Estimates of genetic and phenotypic covariance functions for postweaning growth and mature weight of beef cows

J. Anim. Breed. Genet. 116 (1999), 181 205 Ms. received: 5.2.1999 1999 Blackwell Wissenschafts-Verlag, Berlin ISSN 0931 2668 Animal Genetics and Breeding Unit, University of New England, Armidale NSW 2351, Australia Estimates of genetic and phenotypic covariance functions for postweaning growth and mature weight of beef cows By K. MEYER Introduction Weights of animals, recorded repeatedly during their life, are a typical example of longitudinal data where the trait of interest is changing, gradually but continually, over time. Until recently, such records were frequently analysed fitting a so-called repeatability model, i.e. assuming all records were repeated measurements of a single trait with constant variances. Alternative approaches have been to, somewhat arbitrarily, subdivide the range of ages and consider individual segments to represent different traits in a multi-variate analysis, or to fit a standard growth curve (e.g. Gompertz curve) to the records and analyse the parameters of the growth curve as new traits. Recently, there has been interest in random regression (RR) models for the analysis of such data. Regressing on age at recording (or functions thereof such as orthogonal polynomials), each measurement is used at the age it is taken and there is no need for age corrections. The covariances among RR coefficients then give rise to covariance functions (CF) which describe the covariance structure for the complete range of ages in the data, even pairs of ages for which there are no observations. With potentially infinitely many time points and records per animal, this has been termed an infinite dimensional analysis (KIRKPATRICK et al. 1990). Thus in essence, covariance functions are the infinite-dimensional equivalent to covariance matrices. KIRKPATRICK et al. (1994) demonstrated that a CF analysis allows permanent and temporary environmental effects to be separated. MEYER and HILL (1997) reviewed the application of CF to longitudinal data in animal breeding, and MEYER (1998b) showed that CFs can readily be estimated directly from the data by Restricted Maximum Likelihood (REML), fitting a random regression, animal model. This paper presents a covariance function analysis of mature weight records of beef cows, fitting a random regression model. It attempts to demonstrate the kind of calculations, results and problems which might be encountered with this new type of analysis, but throughout the paper, familiarity with covariance function models is assumed. Material and methods Data Data consisted of January weights of beef cows in two herds in a selection experiment carried out at the Wokalup Research Station in the South West of Western Australia. As outlined by MEYER et al. (1993), this experiment comprised two herds of about 300 cows each. One herd consisted of straightbred Polled Herefords (HEF), the other of a synthetic breed, the so-called Wokalup multibreeds or simply Wokalups (WOK), formed by mating Charolais Brahman bulls with Friesian Angus or Hereford cows. Mating periods (singlesire matings in the paddock) were comparatively short, leading to most calves being born over 7 8 weeks in April and May each year. Animals were weighed on a monthly basis U.S. Copyright Clearance Center Code Statement: 0931 2668/99/1603 0181 $14.00/0

182 K. Meyer Table 1. Characteristics of the data structure for January weights of Polled Hereford and Wokalup cows Age (years) 2 3 4 5 6 7 8 9 10 (months) 19 22 31 34 43 47 55 58 67 71 79 82 91 95 103 107 115 119 Polled Hereford No. records 755 668 514 443 321 234 180 121 75 Mean age (months) 21.0 33.0 45.0 56.9 68.9 80.9 92.9 104.8 116.7 Mean weight (kg) 402.8 488.4 551.9 586.7 597.1 603.2 611.3 604.1 609.0 S.D. (kg) 52.0 64.1 60.9 65.7 64.1 59.2 59.0 58.5 60.6 Wokalup No. records 813 665 513 440 370 289 201 145 94 Mean age (months) 20.8 32.9 44.9 56.8 68.9 80.9 92.8 104.8 116.9 Mean weight (kg) 446.7 521.9 584.0 611.9 626.7 640.9 637.8 632.1 633.9 S.D. (kg) 57.5 70.6 71.8 72.5 72.2 68.8 63.6 64.9 65.6 throughout the experiment, except during the calving season. Weaning usually took place in late November or early December. Selection was for increased growth until weaning, based on October weights early in the experiment and estimated breeding values for 200- day weight and milk later on. With a Mediterranean climate, characterized by Winter and Spring rains and pasture growth, and subsequent almost complete drought in Summer and Autumn, January weighings tended to record cows at their top weight during the year. Records available spanned 1974 to 1990, with cows up to 13 years of age. MEYER (1995) examined genetic parameters for mature weight in these data, considering the records of cows from 3 to 12 years of age. Here, n years of age refers to a cow in her nth year of life with most calvings in April and May, 3-year-olds, for example, were on average 33 months old in January. For this study, data from 2 to 10 years of age were selected. With a tight calving season, this yielded records on cows from 19 to 119 months old, and up to nine weights per cow. Characteristics of the data are summarized in Table 1. Earlier records were excluded to avoid problems of maternal effects. While some carry-over maternal genetic effects on final or 600-day weight, i.e. weight at about 19 months of age, have been found in these data (MEYER et al. 1993), they were assumed negligible for this study. Later data were disregarded as there were only few records per age and weights tended to fluctuate atypically. Figure 1 shows the distribution of records over monthly age classes. In total, this yielded 3320 and 3530 records on 850 and 915 cows for HEF and WOK, respectively. All nine weights were recorded for 50 (HEF) and 57 (WOK) cows, whereas 165 (HEF) and 197 (WOK) animals had only one weight available. Analyses Estimates of phenotypic (P), additive genetic (A) and permanent environmental (R) CFs were obtained by restricted maximum likelihood using program DXMRR (MEYER 1998a). As described in detail by MEYER (1998b), this involved fitting a RR model, with a set of RR coefficients for each source of variation, and estimating the covariances among RR coefficients. Fixed effects in the model of analysis were contemporary groups (CG), defined as paddock-year week (of weighing) age subclasses (CG1) or paddock-year week classes

