect accuracies. For dams, the largest differences between methods were also observed at 205 d, 9.3% (direct), and 15.2% (maternal). The differences between methods for nonparent cattle were greater than for dams for maternal effect accuracies but intermediate between sires and dams for direct effect accuracies. In spite of the less biased results provided by T-M, its use could be problematic when employed in evaluations of large populations due to its greater memory and computation requirements (e.g., 170 and 478% more than S-M-B for a population of 11 million).     

Modelling number of services per conception of Japanese Black cattle by random regression

Repeated records of number of services per conception (NSC) were collected on 607 Japanese Black cows. Data were analysed by random regression (RRM) and multiple trait (MTM) models, considering NSC in each parity as a separate trait. The chosen RRM included additive genetic and permanent environmental effects fitted with a third‐order Legendre polynomials of parity. Heritabilities (h²) estimated by RRM decreased along the NSC trajectory from 0.15 in the first parity to 0.04 in the sixth parity and then increased up to 0.22 in the 10th parity. The corresponding estimates obtained by MTM ranged between 0.04 in parity 9 and 0.13 in parity 1. Permanent environmental proportions (p²) of the total phenotypic variance estimated by RRM showed similar pattern and magnitude to those of h² estimated by the same method. On the contrary, the p² estimated by MTM ranged between 0.04 in the first parity and 0.11 in the 10th parity. Additive genetic (r_G), permanent environmental (r_P) and phenotypic (r_PH) correlations were also estimated. The values estimated by RRM between adjacent parities were higher than those of parities far apart. The corresponding values estimated by MTM were lower than those estimated by RRM with no certain trend. The results indicated that NSC in heifers is more heritable than NSC in cows with different parities. Reproductive traits are economically important traits and hence, they should be considered in breeding goals.     

Estimation of genetic parameters for ultrasound-predicted percentage of intramuscular fat in Angus cattle using random regression models

The present study included 3,358 observations of 675 bulls and heifers from the Iowa State University beef cattle breeding project. Data were collected over a 3-yr period between 1998 and 2000. Each year, cattle were scanned four to six times for ultrasound-predicted percentage of intramuscular fat (UPFAT) and other ultrasound traits, starting at a minimum age of 28 wk. The objective of the current study was to estimate variance components, heritability, and repeatability of UPFAT in young bulls and heifers. Data were subjected to random-regression animal models that included fixed effects of contemporary group, fixed Legendre polynomial of age at measurement, and random regression coefficients on Legendre polynomial of age at measurement for animals' direct genetic and direct permanent environmental effects. Phenotypic and genetic models involving different levels of polynomial fit for the animal component were considered. A model fitting a linear effect of Legendre polynomial of age at a measurement for animal direct genetic and direct permanent environmental effects and a homogeneous error variance described the present data adequately. Heritability of UPFAT ranged from 0.32 at 28 wk of age to a maximum of 0.53 at 63 wk. Repeatability of UPFAT increased from a minimum of 0.60 at ages of 28 to 39 wk to a maximum of 0.80 at ages 61 to 63 wk. Heritability and repeatability of yearling UPFAT were 0.50 and 0.71, respectively. With the exception of minor differences at earlier ages, fitting heterogeneous error variances did not have an effect on genetic parameter estimates for most ages of measurement. The present results showed an optimal heritability and repeatability of UPFAT measures around 52 wk and through at least 63 wk of age. This suggested that differences in UPFAT measures during this period also are good measures of differences in marbling genetic potential of Angus cattle.     

Modeling lactation curves and estimation of genetic parameters in Holstein cows using multiple‐trait random regression models

We compared the goodness of fit of three mathematical functions (including: Legendre polynomials, Lidauer‐Mäntysaari function and Wilmink function) for describing the lactation curve of primiparous Iranian Holstein cows by using multiple‐trait random regression models (MT‐RRM). Lactational submodels provided the largest daily additive genetic (AG) and permanent environmental (PE) variance estimates at the end and at the onset of lactation, respectively, as well as low genetic correlations between peripheral test‐day records. For all models, heritability estimates were highest at the end of lactation (245 to 305 days) and ranged from 0.05 to 0.26, 0.03 to 0.12 and 0.04 to 0.24 for milk, fat and protein yields, respectively. Generally, the genetic correlations between traits depend on how far apart they are or whether they are on the same day in any two traits. On average, genetic correlations between milk and fat were the lowest and those between fat and protein were intermediate, while those between milk and protein were the highest. Results from all criteria (Akaike's and Schwarz's Bayesian information criterion, and ?2*logarithm of the likelihood function) suggested that a model with 2 and 5 coefficients of Legendre polynomials for AG and PE effects, respectively, was the most adequate for fitting the data.     

