Longitudinal random effects models for genetic analysis of binary data with application to mastitis in dairy cattle

A Bayesian analysis of longitudinal mastitis records obtained in the course of lactation was undertaken. Data were 3341 test-day binary records from 329 first lactation Holstein cows scored for mastitis at 14 and 30 days of lactation and every 30 days thereafter. First, the conditional probability of a sequence for a given cow was the product of the probabilities at each test-day. The probability of infection at time t for a cow was a normal integral, with its argument being a function of "fixed" and "random" effects and of time. Models for the latent normal variable included effects of: (1) year-month of test + a five-parameter linear regression function ("fixed", within age-season of calving) + genetic value of the cow + environmental effect peculiar to all records of the same cow + residual. (2) As in (1), but with five parameter random genetic regressions for each cow. (3) A hierarchical structure, where each of three parameters of the regression function for each cow followed a mixed effects linear model. Model 1 posterior mean of heritability was 0.05. Model 2 heritabilities were: 0.27, 0.05, 0.03 and 0.07 at days 14, 60, 120 and 305, respectively. Model 3 heritabilities were 0.57, 0.16, 0.06 and 0.18 at days 14, 60, 120 and 305, respectively. Bayes factors were: 0.011 (Model 1/Model 2), 0.017 (Model 1/Model 3) and 1.535 (Model 2/Model 3). The probability of mastitis for an "average" cow, using Model 2, was: 0.06, 0.05, 0.06 and 0.07 at days 14, 60, 120 and 305, respectively. Relaxing the conditional independence assumption via an autoregressive process (Model 2) improved the results slightly.     

Genetic correlations between production and disease traits during first lactation in Holstein cows

The aim of this study was to estimate genetic correlations between milk yield, somatic cell score (SCS), mastitis, and claw and leg disorders (CLDs) during first lactation in Holstein cows by using a threshold–linear random regression test-day model. We used daily records of milk, fat and protein yields; somatic cell count (SCC); and mastitis and CLD incidences from 46 771 first-lactation Holstein cows in Hokkaido, Japan, that calved between 2000 and 2009. A threshold animal model for binary records (mastitis and CLDs) and linear animal model for yield traits were applied in our multiple trait analysis. For both liabilities and yield traits, additive genetic effects were used as random regression on cubic Legendre polynomials of days on milk. The highest positive genetic correlations between yields and disease incidences (0.36 for milk and mastitis, 0.56 for fat and mastitis, 0.24 for protein and mastitis, 0.32 for milk and CLD, 0.44 for fat and CLD and 0.31 for protein and CLD) were estimated at about the time of peak milk yield (36 to 65 days in milk). Selection focused on early lactation yield may therefore increase the risk of mastitis and CLDs. The positive genetic correlations of SCS with mastitis or CLD incidence imply that selection to reduce SCS in the early stages of lactation would decrease the incidence of both mastitis and CLD.     

Random regression models for genetic evaluation of clinical mastitis in dairy cattle

A genetic analysis of longitudinal binary clinical mastitis (CM) data recorded on about 90 000 first-lactation Swedish Holstein cows was carried out using linear random regression models (RRM). This method for genetic evaluation of CM has theoretical advantages compared to the method of linear cross-sectional models (CSM), which is currently being used. The aim of this study was to investigate the feasibility and suitability of estimating genetic parameters and predicting breeding values for CM with a linear sire RRM. For validation purposes, the estimates and predictions from the RRM were compared to those from linear sire longitudinal multivariate models (LMVM) and CSM. For each cow, the period from 10 days before to 241 days after calving was divided into four 1-week intervals followed by eight 4-week intervals. Within each interval, presence or absence of CM was scored as '1' or '0'. The linear RRM used to explain the trajectory of CM over time included a set of explanatory variables plus a third-order Legendre polynomial function of time for the sire effect. The time-dependent heritabilities and genetic correlations from the chosen RRM corresponded fairly well with estimates obtained from the linear LMVM for the separate intervals. Some discrepancy between the two methods was observed, with the more unstable results being obtained from the linear LMVM. Both methods indicated clearly that CM was not genetically the same trait throughout lactation. The correlations between predicted sire breeding values from the RRM, summarized over different time periods, and from linear CSM were rather high. They were, however, less than unity (0.74 to 0.96), which indicated some re-ranking of sires. Sire curves based on the time-specific breeding values from the RRM illustrated differences in intercept and slope among the best and the worst sires. To conclude, a linear sire RRM seemed to work well for genetic evaluation purposes, but was sensitive for estimation of genetic parameters.     