Covariance functions for growth 183 Fig. 1. Numbers of records ( Polled Hereford, Wokalups) and mean weights (ž: Polled Hereford, : Wokalup) for ages in the data (CG2) only. For the former definition, it was distinguished between three age groups, namely 2-year-olds (19 22 months), 3-year-olds (31 34 months) and older cows. The rationale for CG1 was that CG effects might well affect heifers, young and old cows differently. The population age trend was taken into account by fitting a fixed regression on orthogonal polynomials of age. Preliminary least-squares analyses ignoring animals (i.e. considering contemporary groups and the fixed regression only) were carried out to determine the order of polynomial fit required to model the population trajectory. Random effects fitted for each animal were set(s) of RR coefficients on orthogonal polynomials of age. Model A was a phenotypic model, fitting an overall animal effect only and ignoring any relationships between animals. Model B attempted to separate additive genetic and permanent environmental effects due to the animal by fitting corresponding sets of RR coefficients, and incorporating all information available on relationships between animals, including those not in the data (parents only). Maternal effects were assumed to be absent. This allowed pedigrees to be pruned, i.e. parents without records, unknown pedigree and a single off-spring only were treated as unknown. Temporary environmental effects or measurement errors were taken into account by fitting corresponding random effects. These were assumed to be independently but nonidentically distributed. Seven different measurement error variances (s 2 oi,i = 1, 7) corresponding to ages 19 22 months, 31 34 months, 43 47 months, 55 58 months, 67 71 months, 79 82 months, and 91 119 months were estimated. Analyses were carried out for increasing orders of fit (k) of the RR coefficients. Polynomials up to k = 7 were considered. For model B, additive genetic and permanent environmental RR were fitted to the same order throughout. Note that the first term in a polynomial is a scalar, i.e. for an order to fit of k the exponents of age range from 0 to k 1. Hence, k = 2 gave a linear, k = 3 a quadratic and k = 4 a cubic regression. The fixed regression was

184 K. Meyer fitted to the maximum order for each set of analyses, k = 7 for models A and B and fixed effects CG1 and k = 6 for model B and definition CG2. While this resulted in some overparameterization, it implied that the fixed effects part of the model was constant and thus that likelihood ratio tests (LRT) could be performed to determine the optimum order of fit for the RR coefficients. Coefficient matrices of the CFs (K P, K A or K R for P, A and R, respectively) were estimated as covariance matrices of the corresponding sets of RR coefficients. Additive genetic and permanent environmental RR coefficients (model B) were assumed to be uncorrelated, as is common practice for the respective random effects in finite-dimensional linear model analyses. At outlined by MEYER (1998b), estimation can be carried out forcing these covariance matrices to have reduced rank. Several analyses were performed for each order of fit, increasing the rank (m) of the covariance matrices of RR coefficients one-by-one until full rank was reached (m = k) orm= 4 (model A) or m = 3 (model B). LRTs were carried out to determine the optimum order of fit for each data set and type of analysis. A test for an increased order of fit involves the null hypothesis that the additional (co)variance components are zero, i.e. is at the boundary of the parameter space and thus subject to non-standard conditions (SELF and LIANG 1987). As outlined by STRAM and LEE (1994), the LRT criterion to test whether the variance and k covariances pertaining to the (k+1)th RR coefficient are zero, has a distribution which is a mixture of x 2 distributions with k and k+1 degrees of freedom. Hence, a standard test assuming the LRT criterion has a x 2 distribution with k+1 degrees of freedom would be too conservative. To accommodate this, the error probability (a) was doubled, i.e. (minus twice) differences in likelihood were contrasted to x 2 values corresponding to 2a (and k+1 degrees of freedom). Additional analyses were performed for selected constellations (model, CG definition, order of fit) to further investigate particular results. These included analyses fitting a single measurement error variance only, regressing on age transformed to logarithmic scale instead of age, for WOK fitting separate CGs for each year of age, and for HEF eliminating records for 10-year-old cows. Results Fixed effects Mean square errors (MSE) from least-squares analyses ignoring random effects for different orders of fit of the orthogonal regression modelling the population age trajectory are given in Table 2 for both definitions of CGs. Corresponding estimates of the estimated regression lines and means for ages of data adjusted for estimates of CG effects are displayed in Figure 2 for Polled Herefords (results for Wokalups were similar; not shown). MSEs decreased up to k = 4, i.e. a cubic regression was sufficient to model age effects. Only for Wokalups fitting paddock-year week age subclasses were there small additional reductions in MSEs for k greater than four. With partial confounding of ages and CGs, the latter tended to pick up some of the age effects not explained by an inappropriate regression (i.e. k ³ 4), in particular for CG1. For instance, means adjusted for CG for k = 1, i.e. fitting a scalar term only, showed little age trends (Figure 2). Similarly, with less and larger subclasses, regression lines tended to be steeper for CG2 than for CG1 (particularly evident for k = 2, i.e. a linear regression), as CG effects explained less age effects. For CG2, estimated regression lines and adjusted means for both breeds were virtually the same for all k from four to seven, whereas corresponding lines were more spread out for CG1. This suggests that the subdivision of paddock-year week classes according to age introduced some noise. With MSEs for CG1 only slightly smaller than for CG2, both definitions seemed to account adequately for systematic trends. As shown below, there was indeed little difference in the estimates of CF and resulting covariance components for the

Covariance functions for growth 185 Table 2. Mean square errors from fixed effects analyses for different orders of fit (k) of the population trajectory, and defining contemporary groups as paddock-year week age (CG1) or paddock-year week classes (CG2) P. Hereford Wokalup CG1 CG2 CG1 CG2 No. CG a 188 157 161 113 k = 1 2583.5 4366.0 3424.4 5367.7 k = 2 2553.7 3174.1 3322.2 3823.1 k = 3 2508.8 2620.0 3281.9 3396.1 k = 4 2501.1 2587.8 3273.0 3358.8 k = 5 2501.9 2588.6 3268.7 3358.8 k = 6 2502.5 2588.1 3266.4 3359.0 k = 7 2502.5 2587.7 3265.3 3359.2 a contemporary groups Fig. 2. Estimated population trajectories (top row) and corresponding means for ages adjusting for contemporary groups (bottom row), defining contemporary groups as paddock-year-week-age subclasses (left column) and paddock-year-week classes (right column), respectively. Results for Polled Herefords and increasing orders of polynomial fit k (+ Unadjusted means, R k =1, k=2,*k=3, k=4,žk=5,k=6,žk=7)

