Genetic Parameter Estimation for Milk Yield over Multiple Parities and Various Lengths of Lactation in Danish Jerseys by Random Regression Models

Transcription

1 J. Dairy Sci. 85: American Dairy Science Association, Genetic Parameter Estimation for Milk Yield over Multiple Parities and Various Lengths of Lactation in Danish Jerseys by Random Regression Models Z. Guo, 1 M. S. Lund, P. Madsen, I. Korsgaard, and J. Jensen Department of Animal Breeding and Genetics, Danish Institute of Agricultural Sciences, Research Centre Foulum, PO Box 50, DK-8830 Tjele, Denmark ABSTRACT The objectives of this study were to test for heterogeneity of genetic and environmental variance among completed and extended records from different lactations or different days in milk (DIM) and to build a model that accounts for this heterogeneity. A total of 147, d milk yield records from Danish Jersey cows calving between 1984 and early 1999 from two regions of Denmark were used in this study. Results showed that DIM and parity influenced parameters estimated from an animal model with repeated records. Therefore, the data were analyzed using random-regression models that allow the covariance between measurements to change gradually with DIM and parity. Random regressions were fitted for additive genetic effects and permanent environmental effects using second- or third-order normalized Legendre polynomials for DIM and parity. Variances of random-regression coefficients associated with all orders of the polynomials were significant. Based on these parameter estimates, a covariance function (CF) was defined. The CF showed that the heritability decreases over parities, but within each parity heritability increases with DIM, whereas variance of permanent environmental effects increases over parities and decreases with DIM. Generally, genetic correlations were higher between records with similar DIM and parity. The results indicate that there are problems with the extension procedure used to predict 305-d milk yields. Using the covariance functions estimated in this study, breeding values could be predicted that take into account the covariance structure between records from different parities and different DIM. Received July 2, Accepted December 20, Corresponding author: M. S. Lund; morgens.lund@ agrsci.dk. 1 Current address: CGIL, Department of Animal and Poultry Science, University of Guelph, Guelph, Ontario, Canada N1G 2W1. (Key words: random regression, multiple parity, length of lactation) Abbreviation key: CF = covariance function, DJ = Danish Jersey, LP = Legendre polynomial, NZJ = Zealand Jersey, REP = repeatability model, RR DIM = random regression on DIM model, RR p = random regression on parity model, RR PDIM = random regression on parity and DIM model, USJ = American Jersey. INTRODUCTION In Denmark, breeding values for milk production traits are predicted using an animal model including the first three lactations. The records in each lactation are assumed to be repeated measurements of the same trait (i.e., with a common breeding value and permanent environmental effect). The records used for prediction are either completed 305-d records or noncompleted records extended to 305-d. Including more lactations is desirable, but results in dramatic rank changes of EBV. The simple repeatability model does not fit the data well when later lactations are included in the analysis. Previous studies showed heterogeneity of variance among records from different lactations or with very different DIM (Aamand et al., 1999). This heterogeneity can be taken into account in a multi-trait model, which allows genetic correlations to differ between measurements on the same individual (Meyer and Hill, 1997). In this context, an unstructured (co)variance matrix may be used with the number of traits equal to the number of different measurements, which may result in a highly over-parameterized model and high computational demands (Meyer and Hill, 1997). Alternatively, a canonical decomposition can be used to reduce the number of traits (Wiggans and Goddard, 1997). However, the fixed part of the model must be the same for all records, which is a serious restriction, and the number of records per trait might be small, which leads to estimability problems in the model. An alternative approach is to choose a model that allows the covariance 1596

