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Estimates of genetic parameters for number of stillborns (NSB) in relation to litter size (LS) were obtained with random regression models (RRM). Data were collected from 4 purebred Duroc nucleus farms between 2004 and 2008. Two data sets with 6,575 litters for the first parity (P1) and 6,259 litters for the second to fifth parity (P2-5) with a total of 8,217 and 5,066 animals in the pedigree were analyzed separately. Number of stillborns was studied as a trait on sow level. Fixed effects were contemporary groups (farm-year-season) and fixed cubic regression coefficients on LS with Legendre polynomials. Models for P2-5 included the fixed effect of parity, Random effects were additive genetic effects for both data sets with permanent environmental effects included for P2-5. Random effects modeled with Legendre polynomials (RRM-L), linear splines (RRM-S), and degree 0 B-splines (RRM-BS) with regressions on LS were used. For P1, the order of polynomial, the number of knots, and the number of intervals used for respective models were quadratic, 3, and 3, respectively. For P2-5, the same parameters were linear, 2, and 2, respectively. Heterogeneous residual variances were considered in the models. For P1, estimates of heritability were 12 to 15%, 5 to 6%, and 6 to 7% in LS 5, 9, and 13, respectively. For P2-5, estimates were 15 to 17%, 4 to 5%, and 4 to 6% in LS 6, 9, and 12, respectively. For P1, average estimates of genetic correlations between LS 5 to 9, 5 to 13, and 9 to 13 were 0.53, -0.29, and 0.65, respectively. For P2-5, same estimates averaged for RRM-L and RRM-S were 0.75, -0.21, and 0.50, respectively. For RRM-BS with 2 intervals, the correlation was 0.66 between LS 5 to 7 and 8 to 13. Parameters obtained by 3 RRM revealed the nonlinear relationship between additive genetic effect of NSB and the environmental deviation of LS. The negative correlations between the 2 extreme LS might possibly indicate different genetic bases on incidence of stillbirth. Key words: litter size, pig, random regression, stillbirth doi: 10.2527/jas.2009-2413

Data of 5,390 Merino lambs born from 1986 to 2007 (~6.9 generations) were used to derive genetic parameters and trends for age-specific and overall lamb survival on the underlying scale, as well as for lamb birth weight, using Gibbs sampling. The majority of lambs were descended from lines that were divergently selected for the ability of ewes to rear multiples. The line selected in the upward direction was denoted as the high line (H line), whereas the line selected in the downward direction was the low line (L line). Analyses included the covariance between direct and maternal genetic effects, except where it was not estimable owing to small direct additive variance components, a high incidence of lambs surviving, or both. Direct heritability estimates were 0.02 for lamb survival at birth, 0.12 for lamb survival from birth to tail docking, 0.39 for lamb survival from docking to weaning, 0.28 for overall lamb survival, and 0.17 for birth weight. Corresponding estimates for the maternal genetic effect were 0.26, 0.14, 0.16, 0.14, and 0.29. Dam permanent environmental variance ratios were, respectively, 0.14, 0.09, 0.05, 0.07, and 0.07. Estimates of the direct-maternal genetic correlation were -0.60 for lamb survival from docking to weaning, -0.61 for overall lamb survival, and -0.15 to -0.23 for birth weight. Genetic, maternal genetic, and dam permanent environmental correlations between lamb birth weight and overall or age-specific lamb survival were not different from zero. Expressed relative to overall means, annual direct genetic change in the H line amounted to -0.01% per annum for lamb survival at birth, 0.52% per annum for lamb survival from birth to docking, and 1.3% per annum for lamb survival from docking to weaning. Corresponding trends in the L line were, respectively, -0.01, -0.42, and 0.042% per annum. Maternal genetic trends amounted to, respectively, 0.23, -0.11, and -0.24% per annum in the H line and -0.78, -0.50, and -0.16% per annum in the L line. It was concluded that sustained genetic progress in lamb survival is feasible if directed selection is applied to a correlated trait such as the ability of ewes to rear multiples. Key words: correlation, heritability, lamb survival, linear-threshold model, sheep

The objective of this study was to quantify the effect of heat stress during the life of a pig on its final weight, as a first step toward a genetic evaluation for heat tolerance. Data included carcass weights of 23,556 crossbred pigs [Duroc x (Landrace x Large White)] raised on 2 farms in North Carolina and slaughtered from May 2005 through December 2006. Weather data were available from a nearby weather station. Lifetime of a pig was assumed to be partitioned into 2 periods. During an initial period, the effect of heat stress was assumed to be negligible or compensated for later. During the second period ending in slaughtering, the ADG was assumed to be affected linearly by heat load. Weekly heat load was calculated as degrees of average temperature-humidity index in excess of a threshold (18[degrees]C). The total heat load (H) was the sum of weekly heat loads during the second period. During the months of January to May H was 0; H reached a peak in September. The final BW during the peak of heat stress decreased about 6 kg compared with BW during months of non-heat stress. Weekly and monthly averages of carcass weight generally moved similarly to H. However, there were large fluctuations unrelated to H; the fluctuations were different on the 2 farms. The model included the effects of farm-year of slaughter, sex, age at slaughter, and H, where age at slaughter and H were linear regressions. In analyses, the threshold was varied from 16 to 20[degrees]C, and the second period was varied from 8 to 16 wk. The greatest [R.sup.2] (10.4%) was at the threshold of temperature-humidity index = 18[degrees]C for a period of 10 wk. Varying the threshold and the length of time reduced [R.sup.2] less than 1%. Least squares means of year-month and year-week of carcass weight were calculated using a model with the fixed effects farm-year-month or farm-year-week of slaughter, sex, and age at slaughter (linear covariate), and the random effect of birth litter. Changes in BW of finisher pigs due to heat stress can be quantified by H during the last 10 wk of the life of the pig. Key words: carcass weight, heat stress, pig

The objective of this study was to describe genetic variability of pig carcass weight as a function of heat stress. Data included carcass weights of 23,556 crossbred pigs [Duroc x (Landrace x Large White)] raised on 2 farms in North Carolina and harvested from May 2005 through December 2006. Weather data were obtained from a weather station located about 20 km from the furthest farm. Weekly heat load was calculated as degrees of average temperature-humidity index (THI) in excess of 18[degrees]C. The total heat load (H) was the sum of heat loads for 10 wk before harvest. Variance components were estimated with 3 models: univariate (UNI)--not accounting for heat stress, 2-trait (MT2), and random regression (RR). In all of the models, effects included contemporary group, sex, age at harvest, sire, and litter. In MT2, observations in months in which heat stress was observed ('hot') and not observed ('cold') were treated as separate traits. Heat stress was observed in the months of August to November 2005, as well as July to October 2006. No heat stress was observed in the months of May to July 2005, January to June 2006, and November to December 2006. The RR model added a random regression on heat load for the sire effect. Heat load was adjusted to a scale ranging from 0 (no heat stress) to 5 (greatest heat stress). The heritability estimate [+ or -] SE of carcass weight in UNI was 0.17 [+ or -] 0.01. In MT2, the estimates were 0.14 [+ or -] 0.01 for 'cold' and 0.28 [+ or -] 0.01 for 'hot'; the genetic correlation between carcass weight in 'hot' and 'cold' months was 0.42 [+ or -] 0.13. The heritability estimates obtained with RR were 0.20 [+ or -] 0.11, 0.19 [+ or -] 0.15, and 0.51 [+ or -] 0.17 for H = 0, 2.5, and 5, respectively. The genetic correlation between the performance in 'cold' months (H = 0), and performance under maximum heat load (H = 5) was 0.02, between H = 0 and intermediate heat load (H = 2.5) was 0.52, and between H = 2.5 and H = 5 was 0.86. Rank correlations between EPD derived from the different models ranged from 0.82 to 0.94 between carcass weights under similar H, 0.18 to 0.54 between carcass weights under high and low H, and 0.66 to 0.91 between carcass weights of intermediate and high/low H. Heritability for growth was greater under heat stress. Selection for crossbred performance would be optimal when data for periods both in the absence and presence of heat stress were considered. Key words: carcass weight, genetic parameter, heat stress, pig, random regression