186 K. Meyer two models. Hence, while it does not matter for the estimation of random effects and their covariances which of the fixed effects in the model of analysis account for systematic trends, it is important when interpreting the estimates of the population trajectory. Phenotypic covariance functions Analyses on a phenotypic level only (model A) were conducted to examine the behaviour of estimates under different orders of fit and numbers of parameters fitted (determined by the rank of estimated CF), unencumbered by potential problems of separating additive genetic and permanent environmental effects due to animals. Results from different analyses consisting of the maximum of the log likelihood (log L), the first three eigenvalues of the estimated covariance matrix of RR coefficients and estimates of the measurement error variances are summarized in Table 3. Likelihood values show unequivocally that a repeatability model (k = 1) does not fit the data, even when allowing for heterogeneous measurement error variances. A minimum of k = 4, i.e. a cubic regression on age was necessary to model the data adequately. While log L increased further for k 4, these increases were largely outweighed by the number of additional parameters required and not significant. Some convergence problems are highlighted by likelihood values apparently decreasing with higher orders of fit or rank, which was clearly due to failure of the search procedure to identify the maximum of log L correctly for higher numbers of parameters and close to the boundary of the parameter space (eigenvalues of K P close to zero), even with one or more restarts of the maximization procedure. For HEF, fitting a regression of order k = 7 yielded a significant increase in log L over k = 6 when allowing for a rank of m = 4. Similarly, the LRT criterion for m = 3 and k =6 versus k = 5 in HEF exceeded the corresponding x 2 value for 2a. LRTs were carried out attaching an error probability of 5 % to each test without any adjustment for the multitude of comparisons made. The first eigenvalue of the covariance matrix of RR coefficients was much larger than the second eigenvalue, generally ten to twenty times so, for all orders of fit. Likelihoods and estimates for m = 1 differed little for orders of fit from k = 4 onwards. Fitting a second eigenvalue (m = 2) yielded a significant increase in log L throughout, while the third eigenvalue was only important for k = 6 in HEF. Figure 3 shows estimates of phenotypic standard deviations for the ages in the data, derived from estimates of P and the measurement error variances. The top row displays results from analyses for different orders of fit forcing m = 2. While estimates for k = 1 and k = 2 are clearly different, values for higher orders of fit, especially k 4, are virtually indistinguishable. This demonstrates that modelling of the total variation is robust given a minimum order of fit, with differences in CFs for higher k absorbed by the heterogeneous measurement error variances. The bottom row of Figure 3 shows estimates for k = 7 allowing for different ranks of K P. Again, there was little difference between analyses, not even for a model assuming measurement errors were identically distributed. While the heterogeneous model still fitted the data significantly better, increases in likelihood (over a model fitting a single measurement error variance only) were considerably smaller than for k = 1 (Table 3). For HEF, k = 7 yielded a steep increase in standard deviation for the last four ages in the data, i.e. 10- year-old cows, the more so the higher the rank of K P. This did not match excessive variation within these age classes, neither on the observed scale nor after adjustment for contemporary groups and was thus treated with suspicion together with the associated significant increase in log L for k 4. Hence, a cubic regression (k = 4) with a covariance matrix of rank m = 2 appeared adequate to model phenotypic covariances among weights of 2- to 10-year-old cows in these data.

Covariance functions for growth 187 Table 3. Results from phenotypic covariance function analyses: maximum log likelihood (log L a ), the first three eigenvalues of the estimated phenotypic covariance function (l Pi, i = 1, 3), and estimates of measurement error variances (s 2 oi, i = 1, 7) for different orders of polynomial fit (k) and rank (m) of the estimated coefficient matrices Polled Hereford Wokalup s 1 o1 s 1 o2 s 1 o1 k m p b log L l P0 l P1 l P2 s 1 o0 s 1 o3 s 1 o4 s 1 o5 s 1 o6 log L l P0 l P1 l P2 s 1 o0 s 1 o2 s 1 o3 s 1 o4 s 1 o5 s 1 o6 0 0 1 150[95 1611 878 112[62 2406 0375 0 0 7 107[65 1119 477 0126 0256 0271 0167 0184 0370 059[98 2423 533 0879 0733 0825 0610 0369 0086 1 0 8 81[48 4897 9 441 0141 0046 0033 845 832 0264 49[18 5154 9 517 0877 0684 0682 0128 0358 0018 1 1 09 78[49 5994 52 381 0124 0036 0019 831 819 0187 35[59 5179 86 400 1996 0673 0664 0113 0315 0975 2 0 09 34[24 3348 9 9 477 0052 879 871 807 0912 0355 18[78 4795 9 9 558 0801 0537 0609 0040 0375 0196 2 1 01 21[46 3245 258 9 482 0028 815 897 822 866 0980 12[53 4477 057 9 473 0804 0506 0569 0023 0328 0978 2 2 02 21[95 3246 267 07 423 0023 812 786 814 863 0973 11[97 4465 065 33 360 0828 0507 0523 0092 0323 0943 3 0 00 28[59 3336 9 9 595 0006 823 857 841 0970 0327 14[01 4724 9 9 573 0741 0590 0612 0066 0386 0089 3 1 03 07[21 3239 253 9 475 0951 729 764 813 808 0094 08[29 4528 053 9 505 0734 0441 0569 0043 0309 0091 3 2 05 05[81 3264 254 18 310 0952 707 743 805 806 0095 07[13 4515 056 35 458 0746 0447 0545 0020 0303 0940 3 3 06 05[80 3240 258 18 305 0952 707 742 805 806 0095 07[12 4505 054 35 458 0743 0447 0547 0022 0304 0941 4 0 01 28[59 3338 9 9 595 0006 824 856 846 0972 0330 13[78 4742 9 9 576 0723 0599 0621 0080 0385 0067 4 1 05 07[00 3297 258 9 475 0963 722 754 814 811 0987 07[71 4557 053 9 592 0744 0454 0575 0050 0314 0968 4 2 08 04[16 3188 289 61 461 0919 721 707 779 817 863 05[09 4539 056 79 460 0705 0458 0510 0959 0316 0924 4 3 10 02[79 3181 279 62 468 883 720 702 754 786 775 05[99 4518 067 67 469 0704 0469 0507 0947 0320 0912 5 0 02 28[04 3331 9 9 596 0096 836 865 835 0955 0331 13[78 4750 9 9 574 0727 0591 0618 0080 0385 0067 5 1 07 05[64 3179 259 9 474 0982 703 726 811 757 0029 06[34 4614 195 9 574 0601 0482 0378 0091 0377 0944 5 2 11 00[20 3169 271 017 476 0986 661 712 740 716 775 02[63 4530 192 75 466 0690 0458 0376 0973 0303 0912 5 3 14 00[27 3143 261 016 477 0985 661 714 736 715 780 02[05 4530 101 74 479 0586 0445 0373 0948 0272 852 6 0 03 28[91 3321 9 9 596 0009 839 867 849 0948 0331 13[76 4760 9 9 575 0725 0596 0616 0076 0384 0067 6 1 19 05[38 3163 264 9 474 0099 674 737 806 767 0014 06[16 4643 103 9 570 0585 0480 0381 0002 0376 0943 6 2 14 00[01 3145 275 029 474 0988 653 718 754 701 776 02[11 4627 101 093 528 0623 0382 0401 833 0389 875 6 3 18 5[78 3130 396 068 471 0971 642 729 747 626 522 8[16 4629 115 033 436 0572 0383 0419 835 0359 749 6 6 24 4[35 3134 393 074 373 0919 637 683 725 632 509 7[04 4621 131 061 437 0573 0352 0367 821 0258 618 6 6 18 02[26 3120 286 076 614 36[35 4573 205 076 0946 a 02 049 for Herefords\ and 03 579 for Wokalups b Number of parameters