2 GENETIC PARAMETER ESTIMATES IN DANISH JERSEYS 1597 between measurements to change gradually over time (Van der Werf and Schaeffer, 1997). This can be achieved by fitting a continuous function to the (co)variances [covariance function] (CF) to give a continuous changing of (co)variance of traits measured at different points in time (Kirkpatrick and Heckman, 1989). Kirkpatrick et al. (1990) fitted a normalized orthogonal polynomial to a given (co)variance matrix. They showed a full rank fit, in which the order of the fit is equal to the rank of the given (co)variance matrix, and a reduced rank fit, in which the order of the fit is less than the rank of the given (co)variance matrix. The parameters associated with the variance-covariance matrices can be estimated using REML and used in the mixed-model equations as a random regression (Meyer and Hill, 1997; Schaeffer and Dekkers, 1994). Theoretically, any function can be used in random regression. However, the choice of function plays an important role (Guo, 1998; Jamrozik et al., 1997). Orthogonal polynomials are used often, because the correlations between parameters are lower than with other functions. One choice is the normalized Legendre polynomial (Kirkpatrick et al., 1990). The CF can be extended to more dimensions. Veerkamp and Goddard (1998) fitted a model for milk production with random regressions on test day and for different production levels. In the current study, the objectives were to determine whether heterogeneity of variance among parities and extended 305-d production influenced the parameters in simple repeatability models. Models were developed with random regressions on parity and DIM. Data MATERIALS AND METHODS The analyses were performed on records from Danish Jersey (DJ) cows. Records for 305-d milk yield, calving data, and pedigree information were extracted for DJ cows calving between 1984 until early 1999 from the Danish national milk recording database. Records are divided in 10 geographical regions, of which two (regions 1 and 7) were chosen for the analyses. Initially, the raw data extracted from the database had about 140,000 and 150, d milk records for regions 1 and 7, respectively. In the editing process, records from each dataset were deleted according to the following criteria: 1) records without known AI-sire; 2) lactation numbers greater than 7; 3) record was not within defined range (1500 to 22,000 kg for 305-d milk yield, 65 to 850 kg for fat, 60 to 770 kg for protein, and 578 to 1095 d for age at first calving); 4) lactations, which were still in progress; 5) obvious errors in calving or birth dates; 6) if a cow was moved between herds more than twice within a lactation; 7) if the sum of breed proportion of DJ, New Zealand Jersey (NZJ), and American Jersey (USJ) was less than ; 8) if one or more of the previous lactations were missing; 9) if there were fewer than five records in a management group (see model section for definition); and 10) sire has fewer than 14 daughters with records. After using all editing criteria, 8, 9, and 10 were repeated until no more records were deleted. The simple statistics of the final data files used for statistical analyses are in Table 1. For cows in the final datasets, ancestors were traced back in the pedigree as far as parents were known. The pedigree file from region 1 included 45,583 animals, of which 1588 were sires, and 450 of those had daughters with records. The pedigree file from region 7 included 54,962 animals, of which 2110 were sires, and 337 of those had daughters with records. For preliminary analyses comparing the effects of parity and DIM on estimated parameters in a simple repeatability model, the edited data files were for each region split into eight subdatasets with different numbers of parities included and different requirements on minimum DIM of the records (Table 2). Incomplete and complete lactations with fewer than 305-DIM, but with at least two test days and at least 45 DIM, were extended to a predicted 305-d yield. The extension procedure, which is also used in the breeding value estimation routine, was described in Pedersen (1980) and Aamand et al. (1999). Models Repeatability model. The 305-d milk yields of different lactations of a cow were treated as the same trait with repeated records. The model used for prediction of