Two simulated data sets and one commercial data set were used to evaluate computing options for models in which the effects attributable to QTL were fit as covariables. The simulated data sets included records on 24,000 animals for 10 traits. Data sets 1 and 2 were simulated with low and high correlations among traits, respectively. The model included an overall mean, 160 covariables as effects attributable to QTL, the random animal genetic effect, and the random residual effect. A commercial data set included records on approximately 110,000 animals for 11 growth, reproduction, and other traits. The model included the effects usually fitted for these traits as well as 116 covariables as effects attributable to QTL; models including the number of covariables varied by trait. Initial computing was by the BLUP90IOD program, which applies iterations on data by using a preconditioned conjugate gradient algorithm with a diagonal preconditioner. Modifications included adding block preconditioners for effects attributable to QTL (BQ) and for traits (BT). With the simulated data sets and the original program, one-trait analyses without the covariables took 7 s, whereas the 10-trait analyses with the covariables took 15 min for a data set with low correlations and 1 h 40 m for a data set with high correlations. The BQ improved the convergence rate but increased the computing time. The BT decreased the computing time from 1.5 times (low correlations) to 7 times (high correlation) at a cost of greater memory requirements. For the commercial data and the complete model, computing took 10.3 h with the unmodified program and was reduced to 6 h with BT. Relative changes in computing time and convergence rate with the commercial data set were close to those of the simulated data set, with low correlations among the traits. The BQ decreased the number of rounds by less than expected. Genetic evaluation with a large number of effects attributable to QTL fit as covariables is feasible. Key words: genetic evaluation, genetic marker, molecular information

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 SM-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.sup.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). Key words: accuracy, beef cattle, growth, linear spline, random regression

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 ([greater than or equal to] 0.99 for BWT and WWT and [greater than or equal to] 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. Key words: beef cattle, growth, multibreed, random regression model, spline

The aim of this study was to investigate the possible superiority of a threshold-linear (TL) approach for calving day (CD) and calving success (CS) analysis in beef cattle over 2 multiple-trait (MT), censored models, considering CD at the first 3 calving opportunities. The CD observations on animals that failed to calve in the latter models were defined as cows being assigned a penalty value of 21 d beyond the last observed CD record within contemporary group (PEN model) or censored CD values that were randomly obtained from a truncated normal distribution (CEN-model). In the TL model, CD records were treated as missing if a cow failed to calve, and parameters were estimated in a TL analysis including CS traits (TLMISS-model). The models included the effects of contemporary group (herd x year of calving x mating management), age at calving, physiological status at mating (lactating or nonlactating cow), animal additive genetic effects, and residual. Field data included 6,763 calving records obtained from first, second, and third parities of 3,442 spring-calving Uruguayan Aberdeen Angus cows. Models were contrasted using a data splitting technique, analyzing correlations between predicted breeding values (PBV) for each pair of subsamples, by rank correlations between PBV obtained with the different models, and by inspecting percentage of sires selected in common using the different approaches at 10 and 25% hypothetical percentages of animals selected. Breeding value correlations of CD between the subsamples for the TLMISS approach were greater (0.67 to 0.68) than correlations for the censored MT models (0.49 to 0.54). Average correlations between PBV of CD in 1 subsample obtained by CEN (PEN, TLMISS) and PBV of CS in the other subsample were -0.53 (-0.55, -0.60) in the first calving opportunity (CO), -0.54 (-0.58, -0.63) in the second CO, and -0.50 (-0.49, -0.58) in the third CO. Rank correlations between PBV for CD in PEN and CEN were high (0.93 to 0.97), but correlations of either method with PBV of CD in TLMISS ranged from 0.50 to 0.71. Common identification of bulls for the top 10% of sires (25% of sires), when selected with PEN/CEN models or the TLMISS model, varied between 50 (44%) and 60 (52%). The use of the TL animal model for genetic evaluation seems attractive for genetic evaluation of fertility traits in beef cattle. Key words: beef cattle, calving day, calving success, censored record, model comparison, threshold-linear model

Calving records (n = 6,763) obtained from first, second, and third parities of 3,442 spring-calving, Uruguayan Aberdeen Angus cows were used to estimate heritabilities and genetic correlations for the linear trait calving day (CD) and the binary trait calving success (CS), using models that considered CD and CS at 3 calving opportunities as separate traits. Three approaches were defined to handle the CD observations on animals that failed to calve: 1) the cows were assigned a penalty value of 21 d beyond the last observed CD record within contemporary group (PEN); 2) the censored CD values were randomly obtained from a truncated normal distribution (CEN); and 3) the CD records were treated as missing, and the parameters were estimated in a joint threshold-linear analysis including CS traits (TLMISS). The models included the effects of contemporary group (herd x year of calving x mating management), age at calving (3 levels), physiological status at mating (nonlactating or lactating), animal additive genetic effects, and residual. Estimates of heritability for CD traits in the PEN and CEN data sets ranged from 0.20 to 0.31, with greater values in the first calving opportunity. Genetic correlations were positive and medium to high in magnitude, 0.57 to 0.59 in the PEN data set and 0.38 to 0.91 in the CEN data set. In the TLMISS data set, heritabilities ranged from 0.19 to 0.23 for CD and 0.37 to 0.42 for CS. Genetic correlations between CD traits varied between 0.82 and 0.88; between CS traits, genetic correlations varied between 0.56 and 0.80. Negative (genetically favorable), medium to high genetic correlations (-0.54 to -0.91) were estimated between CD and CS traits, suggesting that CD could be used as an indicator trait for CS. Data recording must improve in quality for practical applications in genetic evaluation for fertility traits. Key words: beef cattle, calving day, calving success, censored record, genetic parameter, threshold-linear model