188 K. Meyer Fig. 3. Estimates of the phenotypic standard deviation for different orders of fit (k) and rank 2 (top row; + k =1, k=2, * k=3, k=4, Ž k=5, k=6, ž k= 7) and for different ranks (m) for an order of fit 7 (bottom row; + m = 1, m = 2, * m = 3, m = 4, ž m = 7, r m = 7 with single measurement error variance) for Polled Herefords (left column) and Wokalups (right column), respectively Similarity of estimated measurement error variances for later ages suggested that less than seven components might suffice, but this was not investigated. Estimated CF were F 2780.3 32.3 1132.7 803.6J F 1 J G G G P(a i,a j )=(1 a i a 2 i a 3 G 32.3 1711.7 525.3 1408.4 a G G G j G i) G 1132.7 525.3 614.7 746.1G Ga 2 G j G G G G f 803.6 1408.4 746.1 1376.0j f j with eigenvalues 4 340 and 364 (Table 3) for HEF and F 3261.3 257.7 826.3 839.5JF 1 J G GG P(a i,a j )=(1 a i a 2 i a 3 G 257.7 510.7 32.3 173.5 a G GG j G i) G 826.3 32.3 211.6 228.8GGa 2 G j G GG G f 839.5 173.5 228.8 333.4jf j with eigenvalues 5639 and 164 for WOK, and a i and a j denoting ages standardized to the a 3 j a 3 j

Covariance functions for growth 189 Fig. 4. Estimates of phenotypic covariances for Wokalups for ages in the data, fitting a regression of order 4 (cubic) with rank 2 and allowing for heterogeneous measurement variances interval 1 to 1 for which Legendre polynomials are defined. Figure 4 shows P evaluated for WOK for the ages in the data, with the spike along the diagonal depicting measurement error variances. Genetic covariance functions Results from corresponding analyses (defining CG as paddock-year week age subclasses) attempting to partition variation between animals into its additive genetic and permanent environmental components (model B) are given in Table 4. Overall, results from different orders of fit and ranks of covariance matrices of RR followed the same pattern as those from phenotypic analyses above (Table 3). Again, a cubic regression (k = 4) with rank 2 provided the most parsimonious description of the data for both breeds. For WOK, log L increased little for higher orders of fit. For HEF estimates of A and thus P for k 4 again gave dubious, greatly increased variances for 10-year-old cows and were accompanied by almost significant increases in log L. Estimates for phenotypic (s P ) and temporary environmental (s o ) standard deviations from analyses under models A and B for k = 4 and m = 2 are shown in Figure 5. On the whole, there was excellent agreement. Phenotypic variances tended to be somewhat higher for the genetic analyses (model B) than for the purely phenotypic analyses (model A), in particular for WOK. This might reflect the inclusion for pedigree information and a resulting reduction in bias of estimates due to selection of animals for growth. There was some discrepancy in estimates of s o for 2-year-old cows, especially for HEF. This was not reflected in corresponding values of s P, however, indicating that differences were picked up by the estimated CFs, i.e. that there was differential partitioning of variation at the youngest ages. As seen previously (Figure 3), there was considerably more fluctuation in estimates of s P for WOK than for HEF. Figure 5 highlights that this was clearly due to marked jumps in estimates of s o between years of age. Estimates of s o for cows 8 years and older for both breeds converged to essentially the same value though. This is consistent with the concept of a measurement error, and the resulting temporary environmental variation, which affects all records for a certain trait equally, regardless of age at recording. Conversely, this suggests that different estimates for earlier ages might pertain to somewhat different trait(s). Alternatively, it might be surmised that estimates have been biased by fitting CFs of insufficient order because the data might not have enough power to support alternative hypotheses of an order of fit higher than k = 4. However, as shown in Table 4, the pattern