breeding values (Aamand et al., 1999) was used for all subdatasets (Table 2) and will subsequently be referred to as REP. The model equation was: y = X 1 β 1 + Z 1pe pe + Z 1a a + e 1 where y isan 1 vector of 305-d milk yields. X 1 is the design matrix relating fixed effects in β 1 to y, where β 1 =[MG, LG, TP, CA, (LG TP), CYM, (LG CYM), PCI, (PCI LG TP), b DJ,b USJ,b NZJ,b het, b het(dj,usj),b het(dj,nzj),b het(usj,nzj),b CCI ]. MG is a vector of management group effects, where management groups were defined by clustering calvings close in time within herd and first lactation versus later lactations. The clustering was performed using an algorithm developed by Schmitz et al. (1991). LG is a vector of lactation group effects, where the lactation groups were defined as 1 = 1st parity, 2 = 2nd parity, 3 = 3rd parity, 4 = 4th parity, 5 = later parities (up to 7th). TP is a vector of

3 1598 GUO ET AL. Table 1. Simple statistics of edited data. DIM 1 Milk (kg) DIM Milk (kg) Parity n M SD M SD n M SD M SD 1 28, , , , , , Total 70, , The minimum and maximum DIM are 45 and 305 d. time period effects, where TP were defined as 1 = 1984 to 1988, 2 = 1989 to 1993, and 3 = 1994 to 1999). CA is effects of season of calving in months. CYM is a vector of calving-year-month effects. PCI is previous calving interval in (10-d periods), and b CCI is a regression on the current calving interval expressed in deviation from the mean current calving interval within LG and TP. Over the last 10 yr, genes from other Jersey populations have contributed significantly to the DJ population. The contributing breeds are mainly NZJ and USJ. The additive effects of gene import/crossbreeding and the heterosis due to dominance effects were taken into account through the regression coefficients on the proportion of genes from the respective breeds (b breed ) and regression on the degree of heterozygosity of each cow that is above the average in all cows (b het ). Specifically, b het(breed1,breed2) is the regression on the degree of heterozygosity due to breed1 and breed2. Breed proportions were calculated as follows: each animal s pedigree was traced back to ancestors with unknown parents. If the terminal ancestor was not purebred, it was assigned the average breed proportion of the population birth-year group to which it belonged. Then, moving one generation forwards at the breed, proportions were calculated for each individual conditional on the breed proportions of their parents. Breed proportions of DJ, NZJ, and USJ were used specifically, whereas proportions for other breeds were combined in one class. The total proportion of genes from breeds other than Jersey were and for region 1 and 7, respectively. The random permanent environmental effects are in the vector pe of length q (q equals the number of animals with records), and the random animal effects are in the vector a of length r (r equals the number of animals in the pedigree file). These random effects are linked to the appropriate records via the design matrices Z 1pe and Z 1a. e 1 is a vector of residuals. The distribution of random effects were assumed to be pe 0 I q σpe a N q+r+n 0, 0 Aσa 2 0 e i I n σe 2 1 Random regression on parity model (RR p ). The second model was a single-trait animal model with random regression on parity. In this model, the 305-d milk yields of different parities were not treated as repeated measurements of the same trait, but genetic and environmental covariances were allowed to change gradually using a CF defined by a third-order normalized Legendre polynomial (LP). This model will subsequently be referred to as (RR P ) and was used for the full datasets for the two regions, in which the first seven parities were included and the minimum requirement of length of lactation was 45 DIM (P17_45 in Table 2). The model equation was: y = X 2 β 2 + Z γ2 γ 2 + Z α2 α 2 + e 2 Table 2. Size of the final dataset split with respect to the number of parities and the minimum DIM (Min., DIM) included. Parities 1 to 7 1 to 3 Min., DIM Dataset P17_45 P17_100 P17_200 P17_305 P13_45 P13_100 P13_200 P13_305 Region 1 70,189 67,375 61,037 36,345 58,988 56,783 51,725 30,663 Region 7 77,268 73,972 66,920 40,046 65,751 63,075 57,325 34,140