Properties of random regression models using linear splines

Properties of random regression models using linear splines (RRMS) were evaluated with respect to scale of parameters, numerical properties, changes in variances and strategies to select the number and positions of knots. Parameters in RRMS are similar to those in multiple trait models with traits corresponding to points at knots. RRMS have good numerical properties because of generally superior numerical properties of splines compared with polynomials and sparser system of equations. These models also contain artefacts in terms of depression of variances and predictions in the middle of intervals between the knots, and inflation of predictions close to knots; the artefacts become smaller as correlations corresponding to adjacent knots increase. The artefacts can be greatly reduced by a simple modification to covariables. With the modification, the accuracy of RRMS increases only marginally if the correlations between the adjacent knots are ≥0.6. In practical analyses the knots for each effect in RRMS can be selected so that: (i) they cover the entire trajectory; (ii) changes in variances in intervals between the knots are approximately linear; and (iii) the correlations between the adjacent knots are at least 0.6. RRMS allow for simple and numerically stable implementations of genetic evaluations with artefacts present but transparent and easily controlled.     

Genetic evaluation of growth in a multibreed beef cattle population using random regression-linear spline models

The objective of this study was to examine the feasibility of using random regression-spline (RR-spline) models for fitting growth traits in a multibreed beef cattle population. To meet the objective, the results from the RR-spline model were compared with the widely used multitrait (MT) model when both were fit to a data set (1.8 million records and 1.1 million animals) provided by the American Gelbvieh Association. The effect of prior information on the EBV of sires was also investigated. In both RR-spline and MT models, the following effects were considered: individual direct and maternal additive genetic effects, contemporary group, age of the animal at measurement, direct and maternal heterosis, and direct and maternal additive genetic mean effect of the breed. Additionally, the RR-spline model included an individual direct permanent environmental effect. When both MT and RR-spline models were applied to a data set containing records for weaning weight (WWT) and yearling weight (YWT) within specified age ranges, the rankings of bulls' direct EBV (as measured via Pearson correlations) provided by both models were comparable, with slightly greater differences in the reranking of bulls observed for YWT evaluations (>or=0.99 for BWT and WWT and >or=0.98 for YWT); also, some bulls dropped from the top 100 list when these lists were compared across methods. For maternal effects, the estimated correlations were slightly smaller, particularly for YWT; again, some drops from the top 100 animals were observed. As in regular MT multibreed genetic evaluations, the heterosis effects and the additive genetic effects of the breed could not be estimated from field data, because there were not enough contemporary groups with the proper composition of purebred and crossbred animals; thus, prior information based on literature values had to be included. The inclusion of prior information had a negligible effect in the overall ranking for bulls with greater than 20 birth weight progeny records; however, the effect of prior information for breeds or groups poorly represented in the data was important. The Pearson correlations for direct and maternal WWT and YWT ranged from 0.95 to 0.98 when comparing evaluations of data sets for which the out-of-range age records were removed or retained. Random regression allows for avoiding the discarding of records that are outside the usual age ranges of measurement; thus, greater accuracies are achieved, and greater genetic progress could be expected.     

Alternative parameterizations of the multiple-trait random regression model for milk yield and somatic cell score via recursive links between phenotypes

Multiple-trait random regression models with recursive phenotypic link from somatic cell score (SCS) to milk yield on the same test day and with different restrictions on co-variances between these traits were fitted to the first-lactation Canadian Holstein data. Bayesian methods with Gibbs sampling were used to derive inferences about parameters for all models. Bayes factor indicated that the recursive model with uncorrelated environmental effects between traits was the most plausible specification in describing the data. Goodness of fit in terms of a within-trait weighted mean square error and correlation between observed and predicted data was the same for all parameterizations. All recursive models estimated similar negative causal effects from SCS to milk yield (up to -0.4 in 46-115 days in milk in lactation). Estimates of heritabilities, genetic and environmental correlations for the first two regression coefficients (overall level of a trait and lactation persistency) within both traits were similar among models. Genetic correlations between milk and SCS were dependent on the restrictions on genetic co-variances for these traits. Recursive model with uncorrelated system genetic effects between milk and SCS gave estimates of genetic correlations of the opposite sign compared with a regular multiple-trait model. Phenotypic recursion between milk and SCS seemed, however, to be the only source of environmental correlations between these two traits. Rankings of sires for total milk yield in lactation, average daily SCS and persistency for both traits were similar among models. Multiple-trait model with recursive links between milk and SCS and uncorrelated random environmental effects could be an attractive alternative for a regular multiple-trait model in terms of model parsimony and accuracy.