190 K. Meyer Table 4. Maximum log likelihood (log L a ), the first three eigenvalues of the estimated genetic (A) and permanent environmental (R) covariance function (l Ai and l Ri, i = 1, 3 for A and R, respectively), and estimates of measurement error variances (s 2 oi, i = 1, 7) for different orders of polynomial fit (k) and rank (m) of the estimated coefficient matrices; analyses fitting paddock-year week age subclasses k m p b log L l A1 l A2 l A3 l R1 l R2 l R3 s 2 o1 s 2 o2 s 2 o3 s 2 o4 s 2 o5 s 2 o6 s 2 o7 Polled Hereford 1 1 3 275.623 1480.2 1224.0 1128 1 1 9 240.234 1404.7 884.4 566 1258 1399 1415 1281 1307 1495 2 1 11 107.215 2460.3 0 3862.8 0 472 1233 1162 1128 930 912 1288 2 2 13 104.443 2924.2 86.9 3378.6 0 448 1238 1159 1119 920 910 1282 3 1 13 60.599 1924.5 0 0 2972.4 0 0 526 1144 984 975 889 967 1280 3 2 17 46.740 2335.7 67.4 0 2106.5 356.3 0 439 1135 928 886 907 970 1082 3 3 19 46.740 2342.8 67.5 0 2106.6 356.2 0 439 1135 928 886 907 969 1081 4 1 15 53.766 2070.5 0 0 2818.5 0 0 572 1088 940 954 902 968 1223 4 2 21 28.098 2459.8 110.6 0 2049.5 299.4 0 343 1057 813 849 899 890 1046 4 3 25 28.002 2460.0 105.8 15.9 2038.4 302.7 5.3 344 1055 813 850 898 886 1032 5 1 17 53.103 1992.1 0 0 2870.8 0 0 563 1103 948 955 918 975 1201 5 2 25 26.594 2321.9 104.1 0 2177.5 320.9 0 354 1066 819 841 900 915 969 5 3 31 23.326 2393.4 98.6 20.1 2097.1 325.6 68.0 328 994 813 818 857 889 871 6 1 19 52.367 1951.5 0 0 2898.1 0 0 553 1089 966 969 912 982 1200 6 2 29 23.086 2263.8 99.0 0 2258.0 324.8 0 364 1085 779 830 862 910 899 6 3 37 18.251 2374.0 101.7 79.0 2126.5 325.1 62.8 324 1017 760 806 837 821 761 7 1 21 52.338 1909.3 0 0 2910.1 0 0 553 1091 960 970 917 981 1204 7 2 33 23.023 2254.3 104.8 0 2262.2 317.6 0 360 1088 766 831 855 919 904 7 3 43 15.007 2452.9 132.6 89.3 2038.1 351.2 118.0 331 1047 743 798 832 781 610

Covariance functions for growth 191 Table 4. continued k m p b log L l A1 l A2 l A3 l R1 l R2 l R3 s 2 o1 s 2 o2 s 2 o3 s 2 o4 s 2 o5 s 2 o6 s 2 o7 Wokalup 1 1 3 268.419 2745.3 1090.0 1326 1 1 9 194.927 2176.8 961.4 623 2010 1890 2015 1632 1690 1405 2 1 11 86.807 4265.0 0 2485.8 0 609 1978 1765 1802 1224 1430 1113 2 2 13 81.572 4968.1 90.6 1737.9 6.8 527 1985 1751 1797 1243 1386 1097 3 1 13 63.218 5541.8 0 0 824.8 0 0 640 1889 1595 1708 1145 1418 1101 3 2 17 55.196 4433.1 71.4 0 1423.8 143.0 0 524 1908 1585 1660 1132 1400 1067 3 3 19 54.976 4492.0 58.3 12.5 1385.5 141.8 15.7 486 1916 1585 1656 1125 1393 1063 4 1 15 58.530 5836.3 0 0 644.2 0 0 641 1825 1535 1724 1176 1396 1127 4 2 21 50.981 4522.8 42.1 0 1414.2 164.1 0 534 1854 1529 1669 1156 1389 1085 4 3 25 50.121 4649.6 105.0 18.3 1362.6 75.0 33.5 508 1850 1512 1664 1147 1378 1054 5 1 17 58.157 5834.2 0 0 647.2 0 0 647 1839 1544 1715 1172 1403 1107 5 2 25 48.728 4673.7 73.5 0 1338.7 167.7 0 560 1814 1555 1623 1082 1422 1032 5 3 31 47.187 4703.1 95.0 27.8 1335.2 140.2 44.6 512 1809 1532 1615 1075 1395 950 6 1 19 57.096 5677.5 0 0 791.2 0 0 673 1848 1536 1635 1162 1379 1107 6 2 29 46.281 4583.3 74.5 0 1428.8 189.4 0 564 1740 1535 1516 1086 1361 1037 6 3 37 44.108 4587.1 97.3 28.8 1421.1 167.1 57.8 501 1749 1505 1504 1078 1318 947 7 1 21 57.213 5831.3 0 0 834.9 0 0 669 1833 1624 1622 1135 1419 1099 7 2 33 44.925 4811.8 82.1 0 1337.8 175.2 0 566 1778 1523 1530 1006 1423 963 7 3 43 39.192 4635.7 172.1 63.4 1417.5 179.5 93.7 537 1734 1474 1510 981 1322 764 a +13 100 for Herefords, and +14 600 for Wokalups b Number of parameters

192 K. Meyer Fig. 5. Estimates of phenotypic (s P ) and temporary environmental (s M ) standard deviations from phenotypic (A) and genetic (B) analyses. Polled Hereford: s P A, s P B, t s M A, r s M B Wokalup: Ž s P A, ž s P B, T s M A, R s M B of estimates of s 2 o explanation. was essentially the same up to k = 7, making the latter an unlikely Definition of contemporary groups Results from corresponding analyses assuming contemporary group effects affected cows of all ages equally (CG2) are given in Table 5. While estimates of eigenvalues and measurement error variances (not shown) were similar to those fitting CG1 (Table 4), LRTs gave slightly different answers, suggesting that a quadratic CF (k = 3) with rank 2 was sufficient for WOK. For HEF though, log L increased significantly up to k = 4 for m = 2 and k =5 for m = 3, the latter being due to a third eigenvalue for R of 56. Figure 6 compares estimates of s P, s o, s A (additive genetic standard deviation) and s R (permanent environmental standard deviation) for the two contemporary group definitions, derived for the ages in the data from the respective estimates of CFs and s 2 oi, for k = 4 and m = 2. Again, differences in estimates from the two models were relatively small and did not exhibit a distinctive pattern. For both breeds, estimates of s A increased unexpectedly for the last ages in the data. While this increase was only slight and accompanied by an increase in estimates of s R for WOK, it was very steep and associated with a corresponding decline in s R for HEF. With s P almost constant (some increase for ages 115 to 119 months), the latter clearly reflected a strong sampling correlation between estimates of s A and s R. For HEF and CG2, an order of fit of k = 5 with rank m = 3 yielded a significant increase in log L over k = 4 with m = 3. Estimated standard deviations for k = 5 exhibited an even steeper increase in s A than for k = 4 and a resulting steep increase in s P for the last ages in the data (not shown). This was also evident for corresponding estimates for CG1, i.e. did

