Introduction 1998c. In particular, RR models accommodate Material and methods

Livestock Production Science 65 2000 19–38 www.elsevier.com locate livprodsci Random regressions to model phenotypic variation in monthly weights of Australian beef cows ,1 Karin Meyer Animal Genetics and Breeding Unit , University of New England, Armidale NSW 2351, Australia Received 7 July 1999; received in revised form 1 December 1999; accepted 1 December 1999 Abstract Weights of beef cows recorded on a monthly basis were analysed using a random regression model. Data originated from a selection experiment in Western Australia, involving two herds of about 300 cows, Polled Herefords and a four breed synthetic, the so-called Wokalups. Weights were subject to large seasonal effects. Short mating periods and thus tight calving seasons led to substantial confounding between age and season at weighing. Records between 19 and 84 months were considered, up to 62 per cow, yielding 27 728 and 29 033 records for 922 and 1020 cows, respectively. Only phenotypic random regressions for animal effects, ignoring relationships, were considered. Covariances between regression coefficients and error variances were estimated by restricted maximum likelihood. A variety of models, involving random regressions on orthogonal polynomials of age, on segmented polynomials and on sine and cosine functions and different assumptions about the structure of error variances, were considered. Analyses identified a distinct cyclic, seasonal pattern of variation, both between animals and for temporary environmental effects. This could only partially be attributed to scale effects. Orthogonal polynomials proved well capable of modelling such sinuousity but required a high order of fit and thus a large number of parameters. Alternative curves utilising the known periodicity 12 months provided more parsimonious parameterisations. Due to the high degree of confounding between age and season of recording their contributions to the total variance could not be separated.  2000 Elsevier Science B.V. All rights reserved. Keywords : Random regression; Growth curve; Modelling; Mature weights; Beef cattle

1. Introduction 1998c. In particular, RR models accommodate

‘repeated’ records for traits which change, gradually Random regression RR models and the resulting and continually, over time, and do not require covariance functions CF have recently been recog- stringent assumptions about constancy of variances nised as ideally suited to the analysis of longitudinal and correlations. RR models for the analysis of data data in animal breeding and the description of the from animal breeding schemes have first been pro- resulting covariance structure Hill, 1998; Meyer, posed to model test day production records of dairy cows Schaeffer and Dekkers, 1994. Applications so far have concentrated in this area, e.g. Jamrozik and Tel.: 161-26-773-3331; fax: 161-26-773-3266. Schaeffer 1997, Jamrozik et al. 1997, Kettunen et E-mail address : kmeyerdidgeridoo.une.edu.au K. Meyer 1 A joint unit with NSW Agriculture. al. 1998, Van der Werf et al. 1998, Tijani et al. 0301-6226 00 – see front matter  2000 Elsevier Science B.V. All rights reserved. P I I : S 0 3 0 1 - 6 2 2 6 9 9 0 0 1 8 3 - 9 20 K . Meyer Livestock Production Science 65 2000 19 –38 1999, Rekaya et al. 1999, Veerkamp and Thomp- growth and its variation parsimoniously. While we son 1999 and Brotherstone et al. 2000. are ultimately interested in a genetic analysis, as a Other traits of prime interest in livestock pro- first step, only phenotypic animal effects and growth duction which are measured over time and for which curves are considered. a simple repeatability model, assuming constant variances and correlations, clearly does not hold are those related to growth of animals. In standard

2. Material and methods