Genetic parameters for serial, automatically recorded milkability and its relationship to udder health in dairy cattle

Serial measurements of three milkability traits from two commercial dairy farms in Germany were used to estimate heritabilities and breeding values (BVs). Overall, 6352 cows in first, second and third lactations supplied 2 188 810 records based on daily values recorded from 1998 to 2003. Only the records between day 8 and day 305 after calving were considered. The estimated genetic correlations between different parities within the three milkability traits ranged from rg = 0.88 to 0.98, i.e. they were sufficiently high to warrant a repeatability model. The resulting estimated heritability coefficients were h2 = 0.42 for average milk flow, h2 = 0.56 for maximum milk flow and h2 = 0.38 for milking time. We analysed the genetic correlation between milkability and somatic cell score (SCS) and between milkability and the liability to mastitis, respectively, as the optimum milk flow for udder health is not well defined. There were 66 146 records with information on somatic cell count. Furthermore, 23 488 days of medical treatment for udder diseases were available, resulting in 2 600 302 days of observation in total. Heritabilities for the liability to mastitis, estimated with a test-day threshold model, were h2 = 0.19 and h2 = 0.13, depending on the data-recording period (first 50 days of lactation and first 305 days of lactation, respectively). With respect to the relationship between milkability and udder health, the results indicated a slight and linear correlation insofar as one can assume: the higher the milk flow, the worse the udder health. For this reason, bulls and cows with high BVs for milk flow should be excluded from breeding to avoid a deterioration of udder health. The establishment of a special data-recording scheme for functional traits such as milkability and mastitis on commercial dairy farms may be possible according to these results.     

Genetic analysis for mastitis resistance and milk somatic cell score in French Lacaune dairy sheep

Genetic analysis for mastitis resistance was studied from two data sets. Firstly, risk factors for different mastitis traits, i.e. culling due to clinical or chronic mastitis and subclinical mastitis predicted from somatic cell count (SCC), were explored using data from 957 first lactation Lacaune ewes of an experimental INRA flock composed of two divergent lines for milk yield. Secondly, genetic parameters for SCC were estimated from 5 272 first lactation Lacaune ewes recorded among 38 flocks, using an animal model. In the experimental flock, the frequency of culling due to clinical mastitis (5%) was lower than that of subclinical mastitis (10%) predicted from SCC. Predicted subclinical mastitis was unfavourably associated with the milk yield level. Such an antagonism was not detected for clinical mastitis, which could result, to some extent, from its low frequency or from the limited amount of data. In practice, however, selection for mastitis resistance could be limited in a first approach to selection against subclinical mastitis using SCC. The heritability estimate of SCC was 0.15 for the lactation mean trait and varied from 0.04 to 0.12 from the first to the fifth test-day. The genetic correlation between lactation SCC and milk yield was slightly positive (0.15) but showed a strong evolution during lactation, i.e. from favourable (-0.48) to antagonistic (0.27). On a lactation basis, our results suggest that selection for mastitis resistance based on SCC is feasible. Patterns for genetic parameters within first lactation, however, require further confirmation and investigation.     

Genetic parameters for paratuberculosis infection and effect of infection on production traits in Israeli Holsteins

Paratuberculosis (Johne's disease) is an infectious enteric disease in dairy cattle and other species that causes substantial economic loss worldwide. In this study, two recursive Gaussian-threshold models were employed in order to infer the effects of Johne's disease on milk yield, fat yield, and protein yield while simultaneously estimating genetic parameters (i.e. heritability and genetic correlation) in an Israeli Holstein population. Disease diagnosis was based on ELISA serum antibody tests. Data were available for 4694 daughters of 361 sires; 3.5% were positive; and 1.6% were suspect for the disease test. Disease status was coded either as a binary character (negative vs. positive) or as an ordered categorical trait (negative, suspect, and positive) in the two recursive models and as a binary trait in a linear model. Among sires with ≥ 50 daughters, predicted probability of Mycobacterium avium ssp. paratuberculosis (MAP) infection in future daughters ranged from <1% to 16.5%. Heritability estimates for Johne's disease were near 0.15, confirming a genetic contribution to disease susceptibility. Genetic correlation estimates for Johne's disease with the three yield traits were 0.15-0.22. Residual correlations for Johne's disease with the yield traits were between -0.01 and -0.10. For the linear regression model, yield losses associated with a positive disease diagnosis during 305 days of lactation were 300 kg milk and around 10 kg for fat and protein. Yield loss estimates from the recursive models were 25-50% less than linear model estimates. Recursive modeling has theoretical advantages over linear models for these phenotypes, but the estimated genetic parameters in this study did not differ significantly between the two types of models.     