The aim of this study was to estimate the genetic correlations between 2 purebred Duroc pig populations (P1 and P2) and their terminal crossbreds [C1 = P1 x (Landrace x Large White) and C2 = P2 x (Landrace x Large White)] raised in different production environments. The traits analyzed were backfat (BF), muscle depth (MD), BW at slaughter (WGT), and weight per day of age (WDA). Data sets from P1, P2, C1, and C2 included 26,674, 8,266, 16,806, and 12,350 animals, respectively. Two-trait models (nucleus and commercial crossbreds) for each group included fixed (contemporary group, sex, weight, and age), random additive (animal for P1 and P2 and sire for C1 and C2), random litter, and random dam (C1 and C2 only) effects. Heritability estimates ([+ or -]SE) for BF were 0.46 [+ or -] 0.04, 0.38 [+ or -] .02, 0.32 [ + or -] 0.02, and 0.33 [+ or -] 0.02 for P1, P2, C1, and C2, respectively. Heritability estimates for MD were 0.31 [+ or -] 0.01, 0.23 [+ or -] 0.02, 0.19 [+ or -] 0.01, and 0.12 [+ or -] 0.01 for P1, P2, C1, and C2, respectively. The estimates for WGT and WDA were 0.31 [+ or -] 0.01, 0.21 [+ or -] 0.02, 0.16 [+ or -] 0.01, and 0.18 [+ or -] 0.01 and 0.32 [+ or -] 0.01, 0.22 [+ or -] 0.02, 0.16 [+or -] 0.01, and 0.19 [+ or -] 0.01, respectively. Genetic correlations between purebreds and crossbreds for BF were 0.83 [+or -] 0.09 (P1 x C1) and 0.89 [+or-] 0.05 (P2 x C2), for MD 0.78 [+or -] 0.05 (P1 x C1) and 0.80 [+ or -] 0.08 (P2 x C2). For WGT and WDA, the correlations were 0.53 [+ or -] 0.08 (P1 x C1), 0.80 [+ or -] 0.10 (P2 x C2), and 0.60 [+ or -] 0.07 (P1 x C1) and 0.79 [+ or -] 0.09 (P2 x C2), respectively. (Co)variances in crossbreds were adjusted to a live BW scale. Compared with purebreds, the genetic variances in crossbreds were lower, and the residual variances were greater. Sire variances in crossbreds were approximately 20 to 30% of the animal variances in purebreds for BF and MD but were 13 to 25% for WGT and WDA. The efficiency of purebred selection on crossbreds, assessed by EBV prediction weights, ranged from 0.43 to 0.91 for line 1 and 0.70 to 0.92 for line 2. When nucleus and commercial environments differ substantially, the efficiency of selection varies by line and traits, and selection strategies that include crossbred data from typical production environments may therefore be desirable. Key words: crossbreds, genetic correlation, genotype x environment interaction, pig, plasticity, variance component

Two methods to jointly model age of dam (AOD) and age of animal in random regression analyses of growth in Gelbvieh cattle were examined. The first method (M1) was analogous to the multiple-trait analysis and consisted of AOD as a nested class variable and a cubic polynomial regression on age nested within birth, weaning, and yearly weights. The second method (M2) used two-dimensional splines, with age knots at 150, 205, 270, 340, and 390 d. The AOD knots were placed at 725, 1,464, and 2,189 d. These selected knots were used to form a two-dimensional grid containing 15 knots, each representing a specific age and AOD combination. A data set containing Gelbvieh growth records was split along contemporary groups into two data sets. Data set 1 contained 316,078 records and was used for prediction by mixed-model equations. Data set 2 contained 164,167 records and was used for cross validation. In the complete data set, only 90 and 30% of animals with birth weight had records on weaning and yearling weights, respectively. Models were evaluated based on [R.sup.2], average squared error (ASE), percent bias, and plots of solutions. The ASE for weights associated with birth weight, weaning weight, and yearling weight for Ml were 15, 505, and 703 [kg.sup.2]. With M2, large jumps in fixed-effect estimates were observed outside the two-dimensional grid. To eliminate this problem, weighted one-dimensional splines were used for extrapolation beyond the two-dimensional grid. For M2 with weighted spline extrapolation, the ASE were 15, 542, and 777 [kg.sup.2] for birth weight, weaning weight, and yearling weight, respectively. Creation of optimal two-dimensional splines is difficult when data are clustered. Despite such difficulties, the two-dimensional spline was capable of jointly and continuously modeling AOD and age of animal. Key Words: Cross Validation, Polynomial Regression, Two-Dimensional Spline

To study the genetic relationship between three grouped reasons for sow removal (SR) in consecutive parities, accounting for censoring, 13,838 records from Large White sows were analyzed. Data were from seven pure-line farms having, on average, 5.9% unknown SR. Three traits were subjectively defined, each corresponding to a classification of SR (reproductive [RR], nonreproductive [RN], and others [RO]). Records for each trait could take one of five categories, according to parity at removal (0 to 4 or later). A multivariate linear censored model was implemented. The model to estimate (co)variance components and parameters included the effects of year-season, region, contemporary group, and additive genetic effects. The most common SR was related to reproduction (48.5%). Diseases of different origin and cause, old age/parity, and sow death or loss accounted for about 18, 7, and 4% of total culls, respectively. Estimates of variance components showed heterogeneity of additive genetic and residual variances for the three traits. Estimates of heritability were 0.18, 0.13, and 0.15 for RR, RN, and RO, respectively. Genetic correlations between removal codes were high ([greater than or equal to] 0.90). Results suggest sizeable additive genetic variances exist for parity at removal and different codes of removal. Different SR reasons seem to operate similarly or as a closely related genetic trait associated with fitness. In particular, RN and RO seem to be genetically indistinguishable. Data structure, definition, and volume are major limitations in studies of sow survival. A multiple-trait censored model is preferred to evaluate reasons of sow disposal. Grouped removal causes seem to be strongly genetically correlated but with heterogeneous variances, suggesting that combining all removal causes and treating the trait as parity at disposal is an alternative approach. Key Words: Sow Culling, Sow Disposal, Sow Removal, Survival, Swine

Records of on-test ADG of Large White gilts were analyzed to estimate variance components of direct and associative genetic effects. Models included the effects of contemporary group (farm-barn-batch), birth litter, pen group, and direct and associative additive genetic effects. The area of each pen was 14 [m.sup.2]. The additive genetic variance was a function of the number of competitors in a group, the additive relationships between the animal performing the record and its pen mates, and the additive relationships between pen mates. To partially account for differences in the number of pen mates, a covariable ([q.sub.i] = 1, 1/n, or [1/n.sup.1/2]) was added to the associative genetic effect. There were 4,946 records from 2,409 litters and 362 pen groups. Pen group size ranged from 12 to 16 gilts. Analyses by REML converged very slowly. A grid search showed that the likelihood function was almost flat when the additive genetic associative effect was fitted. Estimates of direct and associative heritability were 0.15 and 0.03, respectively. Within the BLUPF90 family of programs, the mixed-model equations can be set up directly. For variance component estimation, simple programs (REMLF90 and GIBBSF90) worked without modifications, but more optimized programs did not. Estimates obtained using the three values of [q.sub.i] were similar. With the data structure available for this study and under an environment with relative low competition among animals, accurate estimation of associative genetic effects was not possible. Estimation of competitive effects with large pen size is difficult. The magnitude of competition effects may be larger in commercial populations, where housing is denser and food is limited. Key Words: Competition, Genetic Effects, Growth, Swine