analyses, this has so far generally been taken into account by treating records in different age intervals 2.1. Data as different traits. For example, genetic evaluation of beef cattle often considers four different growth Data consisted of weights of cows collected as traits, namely birth, weaning, yearling and final part of a selection experiment at the Wokalup weights, with the latter three defined as weights agricultural research station, located in the South of taken, say, from 80 to 300 days, 301 to 500 days and Western Australia, approximately 140 kilometers 501 to 700 days of age, respectively. Clearly, this is South of Perth. Details of the experiment and somewhat arbitrary and a RR CF model using each management strategies at the research station have record at the age it was taken would be more been described by Meyer et al. 1993. The experi- appropriate. Indeed, Kirkpatrick et al. 1990 de- ment comprised two herds of about 300 cows each, a veloped their ‘‘infinite-dimensional’’, CF model with purebred Polled Hereford PH herd and a herd of a the analysis of growth traits in mind. four-breed synthetic, formed by mating Charolais 3 RR models have been used for some time in the Brahman bulls to Friesian 3 Angus or Friesian 3 analysis of human growth curve data. Laird and Hereford cows, the so-called ‘Wokalups’ WOK. Ware 1982 outlined a RR mixed model for longi- Selection for improved preweaning growth rate tudinal data which comprised both growth curve began in 1978. models and repeatability models as special cases. In Management of the herds was characterised by animal breeding, selected growth curves, such as short mating periods of seven to eight weeks, Gompertz or Richards functions, have commonly resulting in the bulk of calvings in April and May been fitted to growth records. To date, however, this each year. Weaning took place in December. Except has generally been done independently to the estima- during the calving season, cows and calves were tion of fixed and random effects, i.e. not within a weighed at monthly intervals throughout the experi- linear mixed model framework. Anderson and ment. While the research station has reasonable Pedersen 1996 showed how RR can be employed rainfalls annual average 979 mm, the climate is in modelling growth curves of pigs on a phenotypic Mediterranean with high levels of precipitation dur- level. Meyer 1999 presented an application to ing winter and an annual summer drought, from mature cow weight records in beef cattle, contrasting November to April the following year. This resulted analyses fitting RR coefficients for phenotypic ani- in strong seasonal variation in pasture availability mal effects only to those attempting to partition and, consequently, weight of cows. growth curves into their genetic and environmental After basic edits, there were 87 516 cow weights components. recorded between 1977 and 1990. Records selected Data for the latter study Meyer, 1999 originated were weights of cows aged between 19 and 84 from a selection experiment which involved monthly months. Earlier records were excluded to minimise weighing of animals. However, only January the influence of maternal effects which previous weights, i.e. single, annual measurements were uti- analyses found to have some carry-over effect after lised to avoid problems associated with strong weaning, on yearling and, to a lesser extent, final seasonal fluctuations in growth. This paper presents a weights Meyer et al., 1993. While some cows had RR analysis of these data using weights recorded weights recorded up to 12 or even 13 years of age, throughout the year, and examines the scope of numbers of records decreased with age. Weights at different RR curves to model cyclic patterns of later ages were thus disregarded to eliminate prob- K . Meyer Livestock Production Science 65 2000 19 –38 21 lems associated with small numbers per age and as the independent covariable or ‘meta-meter’. This contemporary group encountered in earlier analyses involved fitting a set of regression coefficients on age considering January weights up to 119 months or functions thereof for each animal as random Meyer, 1999. Similarly, small numbers of records effects. Different assumptions on