Covariance functions for growth 193 Table 5. Results from analyses not fitting age classes: maximum log likelihood (Log L a ) and the first three eigenvalues of the estimated genetic (l Ai, i =1, 3) and permanent environmental (l Ri, i=1, 3) covariance function, for different orders of polynomial fit (k) and (m) of the estimated coefficient matrices Polled Hereford Wokalup k m p b log L l A1 l A2 l A3 l R1 l R2 l R3 log L l A1 l A2 l A3 l R1 l R2 l R3 1 1 9 234.81 1430 889 148.09 2171 950 2 2 13 101.08 3179 89 3036 0 41.88 4793 94 1660 0 3 1 13 53.55 2140 0 0 2809 0 0 25.66 3835 0 0 2091 0 0 3 2 17 38.80 2505 65 0 1976 361 0 16.27 4342 60 0 1450 127 0 3 3 19 38.81 2517 67 1 1965 358 0 16.26 4289 60 0 1499 124 0 4 1 15 45.52 2304 0 0 2623 0 0 20.95 3923 0 0 2079 0 0 4 2 21 19.55 2582 97 0 1972 323 0 12.64 4374 52 0 1493 106 0 4 3 25 18.85 2585 93 55 1938 313 0 11.97 4385 89 26 1501 57 27 5 1 17 44.85 2179 0 0 2757 0 0 20.63 4052 0 0 2012 0 0 5 2 25 17.11 2337 115 0 2228 336 0 9.78 4316 59 0 1623 134 0 5 3 31 9.82 2539 110 46 1977 336 56 8.99 4418 77 11 1481 143 46 6 1 19 44.30 2133 0 0 2709 0 0 20.06 5615 0 0 752 0 0 6 2 29 15.61 2399 97 0 2157 325 0 7.09 4368 54 0 1502 186 0 6 3 37 5.08 2544 97 62 1955 332 134 5.27 4398 66 31 1471 169 56 a +13 310 for Herefords, and +14 880 for Wokalups b Number of parameters

194 K. Meyer Fig. 6. Estimates of phenotypic (s P ), additive genetic (s A ), permanent environmental (s R ) and temporary environmental (s M ) standard deviations (in kg) from analyses fitting paddock-year-week-age subclasses (CGI; black symbols) and paddock-year-week subclasses (CG2; open symbols), for Polled Herefords (left) and Wokalups (right) ž: s P CG1, : s P CG2, R: s A CG1, r: s A CG2, T: s R CG1, t: s R CG2, Ž: s M CG1, : s M CG2 not depend on the definition of CG, and was similar to that observed above for estimates of P and orders of fit of k 4. Comparison with previous estimates Previous analyses of the same data using the approach of MEYER and HILL (1997) had also concluded that orders of fit of k = 4 and k = 3 were appropriate for HEF and WOK, respectively (MEYER 1997). However, those analyses fitted age in years rather than months and did not impose any restrictions on the rank of estimated CFs. Furthermore, there was a separate measurement error for each year of age and the fixed effects part of the model of analysis was multivariate, i.e. a different CG effect was fitted for each year of age. This was deemed sufficient to model age trends, i.e. analyses die not fit a fixed regression to model the population trajectory. As shown in Figure 7, for HEF, estimates of both s P and s o agreed reasonably well with those obtained by MEYER (1997). For WOK, however, estimates of s o obtained earlier were quite different from those in the present study, being almost constant (ranging from 34 to 39 kg) across all ages with a slight increase for older cows. As could be seen in Figure 6, estimates of s o in this data set were more affected by the definition of CGs than those for HEF, in particular for earlier ages. Repeating the present analysis fitting separate CGs for each year of age, however, did not change estimates of s o substantially (results not shown). This suggests that differences in estimates were due to the explicit modelling of the population trajectory in this study. As for previous CF analyses, estimates of s P for both breeds were substantially higher

Covariance functions for growth 195 Fig. 7. Estimates of phenotypic (s P ) and temporary environmental (s o ) standard deviations (in kg) from analyses fitting joint year-paddock-week contemporary groups (CG2, black symbols: ž s P, Ž s o ) for all ages, and previous analyses (Meyer, 1997) fitting separate contemporary groups for each year of age (CGM, open symbols: s P, s o ) together with estimates of s P from univariate analyses for each year of age (+), from Meyer (1997), for Polled Herefords (left) and Wokalups (right) than those from univariate analyses for each year of age (MEYER 1997), especially for cows older than 6 years. In part, this could be due to reduced variation amongst the subset of cows surviving to a given age (selection bias). Erratic estimates for extreme ages As emphasized above, the steep incline in estimated variances for 10-year-old cows in HEF was suspected to be caused by some unidentified problems in the data. A similar situation was encountered earlier in an analysis of the WOK data considering records for 2- to 6- year-old cows only (MEYER 1998b). To investigate the problem further, analyses for HEF were repeated eliminating records for ages 115 to 119 months. As shown in Figure 8, this reduced the incline in estimates of s A and decline in estimates of s R to some extent, but essentially displayed the same pattern as for the full data set. As before, estimates of s P (not shown) for the two analyses agreed well. Results shown were for an order of fit of k = 4. For k = 5, estimates for all data were even more distorted (s P = 64, s A = 54 and s R = 20 for 119 months versus 62, 48 and 25, respectively for k = 4), while estimates of s A and s R reversed roles, inclining for s R for the last 3 years of data (to 48 kg for 119 months) and declining for s A (to 30 kg for 119 months). Note that estimates of standard deviations for 10-year-olds obtained from CFs estimated from the subset of data required extrapolating, an exercise which should be regarded with caution, as estimated CFs strictly speaking are only valid for the range of ages spanned by the data. Nevertheless, results suggested that problems were not inherent to records for the last ages alone and were highly dependent on the order of fit of random regressions.