Genetic parameters for production,health, fertility and longevity traits in dairy cows

Milk production, fertility, longevity and health records, were extracted from databases of two milk recording organisations in the United Kingdom for the first three lactations of the Holstein–Friesian breed. These included data related to health events (mastitis and lameness), voluntarily recorded on a proportion of farms. The data were analysed to calculate disease incidence levels and to estimate genetic parameters for health traits and their relationships with production and other functional traits. The resulting dataset consisted of 124 793 lactations from 75 137 animals of 1586 sires, recorded in 2434 herds. Incidence of health events increased with parity. The overall incidence of mastitis (MAS) and lameness (LAM), defined as binary traits, were 17% and 16%, respectively. Heritability estimates for MAS and LAM were 0.04 and 0.02, respectively, obtained from repeatability linear sire models. Heritability estimates of mastitis and lameness as count traits were slightly higher, 0.05 and 0.03, respectively. Genetic correlations were obtained by bivariate analyses of all pair-wise combinations between milk 305-day yield (MY), protein 305-day yield (PY), fat 305-day yield (FY), lactation average log_e transformed lactation average somatic cell count (SCS), calving interval (CI), days to first service (DFS), non-return at 56 days (NR56), number of inseminations (NINS), mastitis (MAS), number of mastitis episodes (NMAS), lameness (LAM), number of lameness episodes (NLAM) and lifespan score (LS). As expected, MAS was correlated most strongly with SCS (0.69), which supports the use of SCS as an indicator trait for mastitis. Genetic correlations between MAS and yield and fertility traits were of similar magnitude ranging from 0.27 to 0.33. Genetic correlations between MAS with LAM and LS were 0.38 and −0.59, respectively. Not all genetic correlations between LAM and other traits were significant because of fewer numbers of lameness records. LAM had significant genetic correlations with MY (0.38), PY (0.28), CI (0.35), NINS (0.38) and LS (−0.53). The heritability estimates of mastitis and lameness were low; therefore, genetic gain through direct selection alone would be slow, yet still positive and cumulative. Direct selection against mastitis and lameness as additional traits should reduce incidence of both diseases, and simultaneously improve fertility and longevity. However, both health traits had antagonistic relationships with production traits, thus genetic gain in production would be slower.     

Relation of Milk Production Loss to Milk Somatic Cell Count

Milk production loss was studied in relation to increased somatic cell count (SCC). Available data were weekly test-day milk yields and SCC (in 1,000 cells/ml), and mastitis incidences. In total, 18,131 records from 274 cows were used. Production loss was determined for test-day kg milk, kg protein, and kg energy-corrected milk. Least-squares analysis of variance was used to estimate the direct effect of Log10(SCC) on production. The recorded measures of production were first corrected for fixed effects, with adjustment factors estimated from a healthy data-set. The average daily milk yield was 19.7 kg/day in first lactation and 22.0 in later lactations. The geometric mean of SCC was 63.1 in first lactation and 107.2 in later lactations. The incidence of clinical mastitis treated by a veterinarian was 19.8% of the lactations-at-risk. Linear relationships were found between the production parameters and Log10(SCC). Quadratic and cubic effects were evaluated, but were found to contribute little to the overall fit of the models. The individual milk yield loss was 1.29 kg/day for each unit increase in Log10(SCC) for cows in first lactation. Milk yield decreased by 2.04 kg/day per unit Log10(SCC) for older cows. Corresponding values for protein yield were 0.042 and 0.067 kg/day for first and later lactations, respectively.     