4 GENETIC PARAMETER ESTIMATES IN DANISH JERSEYS 1599 where β 2 =[β 1, p ] (where β 1 was defined in a previous model); p is vector of order 7 with effects of parity for considering the general shape of 305-d milk yields over parities; X 2 is a design matrix relating effects in β 2 to y, Z γ2 (of dimension n 3q); Z α2 (of dimension n 3r) are matrices with covariates defined by the first-, second-, and third-order LPs over parities (LP1 (p), LP2 (p), LP3 (p) ) = ¹ ₂, 3/2 p 4 3, 45/8 p /8 (LP1 (p) can be any constant and was actually set to 1); γ 2 is vector of length 3q with random-regression coefficients for the permanent environmental effects; and α 2 is a vector of length 3r with random-regression coefficients related to animal additive genetic effects. The third-order LP was found to be sufficient in preliminary analyses. When a fourth-order LP was added, convergence problems arose from the near singularity of the variance covariance matrices. This means that the variances and covariances of the fourth order are simply a linear combination of the first three and therefore redundant. The random effects were assumed to follow a multivariate normal distribution: γ 2 α 2 e 2 0 I q P N q+r+n 0, 0 A G 2 0, I n σe 2 2 where P 2 and G 2 are 3 3 variance covariance matrices. Random regression on DIM model (RR DIM ). The third model was a single-trait animal model with random regression on DIM. In this model, the 305-d milk yields of different length, were allowed to change gradually using a covariance function defined by a secondorder LP, instead of having been treated as the same trait with repeated measurements the genetic and environmental covariances were allowed to change gradualy over DIM. Preliminary analyses indicated the second-order LP was sufficient. This model will subsequently be referred to as RR DIM and was used for datasets P17_45 for the two regions. The model equation was: y = X 3 β 3 + Z γ3 γ 3 + Z α3 α 3 + e 3, where β 3 =[β 1, b p ], where β 1 was defined in a previous model. b p is a vector of fixed regressions on LPs over DIM (LP1 (DIM), LP2 (DIM) ) = DIM 175 1, 3/2, nested 130 within parity for considering the general shape of extended 305-d milk yields over DIM (DIM [45, 305]); X 3 is a design matrix relating effects in β 3 to y; γ 3 is a vector of length 2q with random-regression coefficients for the permanent environmental effects; α 3 is a vector of length 2r with random-regression coefficients related to animal additive genetic effects; and Z γ3 (of dimension n 2q) and Z α3 (of dimension n 2r) are matrices with the corresponding covariates defined by the first- and second-order LPs over DIM. The random effects were assumed to follow a multivariate normal distribution: γ 3 α 3 e 3 0 I q P N q+r+n 0, 0 A G 3 0, I n σe 2 where P 3 and G 3 are 2 2 variance covariance matrices. Random regression on parity and DIM model (RR PDIM ). A single-trait animal model for random regression was applied to both parity and DIM. In this model, the genetic and environmental covariances between 305-d milk yields were allowed to change gradually over parities and over DIM within parities using a CF. The function was a third-order LP for parity and a second-order LP on DIM. This model will subsequently be referred to as RR PDIM. The model equation was: y = X 4 β 4 + Z γ4 γ 4 + Z α4 α 4 + e 4, where β 4 =[β 1,p,b p ] (where β 1, p, and b p were defined in a previous model); X 4 is a design matrix relating effects in β 4 to y; γ 4 is a vector of length 4q with randomregression coefficients for the permanent environmental effects; α 4 is a vector of length 4r with randomregression coefficients related to animal additive genetic effects; and Z γ4 (of dimension n 4q) and Z α4 (of dimension n 4r) are matrices with the corresponding covariates of LP1 (p), LP2 (p), LP3 (p), and LP2 (DIM). The random effects were assumed to follow a multivariate normal distribution: γ 4 α 4 e 4 0 I q P N q+r+n 0, 0 A G 4 0, I n σe 2 4 where P 4 and G 4 are 4 4 variance covariance matrices. Method The variance-covariance matrices were estimated using an AI-REML algorithm (Jensen et al., 1997) implemented in the DMU package (Madsen and Jensen, 2000). The convergence criteria, which was the norm of the gradient vector (first derivatives of the restricted