the shape of taken during April and May were excluded from the growth curves could be accommodated by choosing analysis. This yielded 27 728 records for 922 PH the functions of age appropriately. Only regression cows and 29 033 records for 1020 WOK cows, coefficients pertaining to phenotypic animal effects respectively, and a maximum of 62 ‘repeated’ re- were considered, i.e. no attempt was made to sepa- cords per cow. Weights ranged from 193 to 836 kg rate overall animal effects into their genetic and for PH and 226 to 946 kg for WOK. permanent environmental components, and relation- ships between animals were ignored. Fixed effects in 2.2. Analyses the model of analysis were CG, defined as above, and a fixed, cubic regression on orthogonal polyno- 2.2.1. Preliminary analyses mials of age to model population age trends. Analy- Preliminary analyses were carried out to character- ses were carried out using a combination of average ise the pattern of variation in the data. Firstly, information and derivative-free REML algorithms as variances within age classes were examined for outlined by Meyer 1998b and implemented in records adjusted for contemporary group CG ef- program D X M RR Meyer, 1998a. fects, defined as paddock-year-week of weighing Analyses yielded estimates of covariances among subclasses, both on the original scale and for data random regression coefficients and estimates of transformed to logarithmic scale. This used estimates variances due to temporary environmental variances, of CG effects obtained from simple least-squares so-called measurement error variances. From these, analyses ignoring animals but fitting a fixed regres- estimates of covariance functions and of variances sion on orthogonal polynomials of age. and covariances among ages in the data were ob- ‘Standard’ univariate REML analyses considering tained. A variety of models involving different records for individual ages were carried out to assess regression curves and assumptions on the distribution the proportion of variation due to animals. Analyses of measurement errors were fitted, more than 100 in fitted phenotypic animal effects as random effects total. Due to computational limitations, most models and ignored any relationships between animals. Thus were examined for PH only, fitting only selected repeated records per animal were required to separate cases for WOK. The majority of analyses were variances due to animals and temporary environmen- carried out for weights as measured. In addition, a tal effects. Hence analyses for the i-th age i 5 small number of analyses considered records trans- 19,84 considered ages i 2 1, i and i 1 1, i.e. in- formed to logarithmic scale. volved records from 18 to 85 months of age, and Models were compared using likelihood ratio tests included any records taken during the calving LRT and by comparing estimated standard devia- season. Fixed effects fitted in these analyses were tions for the ages in the data. When testing whether CG and age three classes. In addition, similar additional regression coefficients should be fitted, i.e. analyses were carried out considering weights re- whether the covariances for the additional coeffi- corded in a particular month only. The model for the cients were different from zero, hypotheses involved latter analyses again fitted animals as random and points at the boundary of the parameter space and, as CG as fixed effects but accounted for age as record- shown by Stram and Lee 1994, conventional LRT ing by fitting it as a linear and quadratic covariable. tend to be too conservative. Where applicable, this All univariate analyses were carried out using pro- was counteracted by doubling the error probability gram ASREML Gilmour et al., 1999. when conducting LRTs. Orthogonal polynomials. The first set of analyses 2.2.2. Random regression analyses fitted RRs on orthogonal polynomials of age, consid- Data at all ages were analysed simultaneously ering orders of fit k from 4 to 22. Polynomials fitting a RR model with age at weighing in months