Random regression models using different functions to model test-day milk yield of Brazilian Holstein cows

We analyzed 152,145 test-day records from 7317 first lactations of Holstein cows recorded from 1995 to 2003. Our objective was to model variations in test-day milk yield during the first lactation of Holstein cows by random regression model (RRM), using various functions in order to obtain adequate and parsimonious models for the estimation of genetic parameters. Test-day milk yields were grouped into weekly classes of days in milk, ranging from 1 to 44 weeks. The contemporary groups were defined as herd-test-day. The analyses were performed using a single-trait RRM, including the direct additive, permanent environmental and residual random effects. In addition, contemporary group and linear and quadratic effects of the age of cow at calving were included as fixed effects. The mean trend of milk yield was modeled with a fourth-order orthogonal Legendre polynomial. The additive genetic and permanent environmental covariance functions were estimated by random regression on two parametric functions, Ali and Schaeffer and Wilmink, and on B-spline functions of days in milk. The covariance components and the genetic parameters were estimated by the restricted maximum likelihood method. Results from RRM parametric and B-spline functions were compared to RRM on Legendre polynomials and with a multi-trait analysis, using the same data set. Heritability estimates presented similar trends during mid-lactation (13 to 31 weeks) and between week 37 and the end of lactation, for all RRM. Heritabilities obtained by multi-trait analysis were of a lower magnitude than those estimated by RRM. The RRMs with a higher number of parameters were more useful to describe the genetic variation of test-day milk yield throughout the lactation. RRM using B-spline and Legendre polynomials as base functions appears to be the most adequate to describe the covariance structure of the data.     

Genetic evaluation of mastitis liability and recovery through longitudinal analysis of transition probabilities

Background

Many methods for the genetic analysis of mastitis use a cross-sectional approach, which omits information on, e.g., repeated mastitis cases during lactation, somatic cell count fluctuations, and recovery process. Acknowledging the dynamic behavior of mastitis during lactation and taking into account that there is more than one binary response variable to consider, can enhance the genetic evaluation of mastitis.

Methods

Genetic evaluation of mastitis was carried out by modeling the dynamic nature of somatic cell count (SCC) within the lactation. The SCC patterns were captured by modeling transition probabilities between assumed states of mastitis and non-mastitis. A widely dispersed SCC pattern generates high transition probabilities between states and vice versa. This method can model transitions to and from states of infection simultaneously, i.e. both the mastitis liability and the recovery process are considered. A multilevel discrete time survival model was applied to estimate breeding values on simulated data with different dataset sizes, mastitis frequencies, and genetic correlations.

Results

Correlations between estimated and simulated breeding values showed that the estimated accuracies for mastitis liability were similar to those from previously tested methods that used data of confirmed mastitis cases, while our results were based on SCC as an indicator of mastitis. In addition, unlike the other methods, our method also generates breeding values for the recovery process.

Conclusions

The developed method provides an effective tool for the genetic evaluation of mastitis when considering the whole disease course and will contribute to improving the genetic evaluation of udder health.     

Comparison of non-linear models to describe the lactation curves for milk yield and composition in buffaloes (Bubalus bubalis)

In order to describe the lactation curves of milk yield (MY) and composition in buffaloes, seven non-linear mathematical equations (Wood, Dhanoa, Sikka, Nelder, Brody, Dijkstra and Rook) were used. Data were 116 117 test-day records for MY, fat (FP) and protein (PP) percentages of milk from the first three lactations of buffaloes which were collected from 893 herds in the period from 1992 to 2012 by the Animal Breeding Center of Iran. Each model was fitted to monthly production records of dairy buffaloes using the NLIN and MODEL procedures in SAS and the parameters were estimated. The models were tested for goodness of fit using adjusted coefficient of determination root means square error (RMSE), Durbin–Watson statistic and Akaike’s information criterion (AIC). The Dijkstra model provided the best fit of MY and PP of milk for the first three parities of buffaloes due to the lower values of RMSE and AIC than other models. For the first-parity buffaloes, Sikka and Brody models provided the best fit of FP, but for the second- and third-parity buffaloes, Sikka model and Brody equation provided the best fit of lactation curve for FP, respectively. The results of this study showed that the Wood and Dhanoa equations were able to estimate the time to the peak MY more accurately than the other equations. In addition, Nelder and Dijkstra equations were able to estimate the peak time at second and third parities more accurately than other equations, respectively. Brody function provided more accurate predictions of peak MY over the first three parities of buffaloes. There was generally a positive relationship between 305-day MY and persistency measures and also between peak yield and 305-day MY, calculated by different models, within each lactation in the current study. Overall, evaluation of the different equations used in the current study indicated the potential of the non-linear models for fitting monthly productive records of buffaloes.     