Estimates of direct and maternal genetic parameters in beef cattle were obtained with a random regression model with a linear spline function (SFM) and were compared with those obtained by a multitrait model (MTM). Weight data of 18,900 Gelbvieh calves were used, of which 100, 75, and 17% had birth (BWT), weaning (WWT), and yearling (YWT) weights, respectively. The MTM analysis was conducted with a three-trait maternal animal model. The MTM included an overall linear partial fixed regression on age at recording for WWT and YWT, and direct-maternal genetic and maternal permanent environmental effects. The SFM included the same effects as MTM, plus a direct permanent environmental effect and heterogeneous residual variance. Three knots, or breakpoints, were set to 1,205, and 365 d. (Co)variance components in both models were estimated with a Bayesian implementation via Gibbs sampling using flat priors. Because BWT had no variability of age at recording, there was good agreement between corresponding components of variance estimated from both models. For WWT and YWT, with the exception of the sum of direct permanent environmental and residual variances, there was a general tendency for SFM estimates of variances to be lower than MTM estimates. Direct and maternal heritability estimates with SFM tended to be lower than those estimated with MTM. For example, the direct heritability for YWT was 0.59 with MTM, and 0.48 with SFM. Estimated genetic correlations for direct and maternal effects with SFM were less negative than those with MTM. For example, the direct-maternal correlation for WWT was -0.43 with MTM and -0.33 with SFM. Estimates with SFM may be superior to MTM due to better modeling of age in both fixed and random effects. Key Words: Beef Cattle, Early Growth, Genetic Parameters, Linear Spline Function, Random Regression

Weaning weights from Gelbvieh (GV; n = 82,138) and Limousin (LM; n = 88,639) calves were used to estimate genetic and environmental variance components with models that included different values for the correlation ([lambda] between permanent environmental effects of dams and their daughters. Each analysis included fixed discrete effects of contemporary group, sex of calf, age of dam at calving, and month of calving, a fixed continuous effect of age of calf, random direct and maternal additive genetic effects, permanent environmental effects due to dams, and residual effects. The REML procedure was employed with a 'grid search,' in which the likelihood was computed for a series of values for [lambda]. For both breeds, models that included a nonzero value for A fitted the data significantly better than the model that did not include [lambda]. The maximum restricted likelihood was obtained for A of approximately -0.2 for both breeds. Estimates of residual and direct genetic variances were similar for all values of [lambda], including zero; however, estimates of maternal genetic variance and maternal heritability increased slightly, and maternal permanent environmental variance and the proportion of the maternal variance to the total (phenotypic) variance decreased slightly, when the correlated structure for permanent environmental effects was assumed. As the value of A became more negative, absolute values of the direct-maternal genetic covariance and direct-maternal correlation estimates were decreased. Pearson and rank correlations for direct genetic, maternal genetic, and maternal environmental effects estimated with and without [lambda] were very high (>0.99). These results indicated that the linear relationship between maternal permanent environmental effects of dams and their daughters for weaning weight is negative but low in both breeds. Considering this relationship in the operational model did not significantly affect estimated breeding values, and thus, it may not be important in genetic evaluations. Key Words: Beef Cattle, Genetic Parameters, Maternal Permanent Environmental Correlation, Weaning Weight

Up to 109,447 records of 49,656 Large White sows were used to evaluate the genetic relationship between number of pigs born dead (BD) and number born alive (BA) in first and later parities. Performance data (n = 30,832) for ultrasound backfat (BF) at the end of the test and days to reach 113.5 kg (AD) were used to estimate their relationships with BD and BA at first parity in a four-trait threshold-linear analysis (TL). Effects were year-farm, contemporary group (CG: farm-farrowing year-farrowing month) and animal additive genetic. At first parity, estimates of heritability were 0.09, 0.09, 0.37, and 0.31 for BA, BD, AD, and BF, respectively. The estimate of genetic correlation between BD and litter size was -0.04 (BD-BA). Corresponding values with test traits were both -0.14 (BD-AD, BD-BF). Estimates of genetic correlation between BA and performance traits were 0.08 (BA-AD) and 0.05 (BA-BF). The two test traits were moderately negatively correlated (-0.22). For later parities, a six-trait (BD, BA in three parities) TL model was implemented. The estimates of additive genetic variances and heritability increased with parity for BD and BA. Estimates of heritabilities were: 0.09, 0.10, and 0.11 for BD, and 0.09, 0.12, and 0.12 for BA in parities one to three, respectively. Estimates of genetic correlations between different parities were high (0.91 to 0.96) for BD, and slightly lower (0.74 to 0.95) for BA. Genetic correlations between BD and BA were low and positive (0.02 to 0.17) for BA in Parities 1 and 2, but negative (-0.04 to -0.10) for BA in Parity 3. Selection for increased litter size should have little effect on farrowing piglet mortality. Intense selection for faster growth and increased leanness should increase farrowing piglet mortality of first-parity sows. A repeatability model with a simple correction for the heterogeneity of variances over parities could be implemented to select against farrowing mortality. The genetic components of perinatal piglet mortality are independent of the ones for litter size in the first parity, and they show an undesirable, but not strong, genetic association in second parity. Key Words: Backfat, Farrowing Mortality, Genetic Correlations, Growth, Litter Size

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. Key Words: Growth, Random Regression Model, Splines

Genetic evaluation of growth in Gelbvieh beef cattle was examined by multiple-trait (MTM) and random regression (RRM) analysis. The data set comprised 541,108 animals with 1,120,086 records. Approximately 15% of the animals in the data set had at least one record measured outside of the accepted MTM age ranges for weaning weight (Wwt) and yearling weight (Ywt). Fourteen percent of Wwt records and 19% of Ywt records were measured outside the accepted ranges for MTM analysis, and thus were excluded from MTM evaluations. Two RRM evaluations were performed using cubic Legendre polynomials (RRML) and linear splines (RRMS) with three knots at 1, 205, and 365 d of age. Data Set 1 (dl) utilized all available records, whereas Data Set 2 (d2) included only records measured within MTM ranges (1 d, 160 to 250 d, and 320 to 410 d). The RRML models did not reach convergence until diagonalization was imposed. After diagonalization, it was found that all longitudinal models required fewer iterations to converge than the MTM. Correlations between the MTM, RRML-d2, and RRMSd2 evaluations were [greater than or equal to] 0.99 for all three traits, indicating that these models were equivalent when predicting breeding values from data within the MTM age ranges. Correlations between MTM, RRML-dl, and RRMS-dl were >0.99 for Bwt and >0.95 for Wwt and Ywt. The lower correlations for Wwt and Ywt indicate that the added information does affect breeding value prediction. The RRM has the capability to incorporate records measured at all ages into genetic evaluations at a computing cost similar to the MTM. Key Words: Beef Cattle, Legendre Polynomial, Multiple Trait, Random Regression, Spline

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 [R.sup.2] 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. Key Words: Beef Cattle, Genetic Evaluation, Random Regression