5 1600 GUO ET AL. log-likelihood) weighted by the asymptotic standard errors and number of parameters, was set to However, in some models the criteria were not met, but the parameters and the likelihood changed very little from round to round. The difficulties in convergence were experienced because estimates were on the boundary of the parameter space, in which case the first derivatives are not necessarily zero and, therefore, the convergence criteria used are not valid. Therefore, the same models were rerun with different starting values to ensure that convergence was reached. If the optimization procedure stabilized at the same point of the likelihood surface, we concluded that convergence was met. The random-regression models reduce to repeatability models if variances for regressions on higher-order polynomials are zero. That is, RR PDIM reduces to RR P (or RR DIM ) if the variance of random-regression coefficients associated with LP2 (DIM) (or LP2 (p) and LP3 (p) )inrr PDIM are zero. Equivalently, RR P or RR DIM reduces to REP if the variance of random-regression coefficients associated with LP2 (p) in RR P or LP2 (DIM) and LP3 (DIM) in RR DIM are zero. Therefore, the variances for higherorder regressions were tested for differences from zero using the estimates of variances and the asymptotic standard deviations. Covariance Functions The basic model that describes the CF is that given by Kirkpatrick et al. (1990). Covariances can be calculated between any combinations of points on the trajectory of DIM or parity between genetic effects (φgφ ) or permanent environmental effects (φpφ ), where G and P are the estimated genetic and environmental (co)variance matrices of random-regression coefficients related to animal additive genetic effects, and animal permanent environmental effects φ is a design matrix of LPs evaluated at specific points of the curve. The number of columns in φ is the dimension of G and P, and the number of rows is the number of points for which the covariances are calculated. The phenotypic CF is in matrix notation: Σ = φgφ +φpφ +Iσ e 2, where Σ = phenotypic (co)variance matrix. REP RESULTS Two trends can be observed from the reports of the REP model (Table 3). The first is that when the dataset includes seven lactations, the heritablity is lower than when only three lactations were included. The second trend is that when short extended lactations are included, the heritability is lower as well. It is difficult to compare the heritabilities directly because they are estimated partly on the same dataset. This means that if variance components were estimated on short lactations only, the heritability would probably be much lower. However, the results clearly indicate heterogeneity of variance due to parity and DIM. RR P Variances of random-regression coefficients of genetic and permanent environmental effects on first-, second-, and third-order LP on parity were significantly different from zero (Appendix, RR P ). The phenotypic variance, as well as the ratio of permanent environmental effects, increases over lactations 1 to 7 (Table 4). The heritability is quite constant in the first three to four lactations and then decreases rapidly. Results (Table 5) show that genetic and phenotypic correlations are highest between adjacent lactations and decrease when lactations are further apart. This is especially true in region 1, in which the drop in correlations to lactation 6 and 7 are more pronounced. Table 3. Estimates of phenotypic variance (σ 2 P) and ratios of permanent environmental (PE), additive genetic (h 2 ), and residual (err) variances relative to σ 2 P from model REP. Ratio Ratio Datasets σ 2 P PE h 2 err σ 2 P PE h 2 err P17_45 508, , P13_45 485, , P17_ , , P13_ , , P17_ , , P13_ , , P17_ , , P13_ , ,

6 GENETIC PARAMETER ESTIMATES IN DANISH JERSEYS 1601 Table 4. Estimates of phenotypic variance (σ 2 P) and ratios of permanent environmental (PE) and additive genetic (h 2 ) variances relative to σ 2 P for 305-d milk yield over parities using parameters from model RR P. Parity σ 2 P h 2 PE σ 2 P h 2 PE 1 426, , , , , , , , , , , , , , RR DIM Variances of random-regression coefficients of genetic and permanent environmental effects on first- and second-order LP on DIM were significantly different from zero (Appendix, RR DIM ). The heritability changes with different lengths of lactation (Figures 1 and 2). With short lactations, the heritability is low and increases to the complete lactations. The heritabilities of short lactations in