196 K. Meyer Fig. 8. Estimates of genetic (s A ; open symbols) and permanent environmental (s R ; black symbols) standard deviation (in kg) for Polled Herefords from analyses considering all data (all; squares), eliminating records for ten year old cows (sub; circles) and transforming ages to logarithmic scale (log; triangles) (: s A, all; Ž: s R, all; : s A, sub; ž: s R, sub; r: s A, log; R: s R, log) As noted above, estimates of measurement error variances tended to pick up variation for insufficient orders of fit of CFs. Conversely, estimates of CFs might be biased by not allowing for sufficient heterogeneity in temporary environmental variation. As shown in Figure 7, estimates of s o for the last 3 years of data from previous analyses fitting a separate variance component for each year (MEYER 1997) were indeed quite different than those from this study assuming s o for cows 8- to 10-years-old was the same. Repeating the analysis for HEF for (k = 4 and m = 2) fitting nine measurement error variances gave estimates of s o of 29, 35 and 33 kg for 8, 9 and 10 years of age, respectively, i.e. except for 10 years, the estimates were very similar to those obtained by MEYER (1997). Estimates of A and R and thus s A and s R, however, were virtually unchanged, i.e. incorrect modelling of heterogeneous measurement error variances was not responsible for erratic estimates of A and R for this data set. Similarly, repeating the analysis fitting a fixed regression to the order k = 4 rather than k =7ork= 6, ruled out that higher order terms in modelling the population trajectory were the source of problems. Age on the logarithmic scale As emphasized by MEYER (1998b), under a CF model observations are contributing information to the analysis at the age they are made, and the order and spacing between repeated records for an individual is taken into account. This together with the fact that the orthogonal polynomials chosen, the Legendre polynomials, were symmetric implies that equal weight was given to observations early and late in life. Often, however, later records are less representative of the animal, being subject to accumulated selection and environmental effects. Hence it might be preferable to reduce emphasis for records in the later part of

Covariance functions for growth 197 life. One suitable mechanism is to transform age to logarithmic scale prior to calculating orthogonal polynomials which decreases the spacing of later records. Analyses were repeated for both breeds transforming age to logarithmic scale for orders of fit k = 3, 4, 5, 6, rank m = 3 and CG1. For WOK, there was little change in estimates (not shown). The transformation, alleviated the problem of dramatically changing variance components at the highest ages for HEF though, at least for k ¾ 4; see Figure 8. For k =6 and to a lesser extent k = 5, however, the pattern of steeply increasing estimates of s A and decreasing estimates of s R observed above was evident again. Corresponding LRTs now did not indicate a significant change in log L above k = 3, although comparison of estimates of s P suggested that, as before, a minimum order of fit of k = 4 was required. Results emphasize that a transformation might stabilize fluctuations in the data but does not cure problems. Other comparisons Comparison of models and analyses and orders of fit so far has relied primarily on likelihood values and corresponding likelihood ratio tests and the examination of standard deviations for ages in the data, calculated from estimated covariance functions. The rationale for the latter has been that there can be little confidence in analyses and covariance estimates if variances, especially phenotypic variances, are not estimated correctly. Eigenvalues (and eigenfunctions) of the estimated CF summarize and characterize multivariate relationships. As seen in Tables 3, 4 and 5, dominant eigenvalues tend to fluctuate little, once CFs have been fitted to a sufficient order and rank. Similar to the visual inspection of standard deviations, complete matrices of covariances for the ages in the data were calculated, and the changes in values between different models and orders of fit examined. On the whole, however, there was little insight to be gained over and above that afforded by the criteria investigated above. For analyses giving significantly different results, average changes in variances and covariances tended to be big, with large standard deviations, and when plotted showed a distinctive trend. If any additional parameters fitted did not improve the likelihood substantially, the pattern of changes tended to fluctuate somewhat erratically, and average changes in (co)variance components were only a fraction of the corresponding standard deviations. Corresponding correlations were much less sensitive to differences in models of analysis. This is illustrated in Figure 9 for genetic covariances and correlations in Wokalups, fitting model B and CG2. Increasing the order of fit from k =2tok= 3 caused estimated covariances to be substantially reduced, in particular for later ages (top left of Figure 9). In this case, the mean change in variances (=diagonal elements) was 437 with a standard deviation of 1219 and a range of 3285 to 647 (all figures in kg 2 ). Corresponding numbers for covariances (off-diagonal elements) were 3072757 with a range of 3224 to 645. Resulting estimates of genetic correlations (top right of Figure 9), however, changed only by 0.00220.038 with changes ranging from 0.131 to 0.076 and correlations for 2-year-olds most affected. In contrast, increasing the order of polynomial fit from k =4tok= 6 had, on average, little effect on estimates of covariance components, while causing fairly substantial fluctuations in individual estimates (bottom left of Figure 9). For this case average changes were 7.02165.2 ( 218 to 350) and 3.82120.2 ( 217 to 294) for variances and covariances, respectively. Corresponding changes in genetic correlations (bottom right corner of Figure 9) were on average 0.00420.039 ( 0.198 to 0.071), i.e. slightly bigger and more erratic than those observed above. Final estimates In the light of results presented above, an analysis fitting year paddock week subclasses (CG2), considered to be the same for all ages, with a cubic, orthogonal polynomial (k =4)

198 K. Meyer Fig. 9. Differences in estimates of genetic covariances (left column) and genetic correlations (right column), between analyses for orders of fit k = 3 and k = 2 (top row), and analyses with orders of fit k = 6 and k = 4 (bottom row); data for Wokalups, fitting year-paddock-week contemporary group subclasses to model mean and individual growth curves, and regressing on age transformed to logarithmic scale was considered to be the most suitable model for the data. Examining full and reduced rank covariance matrices of regression coefficients showed that the first two eigenfunctions, i.e. CFs of rank m = 2, sufficed to model permanent environmental variation in individual growth curves sufficiently. For the genetic CF, ranks of m = 3 for HEF and m = 2 for WOK were needed. Together with seven measurement error variances, that resulted in 23 and 21 parameters to be estimated for HEF and WOK, respectively. Estimates of coefficient matrices of covariances and resulting genetic and permanent environmental covariances functions together with estimates of fixed regression coefficients to model the population trajectory are summarized in Table 6. Corresponding estimates of measurement error variances are given in Table 7. As observed in previous analyses, weights for WOK were considerably more variable than for HEF, and WOK expressed more genetic variation. Estimates of genetic, permanent environmental and phenotypic standard deviations together with resulting heritabilities (h 2 ) and repeatabilities (t) for the ages in the data are shown in Figure 10. Analysing January weights for 3-year-old and older cows, fitting a repeatability model with age (in years) as a fixed effect, MEYER (1995) reported estimates of s P and h 2 of 56.5 kg and 0.31 and 66.0 kg and 0.48 for HEF and WOK, respectively. For WOK, the latter agrees well with ĥ 2 in this study fluctuating between 0.42 at 19 months and 0.49 at 119 months. For HEF, ĥ 2 from 0.57 (19 months) to 0.37 (119 months) were somewhat higher than estimated previously for weights, but comparable to an estimate of 0.47 obtained for average mature weight when fitting a modified Gompertz curve (MEYER 1995). Estimates of s o were highest for 3-year-old cows (see Table 7), causing estimates of both h 2 and t to be lowest for these ages. On the whole, however, estimates of t were considerably higher than the value of 0.61 found by MEYER (1995) for both breeds.