Mating and calving records for 47,533 first-calf heifers in Australian Angus herds were used to examine the relationship between days to calving (DC) and two measures of fertility in AI data: 1) calving to first insemination (CFI) and 2) calving success (CS). Calving to first insemination and calving success were defined as binary traits. A threshold-linear Bayesian model was employed for both analyses: 1) DC and CFI and 2) DC and CS. Posterior means (SD) of additive covariance and corresponding genetic correlation between the DC and CFI were -0.62 d (0.19 d) and -0.66 (0.12), respectively. The corresponding point estimates between the DC and CS were -0.70 d (0.14 d) and -0.73 (0.06), respectively. These genetic correlations indicate a strong, negative relationship between DC and both measures of fertility in AI data. Selecting for animals with shorter DC intervals genetically will lead to correlated increases in both CS and CFI. Posterior means (SD) for additive and residual variance and heritability for DC for the DC-CFI analysis were 23.5 [d.sup.2] (4.1 [d.sup.2]), 363.2 [d.sup.2] (4.8 [d.sup.2]), and 0.06 (0.01), respectively. The corresponding parameter estimates for the DC-CS analysis were very similar. Posterior means (SD) for additive, herd-year and service sire variance and heritability for CFI were 0.04 (0.01), 0.06 (0.06), 0.14 (0.16), and 0.03 (0.01), respectively. Posterior means (SD) for additive, herd-year, and service sire variance and heritability for CS were 0.04 (0.01), 0.07 (0.07), 0.14 (0.16), and 0.03 (0.01), respectively. The similarity of the parameter estimates for CFI and CS suggest that either trait could be used as a measure of fertility in AI data. However, the definition of CFI allows the identification of animals that not only record a calving event, but calve to their first insemination, and the value of this trait would be even greater in a more complete dataset than that used in this study. The magnitude of the correlations between DC and CS-CFI suggest that it may be possible to use a multitrait approach in the evaluation of AI and natural service data, and to report one genetic value that could be used for selection purposes. Key Words: Beef Cattle, Genetic Evaluation, Heifer Fertility

Mating and calving records for 51,084 first-parity heifers in Australian Angus herds were used to examine the relationship between probability of calving to first insemination (CFI) in artificial insemination and natural service (NS) mating data. Calving to first insemination was defined as a binary trait for both sources of data. Two Bayesian models were employed: 1) a bivariate threshold model with CFI in AI data regarded as a trait separate from CFI in NS data and 2) a univariate threshold model with CFI regarded as the same trait for both sources of data. Posterior means (SD) of additive variance in the bivariate analysis were similar: 0.049 (0.013) and 0.075 (0.021) for CFI in AI and NS data, respectively, indicating lack of heterogeneity for this parameter. A similar trend was observed for heritability in the bivariate analysis, with posterior means (SD) of 0.025 (0.007) and 0.048 (0.012) for AI and NS data, respectively. The posterior means (SD) of the additive covariance and corresponding genetic correlation between the traits were 0.048 (0.006) and 0.821 (0.138), respectively. Differences were observed between posterior means for herd-year variance: 0.843 vs. 0.280 for AI and NS data, respectively, which may reflect the higher incidence of 100% conception rates within a herd-year class (extreme category problem) in AI data. Parameter estimates under the univariate model were close to the weighted average of the corresponding parameters under the bivariate model. Posterior means (SD) for additive, herd-year, and service sire variance and heritability under the univariate model were 0.063 (0.007), 0.56 (0.029), 0.131 (0.013), and 0.036 (0.007), respectively. These results indicate that, genetically, cows with a higher probability of CFI when mated using AI also have a high probability of CFI when mated via NS. The high correlation between the two traits, along with the lack of heterogeneity for the additive variance, implies that a common additive variance could be used for AI and NS data. A single-trait analysis of CFI with heterogeneous variances for herd-year and service sire could be implemented. The low estimates of heritability indicate that response to selection for probability of calving to first insemination would be expected to be low. Key Words: Beef Cattle, Female Fertility, Threshold Models

The purpose of this study was to compare estimates of genetic parameters for sequential growth of beef cattle using two models and two data sets. Growth curves of Nellore cattle were analyzed using body weights measured at ages 1 (birth weight) to 733 d. Two data samples were created, one with 71,867 records sampled from all herds (MISS), and the other with 74,601 records sampled from herds with no missing traits (NMISS). Records preadjusted to a fixed age were analyzed by a multiple-trait model (MTM), which included the effects of contemporary group, age of dam class, additive direct, additive maternal, and maternal permanent environment. Analyses were by REML, with five traits at a time. The random regression model (RRM) included the effects of age of animal, contemporary group, age of dam class, additive direct, additive maternal, permanent environment, and maternal permanent environment. All effects were modeled as cubic Legendre polynomials. These analyses were also by REML. Shapes of estimates of variances by MTM were mostly similar for both data sets for all except late ages, where estimates for MISS were less regular, and for birth weight with MISS. Genetic correlations among ages for the direct and maternal effects were less smooth with MISS. Genetic correlations between direct and maternal effects were more negative for NMISS, where few sires were maternal grandsires. Parameter estimates with RRM were similar to MTM except that estimates of variances showed more artifacts for MISS; the estimates of additive direct-maternal correlations were more negative with both data sets and approached -1.0 for some ages with NMISS. When parameters of a growth model obtained by RRM are to be used for genetic evaluation, these parameters should be examined for consistency with parameters from MTM and prior information, and adjustments may be required to eliminate artifacts. Key Words: Beef Cattle, Genetic Parameters, Maximum Likelihood, Regression Analysis

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 is prone to numerical problems. Estimates of EPD with RRM may be more accurate than those with MTM only if accurate parameters are applied. Key Words: Beef Cattle, Genetic Analysis, Regression Analysis

Data from two purebred swine lines A (n = 6,022) and B (n = 24,170), and their reciprocal, cross C (n = 6,135), were used to examine gains in reliability of combined purebred and crossbred evaluation over conventional within-line evaluations using crossbred and pureline models. Random effects in the pureline model included additive, parental dominance, and litter. In the crossbred model, effects were as in the pureline model except traits of each line were treated as separate traits and two additive effects were present. The approximate model was the same as the pureline except it was used for all lines disregarding breed differences. The traits in the evaluation were lifetime daily gain (LDG) and backfat. When separate line evaluations were replaced by evaluations with crossbreds, mean reliabilities of predicted breeding values increased by 2 to 9% for purebreds and by 21 to 72% for crossbreds. Rank correlations between these breeding values were > 0.99 for purebreds but 0.85 to 0.87 for crossbreds. Rank correlations between predicted breeding values obtained from crossbred and approximate models were 0.98 to 0.99 for purebreds and 0.96 to 0.98 for crossbreds. When the number of crossbreds was small in comparison to purebreds, the increase in reliability by using the crossbred data and the crossbred model as opposed to purebred models was small for purebreds but large for crossbreds. The approximate model provided very similar rankings to the crossbred model for purebreds but rankings were less consistent for crossbreds. Key Words: Crossbreds, Pigs, Purebreds, Reliability, Selection