Estimation of genetic parameters for test day records of dairy traits in the first three lactations

Application of test-day models for the genetic evaluation of dairy populations requires the solution of large mixed model equations. The size of the (co)variance matrices required with such models can be reduced through the use of its first eigenvectors. Here, the first two eigenvectors of (co)variance matrices estimated for dairy traits in first lactation were used as covariables to jointly estimate genetic parameters of the first three lactations. These eigenvectors appear to be similar across traits and have a biological interpretation, one being related to the level of production and the other to persistency. Furthermore, they explain more than 95% of the total genetic variation. Variances and heritabilities obtained with this model were consistent with previous studies. High correlations were found among production levels in different lactations. Persistency measures were less correlated. Genetic correlations between second and third lactations were close to one, indicating that these can be considered as the same trait. Genetic correlations within lactation were high except between extreme parts of the lactation. This study shows that the use of eigenvectors can reduce the rank of (co)variance matrices for the test-day model and can provide consistent genetic parameters.     

Environmental effects on lactation curves included in a test-day model genetic evaluation

A large number of environmental factors affect the daily milk production of a cow. Lactation curves included in the French test-day model are modelled as a function of days in milk with semi-parametric curves (splines). The proper modelling of environmental effects in the test-day analysis was investigated using test-day records collected from the first three lactations of French Montbéliarde cows from 1988 to 2005. Four lactation-curve effects describing calving month, length of dry period, age at calving and gestation defined within parity-class were fitted. The shape of lactation curves did not depend on year of calving, which can be modelled as a constant over the whole lactation. To reduce computational requirements and time, data were pre-adjusted in a first step for fixed effects with no year interaction, and then used for genetic evaluation. Correlations for each lactation between 305-day estimates of genetic and permanent environment effects computed using pre-adjustment factors obtained at a 4-year interval were virtually one. The use of a two-step procedure had a very limited impact on the estimates of genetic and permanent environment effects. The minimum correlations with values estimated with a one-step procedure were 0.9984 and 0.9974, respectively. The knowledge of systematic environmental effects affecting the cow daily yield through lactation curves offers interesting perspectives to predict future daily milk production.     

Estimates of genetic parameters for Holstein cows for test-day yield traits with a random regression cubic spline model

Genetic parameters were estimated with restricted maximum likelihood for individual test-day milk, fat, and protein yields and somatic cell scores with a random regression cubic spline model. Test-day records of Holstein cows that calved from 1994 through early 1999 were obtained from Dairy Records Management Systems in Raleigh, North Carolina, for the analysis. Estimates of heritability for individual test-days and estimates of genetic and phenotypic correlations between test-days were obtained from estimates of variances and covariances from the cubic spline analysis. Estimates were calculated of genetic parameters for the averages of the test days within each of the ten 30-day test intervals. The model included herd test-day, age at first calving, and bovine somatropin treatment as fixed factors. Cubic splines were fitted for the overall lactation curve and for random additive genetic and permanent environmental effects, with five predetermined knots or four intervals between days 0, 50, 135, 220, and 305. Estimates of heritability for lactation one ranged from 0.10 to 0.15, 0.06 to 0.10, 0.09 to 0.15, and 0.02 to 0.06 for test-day one to test-day 10 for milk, fat, and protein yields and somatic cell scores, respectively. Estimates of heritability were greater in lactations two and three. Estimates of heritability increased over the course of the lactation. Estimates of genetic and phenotypic correlations were smaller for test-days further apart.     