chosen were the so-called Legendre Polynomials 22 K . Meyer Livestock Production Science 65 2000 19 –38 z LP, and in the following ‘‘LPk’’ denotes a model 2 fitting LPs to order k. LP are defined for the range of y 5 F 1 b 1 b a 1 b a 1 O b a 1 e ij ij i 0 i 1 ij i 2 ij i 21r ir ij r 51 2 1 to 1 1. For the jth age a standardised to this j 3 range, a , the r-th polynomials is given as e.g. j Abramowitz and Stegun 1965 with ]] r 2 1 2r 1 1 for a A m ij r ] ]] fa 5 O 21 r j r a 5 4 2 H ir œ 2 2 m 50 a 2 A for a . A ij r ij r r 22m r 2r 2 2m S DS D a 1 j Similarly, a SC polynomial is described by 4 1 z m r coefficients ] ] Œ Œ Hence the first three polynomials are 1 2, 3 2a j ] ]] 2 3 2 Œ Œ y 5 F 1 b 1 b a 1 b a 1 b a and 2 5 8 1 45 8a , respectively. Note that ij ij i 0 i 1 ij i 2 ij i 3 ij j the first term is a scalar, i.e. a polynomial of order k z involves powers of age up to k 2 1. 1 O b a 1 e 5 i 31r ir ij r 51 This gives the model for the j-th weight y ij recorded for animal i at age a a on the stan- ij ij with dardised scale as for a A ij r k 21 a 5 6 3 H ir a 2 A for a . A ij r ij r y 5 F 1 O b fa 1 e 2 ij ij ir ij r ij r 50 Knots were chosen after inspection of results from with F representing the fixed effects pertaining to the preliminary, univariate analyses and RR analyses ij y , b denoting the set of k random regression fitting LPs. Placing knots at the lowest weights Fig. ij ir coefficients for animal i the ‘animal effects’, and e 1 and standard deviations and dividing ages into 12 ij the corresponding residual error. months segments with the first knot at 25 months of Segmented polynomials. Secondly, models in- age resulted in 5 knots, at 25, 37, 49, 61 and 73 volving RR on segmented quadratic SQ or cubic months, respectively. This resulted in segmented SC polynomial functions were examined. Such quadratic and cubic polynomials with 8 and 9 functions, also called piecewise polynomials or, parameters, denoted by SQ8 and SC9. Reducing sometimes, grafted polynomials Fuller, 1969, are segments to 6 months intervals gave 10 knots at 25, the equivalent to spline functions used widely in 31, 37, 43, 49, 55, 61, 67, 67, 73 and 79 months and non-parametric regression analyses. They are linear models SQ13 and SC14 with 13 and 14 regression in the parameters to be estimated and thus suitable coefficients, respectively. In addition, quadratic poly- for RR model analyses in a linear mixed model nomials with knots at 20,32, . . . ,80 months z56, framework. As the name says, segmented polyno- and knots at 20,26, . . . ,80 months z 5 11, yielding mials allow approximation of a function by different models SQ9 and SQ14 respectively, were examined. polynomials in individual segments. Points at which Procedures to estimate the position of knots simul- segments join are commonly referred to as nodes or taneously with the regression coefficients are avail- knots. Segments are required to be continuous and able e.g. Gallant and Fuller 1973. This should have continuous first and, if they exist, non-zero also be feasible in a REML framework of estimation. higher order derivatives. Together with the assump- However, this was not investigated. tion that segments are of the same form, this reduces Fourier series approximation. Preliminary analy- the number of parameters needed to describe the ses identified a cyclic, seasonal pattern in means and overall curves to the order of the polynomial includ- variation in the data. Periodic or waveform functions ing a scalar term plus the number of knots. Thus a are frequently approximated by Fourier series. In SQ polynomial with z knots at ages A to model essence, such series are the sum of a number of sine r weight y recorded at age a is given as and cosine functions. Fourier theory states that any ij ij K . Meyer Livestock Production Science 65 2000 19 –38 23 Fig. 1. Numbers of records dark grey: Polled Hereford, light grey: Wokalups and mean weights d: Polled Hereford, s: Wokalup for ages in the data. period function Ft with period T can be represented sets