the two regions are somewhat different, but for the complete lactations, for which there are more records, the heritabilities are very similar. A few points of the continuum of lengths of lactation were chosen, and the phenotypic variance as well as the proportion of permanent environmental variance is high for the short extended lactations and decreases with DIM (Table 6). The genetic and phenotypic correlations between short and long lactations were quite low in this model (Table 7). RR PDIM Variances of random-regression coefficients of genetic and permanent environmental effects on first- and second-order LP on DIM as well as first-, second-, and third-order LP on parity were significantly different from zero (Appendix, RR PDIM ). Results (Table 8) show that the phenotypic variance, as well as the percentage of permanent environmental variance, increase over parities but decrease with an increase in DIM. The heritabilities show the opposite trend: an increase with increased DIM and a decrease for lactations 6 and 7. The genetic correlation between records from different lengths of lactation vary between 0.43 and 1.0 in first lactation (Figures 3 and 4). For records with similar DIM, the genetic correlation is high. For lactations with 100-d difference in DIM, the genetic correlation is still high at about 0.9. As the difference increases, the correlation decreases rapidly. The three-dimensional curve for region 1 is somewhat sharper than the curve for region 7, indicating that the correlation between lactations of different lengths decreases faster as the difference in lactation length increases. The curves of heritabilities as a function on the DIM within each parity are in Figures 1 and 2. For the first three or four lactations, the curves are very similar and at the same level. For later lactations, the level of the curves are considerably lower. Genetic correlations between records of the same DIM but in different parities were high (Table 9). The phenotypic correlations were also quite high between records with short lactations, but for longer lactations they decreased faster as the difference in DIM increased. The genetic correlation between two records with different DIM is constant regardless of which lactations are compared (Table 10). The other point to be drawn from Table 10 is that the higher DIM a record has, the higher the genetic correlation to a complete record in other parities. Table 5. Estimates of genetic (upper triangle) and phenotypic (lower triangle) correlations between 305-d milk yields between different parities using parameters from model RR P. Parity

7 1602 GUO ET AL. Table 6. Estimates of phenotypic variance (σ 2 P) and ratios of permanent environmental (PE) and additive genetic (h 2 ) variances relative to σ 2 P for 305-d milk yields at selected DIM with model RR DIM. DIM σ 2 P h 2 PE σ 2 P h 2 PE , , , , , , , , , , , , , , DISCUSSION This study established that heterogeneity with respect to genetic and phenotypic variance exists over parities and DIM and that genetic and phenotypic correlations between records depend on the parity and DIM. This heterogeneity influenced the estimated parameters in simple repeatability models when different requirements were set on maximum DIM and which parities were included in the analysis. The covariances between records from different parities and DIM were successfully modeled by fitting random regressions (of additive genetic effects and permanent environmental effects) on parity and DIM. Heritabilities increased with increased DIM and decrease over lactations, while the phenotypic variance, as well as the ratio of permanent environmental variance, increase over parities but decrease with an increase in DIM. Studies (Swalve, 2000), that used random-regression approaches to model milk production all used test-day records, which are not comparable to this study. Several studies have reported results from multi-trait analyses, but only for the first three lactations (Teepker and Swalve, 1988; Swalve and Van Vleck, 1987; Meyer, 1984; Meyer, 1983; Rothschild and Henderson, 1979; Tong et al., 1979). These studies generally also reported constant heritabilities over the first parities and high genetic correlations, results that agree well with the current study. In this study, short lactations had lower heritabilities, which