Covariance functions for growth 199 Table 6. Estimates of coefficient matrices and resulting covariance functions and their eigenvalues for genetic and permanent environmental effects (K A and K R for A and R with l Ai and l Ri, respectively), fitting a cubic regression (k = 4) on orthogonal polynomials of age transformed to logarithmic scale, and forcing covariance functions to have rank m =2 a Polled Hereford Wokalup Covariance functions Element b K A A K R R K A A K R R 11 2206.83 1366.13 1439.92 1176.95 3549.22 2146.43 1192.92 619.70 12 324.00 310.66 421.27 552.50 730.76 733.09 237.59 262.10 13 221.82 416.12 280.37 900.78 315.39 586.54 1.92 68.01 14 9.42 1.25 97.61 348.76 9.87 35.26 5.68 116.88 22 181.08 913.43 185.41 582.06 253.77 387.64 53.54 171.50 23 30.29 34.13 22.29 18.03 81.66 241.90 14.23 135.23 24 17.03 972.12 35.70 302.10 0.97 6.07 10.09 209.76 33 23.52 132.32 229.65 1291.77 30.73 172.87 34.56 194.38 34 8.09 89.74 7.02 77.87 0.71 7.83 26.53 249.29 44 66.63 1457.60 7.44 162.68 0.04 0.84 20.38 445.90 Eigenvalues l A1 l A2 l R1 l R2 l A1 l A2 lr1 l R2 2279.7 134.3 1624.4 238.0 3732.6 101.1 1239.5 60.9 Fixed regression modelling population trajectory c b 0 b 1 b 2 b 3 b 0 b 1 b 2 b 3 88.376 85.317 34.255 9.907 56.461 85.591 22.757 3.945 a Analyses within paddock-year week subclasses and fitting a cubic, fixed regression on orthogonal polynomials of log (age) b row and column number of elements of symmetric matrices c b 0 : intercept, b 1 : linear coefficient, b 2 : quadratic coefficient, and b 3 : cubic coefficient Table 7. Estimates of measurement error variances (s 2 o,inkg 2 ) for analyses fitting a cubic regression (k = 4) on orthogonal polynomials of age transformed to logarithmic scale, and forcing covariance functions to have rank m =2 a Ages (months) 19 22 31 34 43 47 55 58 67 71 79 82 91 119 Polled Hereford 333 1160 849 940 882 940 1061 Wokalup 548 2169 1729 1705 1212 1387 1063 a Analyses within paddock-year week subclasses and fitting a cubic, fixed regression on orthogonal polynomials of log (age) Some of the earliest weights considered here were identical to final weights, defined at the weight at between 500 and 700 days of age closest to the target age of 600 days, considered by MEYER et al. (1993). At average ages of 592 days (HEF) and 584 days (WOK), estimates of h 2 and s 2 P from univariate, single record analyses were 0.36 and 1008 kg 2 (HEF) and 0.31 and 1382 kg 2 (WOK). Estimates for h 2 derived from CFs were considerably higher: 0.57 at 19 and 20 months for HEF, and 0.42 and 19 months and 0.44 at 20 months for WOK. In part, this can be attributed to the fact that maternal effects have been assumed

200 K. Meyer Fig. 10. Estimates of phenotypic (: Polled Hereford, Ž: Wokalup), genetic (r: Polled Hereford, R: Wokalup) and permanent environmental (: Polled Hereford, ž: Wokalup) standard deviations and resulting heritabilities (+: Polled Hereford, *: Wokalup) and repeatabilities (t: Polled Hereford, T: Wokalup) for analyses fitting regressions on age transformed to logarithmic scale absent in this analysis. Corresponding values for s 2 P were more comparable, albeit slightly higher: 1263 and 1119 kg 2 at 19 and 20 months, respectively, for HEF, and 1411 and 1470 kg 2 at 19 and 20 months, respectively, for WOK. Estimates of additive genetic covariances and correlations for ages in the data are shown in Figure 11. While (co)variances for HEF increased steadily until about 5 years of age and then plateaued, estimates for WOK increased sharply for 3- and 4-year-old cows to decline slightly for the latest ages. Correlation estimates amongst 3-year-old and older cows for WOK were close to unity, exhibited as horizontal plane in Figure 11 (bottom right). For HEF, correlation estimates were less consistent, especially for 2-year-old and 10-year-old cows, being as low as 0.23 for 19 and 119 months (while the corresponding estimate for WOK was 0.64). Clearly this is a reflection of the problems in modelling later ages in this data set observed above. Corresponding plots for permanent environmental and phenotypic covariances and correlations are shown in Figures 12 and 13, respectively. Again, estimates for HEF were considerably more variable than for WOK, estimates for environmental covariances and correlations between records at 3 years of age and other ages being consistently low. For WOK, estimates of environmental covariances increased almost linearly throughout life, leading to correlation estimates close to unity among records in subsequent years. Plots of phenotypic covariance matrices (Figure 13, top row) show spikes along the diagonal, depicting measurement error or temporary environmental variances. Otherwise surfaces were smooth for both breeds, indicating that fluctuations in estimates of genetic and permanent environmental covariances for HEF were to a large extent due to sampling variation in partitioning the total variance into its components. Phenotypic correlation

Covariance functions for growth 201 Fig. 11. Estimates of additive genetic covariances (top row) and genetic correlations (bottom row) for ages in the data, for Polled Hereford (left column) and Wokalup (right column), respectively; analyses fitting regressions on age transformed to logarithmic scale Fig. 12. Estimates of permanent environmental covariances (top row) and permanent environmental correlations (bottom row) for ages in the data, for Polled Hereford (left column) and Wokalup (right column), respectively; analyses fitting regressions on age transformed to logarithmic scale estimates were lowest for records taken at 3 years of age for both HEF and WOK. For WOK, this was mainly due to a large measurement error variance at this age. Similarly, a