of RR coefficients were independent or allowing by an infinite series of form for covariances between them. Let ‘‘Ff ’’ denote the Fourier part of 8 with f variables. Using ‘1’ to ` a symbolise that two separate, uncorrelated sets of RR ] Ft 5 1 O a cosrvt 1 b sinrvt 7 r r 2 r 51 coefficients were fitted, ‘‘Pp 1Ff ’’ then denotes a model with a total of p 1 f coefficients and p p 1 1 with v 5 2P T, and a and b the so-called Fourier i i 2 1 f f 1 1 2 covariances between RR coefficients coefficients, which are given by various integrals of to be estimated. Conversely, ‘‘PpFf ’’ no ‘1’ is the Ft over a single period T. The coefficients have an corresponding model allowing for covariances be- interpretation in their own right, representing fre- tween the two types of regression coefficients, quency components of the function. resulting in p 1 f p 1 f 1 1 2 parameters due to For RR analyses, T was assumed to be 12 months, RR. For instance, LP121F4 describes a model i.e v ¯ 0.5235. Approximations involving one, two fitting LP to order k 5 12 together with a Fourier and three sets of sine and cosine functions, i.e. 2, 4 series approximation involving 2 sine and 2 cosine and 6 RR coefficients for each animal, were consid- functions. Since the two sets of RR coefficients are ered. The expansion given by 7 is appropriate for a assumed uncorrelated, the total number of covar- function with a level baseline. To account for age iances between RR coefficients is 12 3 13 2 1 4 3 trends, Fourier series regressions were thus fitted 5 2 5 88. LP12F4 is the same model accounting for jointly with one of the polynomials described above, non-zero covariances with 16 3 17 2 5 136 compo- i.e., in essence, substituting a polynomial in age for nents to be estimated. the scalar term a 2 in 7. Let f denote the Regression on second ‘meta-meter’. In addition, number of regression coefficients for the Fourier analyses were carried introducing a second indepen- series fitted, and ‘‘Pp’’ a polynomial LP or SQ with dent, continuous variable, namely month of record- p regression coefficients on functions of age ga . ij ing m , and fitting an additional set of RR for each This gives the model of analysis ij animal, regressing on month or functions thereof, f 2 p hm . Covariances between RR on months and RR ij y 5 F 1 O b ga 1 O b cosrva ij ij ir ij i p 12r 21 ij on age were assumed to be zero, and, as above, this r 51 r 51 is denoted by ‘1’. Both orthogonal polynomials and 1 b sinrva 1 e 8 i p 12r ij ij Fourier series approximations were used to model Analyses were carried out either assuming the two the effect of month. The former were fitted to orders 24 K . Meyer Livestock Production Science 65 2000 19 –38 2, 6 and 10. With records for April and May not other ages from this estimate and the estimated included in the analysis, there were 10 different regression line. Orders of polynomial fit of 5, 7, . . . , 2 months, i.e. the latter represented a full order fit. 17, 19 were examined. Finally, s were assumed to e i Models including a regression on months are denoted change in a cyclic fashion over the year. Hence 12 2 as ‘‘Pp 1M Qq’’ with ‘‘Pp’’ as above the polynomial components were fitted, assuming s pertained to e 1 ] 2 of age with p coefficients, and ‘‘Qq’’ standing for the 25, 37, 49, 61 and 73 months of age, s to 26, e 2 2 function involving q parameters which was used to 38, . . . , 74 months, . . . , s to 19, 31, . . . , 79 e 7 2 model month. For weight y recorded at age a in months, . . . , and s to 24, . . . , 84 months. This ij ij e 12 month m , error structure is denoted as ‘‘E12C’’. ij Table 1 summarises the random regression models p q examined and abbreviations used to describe them. y 5 F 1 O b ga 1 O b hm 1 e 9 ij ij ir ij i p 1r ij ij r 51 r 51 For example, LP121M F2 is a model fitting a RR 3. Results ] on orthogonal polynomials of age to order k 5 12 and a RR on the first set of sine and cosine functions 3.1. Characteristics of the data in a Fourier series, with 12 3 13 2 1 2 3 3 2 5 81 covariances between