Effects of stage of pregnancy on variance components,daily milk yields and 305-day milk yield in Holstein cows,as estimated by using a test-day model

Pregnancy and calving are elements indispensable for dairy production, but the daily milk yield of cows decline as pregnancy progresses, especially during the late stages. Therefore, the effect of stage of pregnancy on daily milk yield must be clarified to accurately estimate the breeding values and lifetime productivity of cows. To improve the genetic evaluation model for daily milk yield and determine the effect of the timing of pregnancy on productivity, we used a test-day model to assess the effects of stage of pregnancy on variance component estimates, daily milk yields and 305-day milk yield during the first three lactations of Holstein cows. Data were 10 646 333 test-day records for the first lactation; 8 222 661 records for the second; and 5 513 039 records for the third. The data were analyzed within each lactation by using three single-trait random regression animal models: one model that did not account for the stage of pregnancy effect and two models that did. The effect of stage of pregnancy on test-day milk yield was included in the model by applying a regression on days pregnant or fitting a separate lactation curve for each days open (days from calving to pregnancy) class (eight levels). Stage of pregnancy did not affect the heritability estimates of daily milk yield, although the additive genetic and permanent environmental variances in late lactation were decreased by accounting for the stage of pregnancy effect. The effects of days pregnant on daily milk yield during late lactation were larger in the second and third lactations than in the first lactation. The rates of reduction of the 305-day milk yield of cows that conceived fewer than 90 days after the second or third calving were significantly (P<0.05) greater than that after the first calving. Therefore, we conclude that differences between the negative effects of early pregnancy in the first, compared with later, lactations should be included when determining the optimal number of days open to maximize lifetime productivity in dairy cows.     

Modeling relationships between calving traits: a comparison between standard and recursive mixed models

Background

The use of structural equation models for the analysis of recursive and simultaneous relationships between phenotypes has become more popular recently. The aim of this paper is to illustrate how these models can be applied in animal breeding to achieve parameterizations of different levels of complexity and, more specifically, to model phenotypic recursion between three calving traits: gestation length (GL), calving difficulty (CD) and stillbirth (SB). All recursive models considered here postulate heterogeneous recursive relationships between GL and liabilities to CD and SB, and between liability to CD and liability to SB, depending on categories of GL phenotype.

Methods

Four models were compared in terms of goodness of fit and predictive ability: 1) standard mixed model (SMM), a model with unstructured (co)variance matrices; 2) recursive mixed model 1 (RMM1), assuming that residual correlations are due to the recursive relationships between phenotypes; 3) RMM2, assuming that correlations between residuals and contemporary groups are due to recursive relationships between phenotypes; and 4) RMM3, postulating that the correlations between genetic effects, contemporary groups and residuals are due to recursive relationships between phenotypes.

Results

For all the RMM considered, the estimates of the structural coefficients were similar. Results revealed a nonlinear relationship between GL and the liabilities both to CD and to SB, and a linear relationship between the liabilities to CD and SB.Differences in terms of goodness of fit and predictive ability of the models considered were negligible, suggesting that RMM3 is plausible.

Conclusions

The applications examined in this study suggest the plausibility of a nonlinear recursive effect from GL onto CD and SB. Also, the fact that the most restrictive model RMM3, which assumes that the only cause of correlation is phenotypic recursion, performs as well as the others indicates that the phenotypic recursion may be an important cause of the observed patterns of genetic and environmental correlations.     

Genetic relationship of discrete-time survival with fertility and production in dairy cattle using bivariate models

Bivariate analyses of functional longevity in dairy cattle measured as survival to next lactation (SURV) with milk yield and fertility traits were carried out. A sequential threshold-linear censored model was implemented for the analyses of SURV. Records on 96 642 lactations from 41 170 cows were used to estimate genetic parameters, using animal models, for longevity, 305 d-standardized milk production (MY305), days open (DO) and number of inseminations to conception (INS) in the Spanish Holstein population; 31% and 30% of lactations were censored for DO and INS, respectively. Heritability estimates for SURV and MY305 were 0.11 and 0.27 respectively; while heritability estimates for fertility traits were lower (0.07 for DO and 0.03 for INS). Antagonist genetic correlations were estimated between SURV and fertility (-0.78 and -0.54 for DO and INS, respectively) or production (-0.53 for MY305), suggesting reduced functional longevity with impaired fertility and increased milk production. Longer days open seems to affect survival more than increased INS. Also, high productive cows were more problematic, less functional and more liable to being culled. The results suggest that the sequential threshold model is a method that might be considered at evaluating genetic relationship between discrete-time survival and other traits, due to its flexibility.     