Records on lifetime daily gain and backfat from two purebred lines A (n = 6,022), B (n = 24,170), and their reciprocal crosses C (n = 6,135) were used to estimate genetic parameters using within-line and terminal-cross models. The models that were fitted included fixed (contemporary group and sex), random additive A and(or) random additive B, random dominance, and random litter effects. Model for purebreds included only one additive effect, whereas the model for crossbreds included two additive effects. End weight was included as a covariable for backfat. Heritability estimates for lifetime daily gain were 0.26, 0.28, and 0.23 with within-line models for lines A, B, and C, respectively, and 0.26, 0.30, and 0.27 with the crossbred model, respectively. Heritability estimates for backfat were 0.52, 0.35, and 0.29 with within-line models for lines A, B, and C, respectively, and 0.51, 0.38, and 0.29 with the crossbred model, respectively. The genetic correlations between purebreds and crossbreds ([r.sub.pc]) for lifetime daily gain were 0.99 (A-C) and 0.62 (B-C); for backfat the correlations were 0.32 (A-C) and 0.70 (B-C). The amount of dominance variance from the crossbred model expressed as a proportion of phenotypic variance for lifetime daily gain was 0.39, 0.16, and 0.29 for lines A, B, and C respectively. Dominance variance for backfat was estimated as 0. A joint evaluation of purebreds and crossbreds would be most efficient with the crossbred model. The dominance variation should be accounted for lifetime daily gain. Key Words: Accuracy, Crossbreds, Dominance, Genetic Correlation, Pigs

Parameters for direct and maternal dominance were estimated in models that included nonadditive genetic effects. The analyses used weaning weight records adjusted for age of dam from populations of Canadian Hereford (n = 467,814), American Gelbvieh (n = 501,552), and American Charolais (n = 314,552). Method R estimates of direct additive genetic, maternal additive genetic, permanent maternal environment, direct dominance, and maternal dominance variances as a proportion of the total variance were 23, 12, 13, 19, and 14% in Hereford; 27, 7, 10, 18, and 2% in Gelbvieh; and 34, 15, 15, 23, and 2% in Charolais. The correlations between direct and maternal additive genetic effects were -0.30, -0.23, and -0.47 in Hereford, Gelbvieh, and Charolais, respectively. The correlations between direct and maternal dominance were -0.38, -0.02, and -0.04 in Hereford, Gelbvieh, and Charolais, respectively. Estimates of inbreeding depression were -0.20, -0.18, and -0.13 kg per 1% of inbreeding for Hereford, Gelbvieh, and Charolais, respectively. Estimates of the maternal inbreeding depression were -0.01, -0.02, and -0.02 kg, respectively. The high ratio of direct dominance to additive genetic variances provided some evidence that direct dominance effects should be considered in beef cattle evaluation. However, maternal dominance effects seemed to be important only for Hereford cattle. Key Words: Dominance, Genetic Variance

Bayesian analysis via Gibbs sampling, restricted maximum likelihood (REML), and Method R were used to estimate variance components for several models of simulated data. Four simulated data sets that included direct genetic effects and different combinations of maternal, permanent environmental, and dominance effects were used. Parents were selected randomly, on phenotype across or within contemporary groups, or on BLUP of genetic value. Estimates by Bayesian analysis and REML were always empirically unbiased in large data sets. Estimates by Method R were biased only with phenotypic selection across contemporary groups; estimates of the additive variance were biased upward, and all the other estimates were biased downward. No empirical bias was observed for Method R under selection within contemporary groups or in data without contemporary group effects. The bias of Method R estimates in small data sets was evaluated using a simple direct additive model. Method R gave biased estimates in small data sets in all types of selection except BLUP. In populations where the selection is based on BLUP of genetic value or where phenotypic selection is practiced mostly within contemporary groups, estimates by Method R are likely to be unbiased. In this case, Method R is an alternative to single-trait REML and Bayesian analysis for analyses of large data sets when the other methods are too expensive to apply. Key Words: Bayesian Theory, Estimation, Selection, Variance

Records from the Hereford Associations of the United States (USA), Canada, and Uruguay were used to estimate genetic and phenotypic variances and covariances for weaning weight. Estimation was done using a complete animal model, relatively large data sets, and the same methodology for the three countries in order to determine whether genetic parameters for weaning weight were homogeneous across environments. Data were composed of 2,322,722,487,661, and 102,986 edited weaning weight records for USA, Canada, and Uruguay, respectively. Ten samples were obtained from each country by eliminating data from small herds with fewer than 500 records, selecting herds at random from the entire data set after removing the small herds, and then retaining the direct-sire-connected contemporary groups within each sample. The final sample sizes ranged from 9,832 to 46,377 records. An accelerated EM-REML algorithm was used in estimating the (co)variance components in each sample. The estimates were pooled by calculating the arithmetic mean of the 10 samples from within each country. Direct and maternal (in parentheses) heritability estimates were .24 (.16), .20 (.16), and .23 (.18) for USA, Canada, and Uruguay, respectively. Maternal heritabilities reported here are nearly 50% smaller than the values currently used in national genetic evaluation for the breed, which were estimated using sire-maternal grandsire models. Covariance between direct and maternal was negative in all countries, accounting for 6, 8, and 10% of the total phenotypic variation, and the total dam effect was 32.5, 37.0, and 34.0% in USA, Canada, and Uruguay, respectively. Total heritabilities were similar among the countries, with values of .19, .19, and .17 for the three respective countries. The similarity of genetic and environmental parameters across the three countries suggests that joint genetic evaluation is feasible across environments provided that the genotype x environment interaction is negligible and can be ignored. Key Words: Beef Cattle, Genetic Parameters

Several models were evaluated in terms of predictive ability for calving difficulty. Data included birth weight and calving difficulty scores provided by the American Gelbvieh Association from 26,006 calves born to first-parity cows and five simulated populations of 6,200 animals each. Included in the model were fixed age of dam x sex interaction effects, random herd-year-season effects, and random animal direct and maternal effects. Bivariate linear-threshold and linear-linear models for birth weight/calving ease and univariate threshold and linear models for calving ease were applied to the data sets. For each data set and model, one-half of calving ease records were randomly discarded. Predictive ability of the different models was defined with the mean square error (MSE) for the difference between a deleted calving ease score and its prediction obtained from the remaining data. In terms of correlation between simulated and predicted breeding values, the threshold models had a 1% advantage for direct genetic effects and 3% for maternal genetic effects. In simulation, the average MSE was .29 for linear-threshold, .32 for linear-linear, .37 for threshold, and .39 for linear model. For the field data set, the MSE was .31, .33, .39, and .40, respectively. Although the bivariate models for calving ease/birth weight were more accurate than univariate models, the threshold models showed a greater advantage under the bivariate model. For the purpose of genetic evaluation for calving difficulty in beef cattle, the use of the linear-threshold model seems justified. In dairy cattle, the evaluation for calving ease can benefit from recording birth weight. Key Words: Threshold Models, Beef Cattle, Dystocia

Birth weight and calving difficulty were analyzed with Bayesian methodology using univariate linear models, a bivariate linear model, a threshold model for calving difficulty, and a joint threshold-linear model using a probit approach. Field data included 26,006 records of Gelbvieh cattle. Simulated populations were generated using parameters estimated from the field data. The Gibbs sampler was used to obtain estimates of the marginal posterior mean and standard deviation of the (co)variance components, heritabilities, and correlations. In the univariate analyses, the posterior mean of direct heritability for calving difficulty was .23 with the threshold model and .18 with the linear model. Maternal heritabilities were .10 and .08, respectively. In the bivariate analysis, posterior means of direct heritability for calving difficulty were .21 and .18 for the bivariate linear-threshold and linear-linear model, respectively. Maternal heritabilities were .09 and .06, respectively. Direct heritability for birth weight was .25 for the univariate model and .26 for bivariate models. Maternal heritability was .05 for the linear-threshold model and the univariate model and .06 for the bivariate linear model. Genetic correlation between direct genetic effects in both traits was .81 for the linear-threshold model and .79 for the bivariate linear. Residual correlation was .35 for the bivariate linear model and .50 for the bivariate linear-threshold. A simulation study confirmed that the posterior mean of the marginal distribution was suitable as a point estimate for univariate threshold and bivariate linear-threshold models. Key Words: Bayesian Analysis, Threshold Models, Beef Cattle, Dystocia