RR coefficients. Means and numbers of records for the 66 ages in Measurement error variances. All models in- the data are shown in Fig. 1. Low numbers of cluded a random error term, e , representing tempor- records at 25, 37, . . . , 73 months reflect lack of ij ary environmental effects or so-called measurement recording of cow weights during the calving season. errors. These were assumed to be independently Mean weights for both breeds followed a distinct 12 distributed for all analyses. Several alternatives month cycle with an amplitude, i.e. difference be- regarding the number m of different measurement tween highest and lowest mean weights, of about 2 error variances s , i 5 1, m required to model 100 kg. WOK cows were larger than PH cows e i temporary variation in the data adequately were throughout, with differences tending to be bigger at investigated. Thus complete description of the the higher than at lower weights. RR model used required augmenting the abbrevia- Corresponding standard deviations within age tions given above by a term ‘‘.Em’’. For example classes are depicted in Fig. 2 left graph. Records LP20.E15 is a model fitting LP with k 5 20 and 15 for WOK tended to be more variable than for PH. 2 individual s . Firstly, variances were considered to Transforming records to logarithmic value substan- e i be homogeneous, i.e. m 5 1, consistent with the tially reduced this difference, indicating that it was concept of a measurement error affecting all records mainly due to scale effects. Moreover, the trans- equally. Secondly, variances were allowed to be formation emphasized a cyclic pattern in variation at heterogeneous, changing with age. Fitting individual individual ages, again following cycle of about 12 2 s for every six-month interval i.e 19–24, . . . , months. Peaks for PH at 37, 49 and 61 months e i 79–84 months yielded m 5 12 components. Sub- represented ages with few records c.f. Fig. 1 and 2 dividing the first and last interval so that separate s could not be attributed to the same animals. They e i were fitted for the first last and second plus third were thus treated as random fluctuations. Adjusting second-last plus third-last age i.e. 19, 20–21, 22– records for least-squares estimates of CG effects 24, 25–30, . . . , 73–78, 79–81, 82–83, 84 months showed a similar pattern Fig. 2, right graph, with increased the number to m 5 15. Allowing for a differences in variances between ages particularly different variance for each age resulted in a model pronounced for WOK. with m 5 66 error variances to be estimated. Thirdly, With a short mating period and, consequently, 2 it was attempted to model changes in s over time most calves born during April and May each year, e i through a polynomial regression, estimating a single there was a close relationship between ages at measurement error variance at the mean age 0 on weighing and seasonal effects. This is illustrated in the standardised scale and deriving variances for the Fig. 3, showing means and numbers of records K . Meyer Livestock Production Science 65 2000 19 –38 25 Table 1 Random regression models fitted a Abbreviation LPk SQ31z SC4 1 z Ff M Pp Em ] Function Legendre Segmented Segmented Fourier Regression Error Polynomial quadratic cubic approximation on month structure polynomial polynomial Features Order k z knots z knots f 2 sine and Polynomial P m variance cosine functions with p parameters components Equation 2 3 5 8 9 Cases LP4 SQ8 SC9 F2 M LP4 .E1 ] LP6 SQ9 SC14 F4 M LP6 .E12 ] LP8 SQ13 F6 M LP10 .E15 ] LP10 SQ14 M F2 .E66 ] LP12 M F4 .E12C ] LP14 LP16 LP18 LP20 LP22 a Brackets for clarity here only. Fig. 2. Standard deviations within age class for raw data left graph and records adjusted for fixed effects right graph for weights in kg; circles and weights transformed to logarithmic sale log kg 3 100; squares for Polled Herefords closed symbols and Wokalups open symbols. averages for both breeds for individual months of gonal polynomials of age mean growth curve from recording. For instance, most