Genetic parameters for first lactation test-day milk flow in Holstein cows

Genetic parameters for test-day milk flow (TDMF) of 2175 first lactations of Holstein cows were estimated using multiple-trait and repeatability models. The models included the direct additive genetic effect as a random effect and contemporary group (defined as the year and month of test) and age of cow at calving (linear and quadratic effect) as fixed effects. For the repeatability model, in addition to the effects cited, the permanent environmental effect of the animal was also included as a random effect. Variance components were estimated using the restricted maximum likelihood method in single- and multiple-trait and repeatability analyses. The heritability estimates for TDMF ranged from 0.23 (TDMF 6) to 0.32 (TDMF 2 and TDMF 4) in single-trait analysis and from 0.28 (TDMF 7 and TDMF 10) to 0.37 (TDMF 4) in multiple-trait analysis. In general, higher heritabilities were observed at the beginning of lactation until the fourth month. Heritability estimated with the repeatability model was 0.27 and the coefficient of repeatability for first lactation TDMF was 0.66. The genetic correlations were positive and ranged from 0.72 (TDMF 1 and 10) to 0.97 (TDMF 4 and 5). The results indicate that milk flow should respond satisfactorily to selection, promoting rapid genetic gains because the estimated heritabilities were moderate to high. Higher genetic gains might be obtained if selection was performed in the TDMF 4. Both the repeatability model and the multiple-trait model are adequate for the genetic evaluation of animals in terms of milk flow, but the latter provides more accurate estimates of breeding values.     

Evaluation of reference lactation length in Chios dairy sheep

Definition and establishment of a fixed reference lactation length could provide useful tools for on-farm comparison of ewes and flock management as well as genetic evaluations for the breeding programme. The objectives of this study were to (i) evaluate different reference lactation lengths for the Chios dairy sheep and (ii) define the most suitable reference length for the breed. A total of 260 042 test-day milk records from 24 474 ewes in 130 flocks collected between 2003 and 2014 were used; 15 different lactation lengths were evaluated ranging from 120 to 260 days, defined at 10-day intervals as reference for the Chios sheep. The evaluation criteria included: (a) heritability and repeatability of milk yield in each reference lactation, (b) genetic correlation of reference lactation milk yield with actual lactation milk yield and yield at first test-day record and (c) correlated response in reference lactation milk yield from selection based on first test-day milk yield. The latter emulates genetic gains achieved for milk yield based on early lactation selection. Heritability and repeatability estimates of reference lactation milk yield and genetic correlation with actual lactation yield favoured long reference lactations (180 to 230 days). On the contrary, correlation with first test-day record milk yield was higher for short lactations (120 to 170 days). Moreover, selection on first test-day record milk yield would lead to a correlated response in reference yield in 220 days equal to 85% of the highest estimate achieved in the maximum reference length of 260 days (190 days when only considering first lactation milk yield). Based on the results of the present study, an overall reference lactation length for the Chios breed of 220 days post-lambing and a first lactation reference length of 190 days post-lambing are recommended.     

Quantitative genetic models for describing simultaneous and recursive relationships between phenotypes

Multivariate models are of great importance in theoretical and applied quantitative genetics. We extend quantitative genetic theory to accommodate situations in which there is linear feedback or recursiveness between the phenotypes involved in a multivariate system, assuming an infinitesimal, additive, model of inheritance. It is shown that structural parameters defining a simultaneous or recursive system have a bearing on the interpretation of quantitative genetic parameter estimates (e.g., heritability, offspring-parent regression, genetic correlation) when such features are ignored. Matrix representations are given for treating a plethora of feedback-recursive situations. The likelihood function is derived, assuming multivariate normality, and results from econometric theory for parameter identification are adapted to a quantitative genetic setting. A Bayesian treatment with a Markov chain Monte Carlo implementation is suggested for inference and developed. When the system is fully recursive, all conditional posterior distributions are in closed form, so Gibbs sampling is straightforward. If there is feedback, a Metropolis step may be embedded for sampling the structural parameters, since their conditional distributions are unknown. Extensions of the model to discrete random variables and to nonlinear relationships between phenotypes are discussed.