Carcass measurements of 12th-rib fat thickness (CARCFAT), longissimus muscle area (CARCLMA), and weight (CARCWT) on 2,028 Brangus and Brangus-sired fed steers and heifers, as well as yearling weights (YWT) and ultrasound measures of 12th-rib fat thickness (USFAT) and longissimus muscle area (USLMA) on 3,583 Brangus bulls and heifers were analyzed to estimate genetic parameters. Data were analyzed using a six-trait animal model and an average information REML algorithm. The model included fixed effects for contemporary group and breed of dam, covariates for age at slaughter or measurement, and random animal and residual effects. Heritabilities for CARCFAT, CARCLMA, CARCWT, USFAT, USLMA, and YWT were .27 [+ or -] .05, .39 [+ or -] .05, .59 [+ or -] .06, .11 [+ or -] .03, .29 [+ or -] .04, and .40 [+ or -] .04, respectively. Genetic correlations between CARCFAT and USFAT, CARCLMA and USLMA, and CARCWT and YWT were .69 [+ or -] .18, .66 [+ or -] .14, and .61 [+ or -] .11, respectively. The favorable and moderately strong genetic correlations between carcass measurements and similar yearling breeding-animal ultrasound measurements indicate that such measurements of 12th-rib fat and longissimus muscle area are useful in predicting genetic values for carcass leanness and longissimus muscle area. Selection using yearling ultrasound measurements of breeding cattle should result in predictable genetic improvement for carcass characteristics. Inclusion of yearling ultrasound measurements for fat thickness and longissimus muscle area should enhance national cattle evaluation programs. Key Words: Beef Cattle, Genetic Parameters, Ultrasound, Carcasses, Fat Thickness, Longissimus Dorsi, Muscles

The objective of this study was to estimate the dominance variance for postweaning gain in Limousin cattle. Data included 215,326 records of postweaning gain from 205 to 365 d, provided by the North American Limousin Foundation. Parental dominance subclasses were formed and related using the method of Hoeschele and VanRaden. Variance components were estimated using Method R based on six samples of 50%. Fixed effects in the model included contemporary group and covariates for inbreeding and breed composition (percentage Limousin). Heterozygosity was negatively correlated with breed composition (< -.99) and was therefore not included in the model. Two types of contemporary groups used as original groups from the National Cattle Evaluation were partially based on breed composition. Original contemporary groups that were too homogeneous for breed composition were replaced by herd-year-sex classes. Two models were used with the two data sets. Model 1 contained the fixed effects described above and an additive genetic effect. Model 2 included a dominance effect in addition to the effects contained in Model 1. In total, four combinations of contemporary group x model were used. Dominance variance was computed as being four times the estimated parental subclass variance. Estimates for inbreeding depression and breed composition (percentage Limousin) were all small and not greatly affected by inclusion of dominance effects or changes in contemporary groups. Estimates of the additive variance (expressed as percentage of the phenotypic variance) were only slightly affected, with values between 20 and 21%. Dominance estimates were highly affected when passing from original (10%) and to alternative contemporary groups (18%). Such large values may indicate that dominance is important for postweaning gain. Results showed the advantage of an individual dominance approach based on sire-dam combinations; therefore, expected gains through the use of specific combination ability as a part of the mating selection criteria for growth might be high. Key Words: Limousin, Postweaning Interval, Dominance, Genetic Parameters

We used 179,485 Yorkshire reproductive and 239,354 Yorkshire growth records to estimate additive and dominance variances by Method Fraktur R. Estimates were obtained for number born alive (NBA), 21-d litter weight (LWT), days to 104.5 kg (DAYS), and backfat at 104.5 kg (BF). The singletrait models for NBA and LWT included the fixed effects of contemporary group and regression on inbreeding percentage and the random effects mate within contemporary group, animal permanent environment, animal additive, and parental dominance. The single-trait models for DAYS and BF included the fixed effects of contemporary group, sex, and regression on inbreeding percentage and the random effects litter of birth, dam permanent environment, animal additive, and parental dominance. Final estimates were obtained from six samples for each trait. Regression coefficients for 10% inbreeding were found to be -.23 for NBA, -.52 kg for LWT, 2.1 d for DAYS, and 0 mm for BF. Estimates of additive and dominance variances expressed as a percentage of phenotypic variances were, respectively, 8.8 [+ or -] .5 and 2.2 [+ or -] .7 for NBA, 8.1 [+ or -] 1.1 and 6.3 [+ or -] .9 for LWT, 33.2 [+ or -] .4 and 10.3 [+ or -] 1.5 for DAYS, and 43.6 [+ or -] .9 and 4.8 [+ or -] .7 for BF. The ratio of dominance to additive variances ranged from .78 to .11. Key Words: Pigs, Inbreeding, Dominance, Genetic Parameters

Two data sets from the USDA Livestock and Range Research Laboratory were analyzed to study dominance variance and the influence of dominance relationships. The first consisted of 4,155 birth weight (3,884 weaning weight) records of inbred USDA Line 1 Herefords. The second consisted of 8,065 birth weight (7,380 weaning weight) records from a line-cross experiment with five lines. Two models were used. Both included fixed effects of year-sex of calf and age of dam, and covariates for calving date, inbreeding of animal, and inbreeding of dam. For the second set, additional covariates were line composition and heterozygosity coefficients. Random effects were direct and maternal additive genetic, maternal permanent environment, sire-dam subclass, and residual. Model 1 considered sire-dam subclasses unrelated. Model 2 related sire-dam subclasses with a parental dominance relationship matrix. Variance components were estimated using REML. Differences between estimates with Model 1 and 2 were unimportant except for dominance variance. For the first data set, estimates with Model 2 of relative genetic direct and maternal variances, direct-maternal correlation, permanent environment, and dominance variances for birth weight were .35, .13, -.02, .03, and .25, respectively, and they were .39, .11, .04, .06 and .14 for the second data set. For weaning weight, the first data set estimates were .20, .15, -.37, .19, and .11, respectively, and they were. 16, .20, -.07, .18, and .18 for the second data set. Changes, decreases and increases, in estimates of dominance variances may be due to increased information from relationships and family types other than full-sibs. The assumption of unrelated sire-dam subclasses might not be appropriate for estimation of dominance variance in populations with many dominance relationships among siredam classes. Key Words: Hereford, Dominance, Growth, Genetic Parameters