records for cows 33 least-squares analyses ignoring animals, together months old were taken in January, and most records with means of records adjusted for estimated CG for 38 months olds represented June weights. Fig. 4 effects. Smoothness of the curves and close agree- shows the estimated fixed cubic regression on ortho- ment between the fitted regression and means of 26 K . Meyer Livestock Production Science 65 2000 19 –38 Fig. 3. Numbers of records bars and mean weights d for individual months of recording total number and average over both breeds, respectively. Fig. 4. Estimated population trajectory fixed, quadratic regression on orthogonal polynomials of age from least-squares analysis ignoring random effects together with means for ages of records adjusted for fixed effects except the regression on age, for Polled Herefords d and Wokalups s. records, adjusting for fixed effects other than age, analysis removed any systematic differences in mean emphasize that CG accounted for seasonal fluctua- growth. However, as seen in Fig. 2, CG effects only tions and that the fixed effects in the model of accounted for a small proportion of seasonal fluctua- K . Meyer Livestock Production Science 65 2000 19 –38 27 tions in variance. Hence the pattern of variation tween corresponding estimates of s was as low as P 2 shown in Fig. 2 right graph is what needed to be 0.064 kg with a maximum difference of 0.7 kg at 62 modelled in RR model analyses. months. Increasing k until log + ceased to increase would have been desirable but was disregarded due 3.2. Random regression on orthogonal polynomials to the large number of parameters to be estimated of age simultaneously, as large as 254 at k 5 22, and the resulting computational requirements. Modelling Log likelihoods log + from RR analyses on the liveweight changes in lactating dairy cows using RR original scale are summarised in Table 2, and on orthogonal polynomials, Koenen et al. 1999 corresponding values for data transformed to loga- also found that, based on LRTs, a high order of rithmic scale are given in Table 3. For models polynomial fit was required. assuming homogeneous measurement error variances 2 ‘‘.E1’’, estimates of s are given in addition. 3.3. Modelling measurement error variances e i Which order of fit? Phenotypic standard devia- tions for PH from analyses fitting RRs on orthogonal Similar patterns were observed for analyses fitting polynomials and a single measurement error variance multiple measurement error variances Models Models LPk.E1 are shown in Fig. 5, for orders of LPk.E12, LPk.E15 and LPk.E66. As illustrated in fit ranging from k 5 4 to 22. For previous analyses of Fig. 6, log + increased with the order of fit at January weights only, a cubic polynomial, i.e. k 5 4, approximately the same rate for different assump- sufficed to model growth of cows from 19 to 118 tions about the variance structure of temporary months of age adequately Meyer, 1999. Consider- environmental effects. Plotting log + against the ing records throughout the year, however, this was number of parameters p estimated shows that at 2 clearly inappropriate, with phenotypic standard de- low p a single s m 5 1 provided the best fit to the e i viations s derived from the estimated CF increas- data since for constant p it allowed the highest k. P 2 ing sharply for the last 12 ages in the data, reaching Conversely, individual s m 5 66 were only ad- e i 132.5 kg for 84 months. In contrast, the estimate of vantageous, at equal numbers of parameters fitted, s at 84 months from a univariate analysis was 52.8 for high orders of fit k 17 and large numbers of P kg. parameters p . 