could be due to the extension procedure that introduced extra residual variation. Genetic correlations were high between short lactations in different parities compared with correlations between short and long lactations. Two possible explanations are that some genes are primarily expressed in the beginning of the lactation or that the extension procedure tends to underestimate short lactations, generating a covariance between records of short lactations. If this covariance is not completely picked up by Figure 1. Estimated heritabilities of 305-DIM yield over DIM (region 1). Figure 2. Estimated heritabilities of 305-d milk yield over DIM (region 7).

8 GENETIC PARAMETER ESTIMATES IN DANISH JERSEYS 1603 Table 7. Estimates of genetic (upper triangle) and phenotypic (lower triangle) correlations between 305-d milk yields at selected DIM with covariance functions from model RR DIM. DIM the covariance between permanent effects, it could bias the genetic covariance upwards. In the random-regression models, the residual variances were assumed to be constant across DIM and/or parities. If heterogeneity of variance related to DIM and/or parities exist in the residuals, this will be included in the permanent environmental terms. Therefore, the permanent environmental variances cannot be interpreted in the traditional way. Our results indicate that genetic progress in later lactations will be smaller than expected from the re- peatability model for two reasons. First, the heritability decreases over parities, and second, the genetic correlation between early and late parities are considerably lower than one. The results indicate that there are problems with the extension procedure used to predict 305- d milk yields. Using the CF estimated in this study, breeding values could be predicted that take into account the covariance structure between 305-d milk records from different lactations and different DIM. Some of these problems would be overcome by the introduction of random-regression test-day models. However, in Table 8. Estimates of phenotypic variance (σ 2 P) and ratios of permanent environmental (PE) and additive genetic (h 2 ) variances relative to σ 2 P for 305-d milk yields at selected DIM with model RR PDIM. Parity DIM σ 2 P h 2 PE σ 2 P h 2 PE , , , , , , , , , , , , , , , , , , , , , , , , , ,027, , , , , , , , ,057, , , , , , , ,014, ,102, , , , , , , ,129, ,220, , , , , , ,

9 1604 GUO ET AL. Figure 3. Estimated genetic correlation between DIM (parity 1, region 1, RR PDIM ). Figure 4. Estimated genetic correlation between DIM (parity 1, region 7, RR PDIM ). such models it will still be necessary to use some lactation records such as the 305-d milk yield to predict breeding values. This is especially true for old records for which test-day measurements may have been dis- carded. For such models with a mixture of records, the ideas from the current project could be useful. Table 9. Estimates of genetic (upper triangle) and phenotypic (lower triangle) correlations between extended 305-d milk yields with the same DIM but in different parities. DIM Parity

10 GENETIC PARAMETER ESTIMATES IN DANISH JERSEYS 1605 Table 10. Estimates of genetic correlation between 305-d milk yields at DIM of 305 d vs. those of 45, 100, 200, and 250 over parities. P 1 DIM P = Parity. REFERENCES Aamand, G. P., J. Pedersen, U. S. Nielsen, J. Jensen, P. Madsen, J. R. Thomasen, and L. G. Christensen Animal model for ydelse. Report no. 86, Landsudvalget for kvæg. (In Danish). Guo, Z Modelle zur Beschreibung der Laktationskurve des Milchrindes und ihre Verwendung in Modellen zur Zuchtwertschaetzung. Cuvilllier Verlag, Goettingen, Germany. Jamrozik, J., G. J. Kistemaker, J. C. M. Dekkers, and L. R. Schaeffer Comparison of possible covariates for use in random regression model for analyses of test day yields. J. Dairy Sci. 80: Jensen, J., E. A. Mäntysaari, P. Madsen, and R. Thompsen Residual maximum likelihood estimation of (co)variance components in multivariate mixed linear models using average information. J. Ind. Soc. Agric. Stat. 49: Kirkpatrick, M., and N. Heckman A quantitative genetic model for growth, shape and other infinitive-dimensional characters. J. Math. Biol. 27: Kirkpatrick, M., D. Lofsvold, and M. Bulmer Analysis of the inheritance, selection and evolution of growth trajectories. Genetics 124: Madsen, P., and J. Jensen A user s guide to DMU. A package for analysing multivariate mixed models. Version 6, release 4. Danish Institute of Agricultural Sciences, Research Centre Foulum, Tjele, Denmark. Meyer, K Estimates of genetic parameters for milk and fat yield for first three lactations in British Friesian cows. Anim. Prod. 38: Meyer, K Maximum likelihood procedures for estimating genetic parameters for later lactations of dairy cattle. J. Dairy Sci. 66: Meyer, K., and W. G. Hill Estimation of genetic and phenotypic covariance functions for longitudinal or repeated records by restricted maximum likelihood. Livest. Prod. Sci. 47: Misztal, I., T. Strabel, J. Jamorozik, E. A. Mantysaari, and T. H. Meuwissen Strategies for estimating the parameters needed for different test-day models. J. Dairy Sci. 83: Pedersen, J Forængelse af dellaktationer. Report no Danish Institute of Animal Sciences, Copenhagen, Denmark. (In Danish with English summary and subtitles.) Rothschild, M. F., and C. R. Henderson Maximum likelihood estimates of parameters of first and second lactation milk records. J. Dairy Sci. 62: Schaeffer, L. R., and J. C. M. Dekkers Random regressions in animal models for test-day procuction in dairy cattle. 5th World Congr. Genet. Appl. Livest. Prod. 18: Swalve, H. H., and L. D. Van Vleck Estimation of genetic (co)variance for milk yields in first three lactations using an animal model and restricted maximum likelihood. J. Dairy Sci. 70: Swalve, H. H Theoretical basis and computational methods for different test-day genetic evaluation methods. J. Dairy Sci. 83: Schmitz, F., R. W. Everet, and R. I. Quass Herd-year season clustering. J. Dairy Sci. 74: Teepker, G., and H. H. Swalve Estimation of genetic parameters for milk production in the first lactations. Livest. Prod. Sci. 20:

11 1606 GUO ET AL. Tong, A. K. W., B. W. Kennedy, and J. E. Moxley Heritabilities and genetic correlation for first three lactaions from records subject to culling. J. Dairy Sci. 62: Van der Werf, J., and L. R. Schaeffer Random regression in animal breeding. Course notes. CGIL Guelph, June 25 28, Veerkamp, R. F., and M. E. Goddard Covariance function across herd production levels for test day records on milk, fat, and protein yields. J. Dairy Sci. 81: Wiggans, G. R., and M. E. Goddard A computationally feasible test day model for genetic evaluation of yield traits in the United States. J. Dairy Sci. 80: APPENDIX Estimates for (co)variance (diagonal and above) and correlation (below diagonal) structure for random-regression coefficients of breeding values and permanent environment, and their standard errors (in parentheses). RR P Region 1 Ĝ 2 = 155,109.4 (13,006.8) 10,084.0 (9456.2) 25,076.8 (4043.0) (0.1594) 28,490.9 (9344.8) (3569.4) (0.1170) (0.1280) (1972.3) Pˆ 2 = 148,916.3 (10,363.9) 38,917.3 (8470.2) 24,501.5 (3901.8) (0.0590) 82,139.4 (10,293.4) 27,089.5 (4738.4) (0.0833) (0.0548) 19,624.6 (3498.7) Region 7 Ĝ 2 = 144,677.9 (12,425.4) 19,042.4 (7692.9) 25,859.5 (3721.8) (0.2454) 12,101.6 (6929.5) (2883.4) (0.0952) (0.2261) (1849.9) Pˆ 2 = 197,479.3 (10,840.2) 53,871.6 (8308) 40,285.2 (4018.2) (0.0448) 109,803.2 (10,222.6) 34,721.7 (5052.2) (0.0595) (0.0449) 30,188.4 (3913.8) RR DIM Rgion 1 Ĝ 3 = 126,286.2 (9715.0) 18,178.7 (5974.6) (0.0719) 29,385.8 ( ) Pˆ 3 = 144,576.4 (7841.3) 86,700.7 (5505.5) (0.0169) 84,580.0 ( ) Region 7 Ĝ 3 = 108,598.0 (8802.5) ( ) (0.1071) 20,790.3 (4690.8) Ĝ 3 = 179,319.6 (7566.8) 110,614.1 (5172.6) (0.0137) 105,075.6 ( ) RR PDIM Region 1 Ĝ 4 = 125,474.5 (12,576.3) (7044.5) 14,378.5 (3245.7) 16,509.4 (6562.6) (0.1814) 14,680.9 (5519.6) (2207.0) (4433.0) (0.1475) (0.1546) (1256.9) 84.0 (2100.0) (0.0842) (0.2055) (0.1380) 31,322.3 (5368.0) Pˆ 4 = 179,927.5 (11,864.1) 31,670.7 (7905.5) 10,690.3 (3945.7) 87,568.1 (6339.2) (0.0933) 32,923.7 (7905.5) 11,422.0 (2918.5) 19,362.5 (6274.9) (0.1022) (0.0843) (2276.8) (3211.7) (0.0280) (0.1339) (0.1211) 79, (5709.7) Region 7 Ĝ 4 = 103,810.2 (11,083.3) (6140.7) 10,345.4 (2725.5) (5356.1) (0.2051) (5285.2) (1872.9) (3976.6) (0.1799) (0.2906) (1073.4) 3,796.6 (1793.3) (0.1171) (0.2641) (0.2100) 23,139.6 (4849.9) Pˆ 4 = 231,903.8 (12,173.5) 36,475.0 (8297.1) 19,843.1 (4047.4) 112,310.5 (6117.3) (0.0752) 43,780.3 (6566.1) 14,931.4 (3085.4) 19,015.8 (6724.9) (0.0851) (0.0687) 12,433.2 (2465.9) (3352.8) (0.0244) (0.1100) (0.0976) 94,736.4 (5590.6)