Utility of the preconditioned conjugate gradient algorithm with a diagonal preconditioner for solving mixed-model equations in animal breeding applications was evaluated with 16 test problems. The problems included single- and multiple-trait analyses, with data on beef, dairy, and swine ranging from small examples to national data sets. Multiple-trait models considered low and high genetic correlations. Convergence was based on relative differences between left- and right-hand sides. The ordering of equations was fixed effects followed by random effects, with no special ordering within random effects. The preconditioned conjugate gradient program implemented with double precision converged for all models. However, when implemented in single precision, the preconditioned conjugate gradient algorithm did not converge for seven large models. The preconditioned conjugate gradient and successive overrelaxation algorithms were subsequently compared for 13 of the test problems. The preconditioned conjugate gradient algorithm was easy to implement with the iteration on data for general models. However, successive overrelaxation requires specific programming for each set of models. On average, the preconditioned conjugate gradient algorithm converged in three times fewer rounds of iteration than successive overrelaxation. With straightforward implementations, programs using the preconditioned conjugate gradient algorithm may be two or more times faster than those using successive overrelaxation. However, programs using the preconditioned conjugate gradient algorithm would use more memory than would comparable implementations using successive overrelaxation. Extensive optimization of either algorithm can influence rankings. The preconditioned conjugate gradient implemented with iteration on data, a diagonal preconditioner, and in double precision may be the algorithm of choice for solving mixed-model equations when sufficient memory is available and ease of implementation is essential. Key Words: Computer Software, Genetic Analysis

Reliabilities for a multiple-trait maternal model were obtained by combining reliabilities obtained from single-trait models. Single-trait reliabilities were obtained using an approximation that supported models with additive and permanent environmental effects. For the direct effect, the maternal and permanent environmental variances were assigned to the residual. For the maternal effect, variance of the direct effect was assigned to the residual. Data included 10,550 birth weight, 11,819 weaning weight, and 3,617 postweaning gain records of Senepol cattle. Reliabilities were obtained by generalized inversion and by using single-trait and multiple-trait approximation methods. Some reliabilities obtained by inversion were negative because inbreeding was ignored in calculating the inverse of the relationship matrix. The multiple-trait approximation method reduced the bias of approximation when compared with the single-trait method. The correlations between reliabilities obtained by inversion and by multiple-trait procedures for the direct effect were 0.85 for birth weight, 0.94 for weaning weight, and 0.96 for postweaning gain. Correlations for maternal effects for birth weight and weaning weight were 0.96 to 0.98 for both approximations. Further improvements can be achieved by refining the single-trait procedures. Key Words: Beef Cattle, Breeding Value, Maternal Effects, Reliability

Method R is a simple and computationally inexpensive method for estimating (co)variances. The objective of the study was to investigate properties of Method R for estimation of (co)variance components with emphasis on covariance estimation. Theoretical Method R formulas were developed for simplified single-variate and bivariate models. In single-trait models, the curve of the regression of Method R was continuous and monotonic and its slope depended on the amount of information on each animal and on the variance ratio. The curve became steeper as the number of records per animal decreased. For covariance, the curve of the regression was monotonic but not continuous. However, a regression coefficient of 1 still corresponded to the correct covariance. Similar curves were observed in analyses of simulated data sets. Because of the observed discontinuity, algorithms implementing Method R that require a continuous regression curve would not work in models with covariances. An alternative algorithm was based on a transformation matrix obtained by multiplying a matrix of numerators with the inverse of a matrix of denominators of the regression factors. Such an algorithm converged reliably for all models tested. Method R can be modified to estimate covariances in models too large for other methods. Key Words: Covariance, Statistical Analysis, Variance Covariance Matrix

This study compared the accuracy of several models for obtaining genetic evaluations of calving difficulty. The models were univariate threshold animal (TAM), threshold sire-maternal grandsire (TSM), linear animal (LAM), and linear sire-maternal grandsire (LSM) models and bivariate threshold-linear animal (TLAM), threshold-linear sire-maternal grandsire (TLSM), linear-linear animal (LLAM), and linear-linear sire-maternal grandsire (LLSM) models for calving difficulty and birth weight. Data were obtained from the American Gelbvieh Association and included 84,420 first-parity records of both calving difficulty and birth weight. Calving difficulty scores were distributed as 73.4% in the first category (no assistance), 18.7% in the second, 6.3% in the third, and 1.6% in the fourth. Included in the animal models were fixed sex of calf by age of dam subclasses, random herd-year-season effects, and random animal direct and maternal breeding values. Sire-maternal grandsire models were similar to the animal models, with animal and maternal effects replaced by sire and maternal grandsire effects. Models were compared using a data splitting technique based on the correlation of estimated breeding values from two samples, with one-half of the calving difficulty records discarded randomly in the first sample and the remaining calving difficulty records discarded in the second sample. Reported correlations are averages of 10 replicates. The results obtained using animal models confirmed the slight advantage of TAM over LAM (0.69 vs 0.63) and TLAM over LLAM (0.90 vs 0.86). Bivariate analyses greatly improved the accuracy of genetic prediction of direct effects on calving difficulty relative to univariate analyses. Similar ranking of the models was found for maternal effects, but smaller correlations were obtained for bivariate models. For sire-maternal grandsire models, no differences between sire or maternal grandsire correlations were observed for TLSM compared to LLSM, and small differences were observed between TSM and LSM. The threshold model offered advantages over the linear model in animal models but not in sire-maternal grandsire models. For genetic evaluation of calving difficulty in beef cattle, the threshold-linear animal model seems to be the best choice for predicting both direct and maternal effects. Key Words: Threshold Models, Beef Cattle, Dystocia

The objective of this study was to investigate the possibility of genotype x environment interactions for weaning weight (WWT) between different regions of the United States (US) and between Canada (CA), Uruguay (UY), and US for populations of Hereford cattle. Original data were composed of 487,661, 102,986, and 2,322,722 edited weaning weight records from CA, UY, and US, respectively. A total of 359 sires were identified as having progeny across all three countries; 240 of them had at least one progeny with a record in each environment. The data sets within each country were reduced by retaining records from herds with more than 500 WWT records, with an average contemporary group size of greater than nine animals, and that contained WWT records from progeny or maternal grand-progeny of the across-country sires. Data sets within each country were further reduced by randomly selecting among remaining herds. Four regions within US were defined: Upper Plains (UP), Cornbelt (CB), South (S), and Gulf Coast (GC). Similar sampling criteria and common international sires were used to form the within-US regional data sets. A pairwise analysis was done between countries and regions within US (UP-CB vs S-GC, UP vs CB, and S vs GC) for the estimation of (co)variance components and genetic correlation between environments. An accelerated EM-REML algorithm and a multiple-trait animal model that considered WWT as a different trait in each environment were used to estimate parameters in each pairwise analysis. Direct and maternal (in parentheses) estimated genetic correlations for CA vs UY, CA vs US, US vs UY, UP-CB vs S-GC, UP vs CB, and S vs GC were .88 (.84), .86 (.82), .90 (.85), .88 (.87), .88 (.84), and .87 (.85), respectively. The general absence of genotype x country interactions observed in this study, together with a prior study that showed the similarity of genetic and environmental parameters across the three countries, strongly indicates that a joint WWT genetic evaluation for Hereford cattle could be conducted using a model that treated the information from CA, UY, and US as a single population using single population-wide genetic parameters. Key Words: Beef Cattle, Genotype Environment Interaction, International Comparisons