211. Increasing k reduced the obvious upwards bias in Estimates of standard deviations from analyses estimates of s for the latest ages, i.e. the ages with fitting RR on LPs of age allowing for different P the least records. An order of k 5 12 was required numbers of measurement error variances are con- before estimates of s for cows older than 6 years trasted in Fig. 7. While estimates of s differed P e settled around the 60 kg mark. For orders of fit larger substantially for m 5 1, m 5 15 and m 5 66 and log than k 5 12, the overall trend in estimates of s over + increased markedly and significantly with increas- P time changed little. However, curves representing s ing number of variance components fitted, there was P exhibited more and more oscillations as k increased little difference in estimates of between animal k 5 16 and k 5 18 not shown. As shown in Table 2, standard deviations for k 12. Differences in esti- likelihoods increased dramatically with increasing mates of s not shown were then largely due to P order of polynomial fit, accompanied by a steady differences in estimates of s . Corresponding results e 2 decline in the corresponding estimates of s . have been reported by Olori et al. 1999, who found e i With large numbers of records per individual, all no significant differences in estimates of additive increases in k yielded significant improvements in genetic and permanent environmental variances of log +. Augmenting k from 20 to 22 added 43 test day records in dairy cows when fitting different covariances among RR coefficients to be estimated, numbers of measurement error variances. Similarly, increasing log + by 172.5, while there was little Snyder et al. 1999 reported only minor differences difference in the resulting estimates of s . Not only in estimates of variances for daily feed intake in pigs P were the corresponding plots Fig. 5 virtually between models fitting a single measurement error indistinguishable, the mean squared difference be- variance and assuming the log error variance was 28 K . Meyer Livestock Production Science 65 2000 19 –38 Table 2 2 Maximum log likelihood values log + for different analyses, together with estimates of the measurement error variance s for models e fitting a single component a 2 b c a 2 b c Model p log + s r p Model p log + s r p e e Polled Hereford LP8F2.E1 56 293 467.0 220.5 LP4.E1 11 298 495.9 401.1 LP8F4.E1 79 293 384.5 215.3 LP6.E1 22 296 451.6 316.1 LP8F6.E1 106 293 345.5 213.3 LP8.E1 37 294 566.6 260.7 LP10F2.E1 79 292 289.9 188.5 LP10.E1 56 293 540.0 220.1 LP10F4.E1 106 292 196.9 184.1 LP12.E1 79 292 943.8 208.1 LP12F2.E1 106 291 524.1 167.9 LP14.E1 106 292 443.4 191.2 LP12F4.E1 137 291 420.6 162.7 LP16.E1 137 291 710.1 171.2 15 136 LP14F2.E1 137 291 064.7 157.4 LP18.E1 172 291 077.3 156.4 16 169 LP121F2.E1 82 291 720.5 169.4 LP20.E1 211 290 747.7 148.4 19 210 LP121F4.E1 89 291 662.8 165.0 LP22.E1 254 290 575.2 143.0 SQ13F2.E1 121 291 249.0 163.0 LP7.E12 40 294 861.2 6 39 LP61M LP4.E1 32 295 755.8 278.6 ] LP8.E12 48 294 183.3 6 47 LP81M LP4.E1 47 293 646.3 222.2 ] LP9.E12 57 294 068.4 LP81M LP6.E1 58 293 526.1 218.2 ] LP10.E12 67 293 295.8 LP81M LP8.E1 73 293 503.6 216.1 816 70 ] LP7.E15 43 294 713.1 LP101M LP6.E1 77 292 373.1 187.0 ] LP8.E15 51 293 915.7 LP121M LP6.E1 100 291 658.5 166.6 ] LP10.E15 70 292 977.6 LP121M LP10.E1 134 291 612.8 163.9 1117 127 ] LP11.E15 81 292 609.3 LP201M LP6.E1 232 290 642.7 143.4 1815 228 ] LP12.E15 93 292 358.6 LP201M LP10.E1 266 290 575.2 139.7 1817 257 ] LP13.E15 106 292 112.1 LP121M F2.E1 82 291 728.8 169.9 ] LP14.E15 120 291 884.5 LP141F2.E1 109 291 272.9 159.7 1312 108 LP15.E15 135 291 582.9 14 134 LP141F2.E12C 120 290 890.7 1312 119 LP16.E15 151 291 295.5 LP141M F2.E12C 120 290 889.5 1312 119 ] LP17.E15 168 290 994.9 SQ131M F2.E1 95 291 338.8 163.6 1212 94 ] LP18.E15 186 290 676.0 16 183 SQ131M F2.E12C 106 290 892.7 1212 105 ] LP7.E66 94 294 288.8 SQ131F2.E1 95 291 340.9 163.6 LP10.E66 121 292 685.8 SQ131M F2.E1 95 291 338.3 163.6 ] LP12.E66 144 291 865.5 Wokalup LP14.E66 171 291 431.8 LP14.E1 106 2100 396.1 233.1 LP17.E66 219 290 576.5 16 218 LP15.E1 121 299 881.1 217.8 LP18.E66 237 290 357.4 LP22.E1 254 298 357.6 172.0 21 253 LP20.E66 276 290 068.0 19 275 LP14.E66 171 299 350.4 13 170 LP20.E12C 222 290 429.6 18 219 LP15.E66 186 298 918.3 14 185 SQ8.E1 37 294 478.3 262.4 7 36 LP17.E66 219 298 326.3 SQ9.E1 46 295 549.4 284.0 8 45 LP20.E66 276 297 807.1 SQ13.E1 92 291 925.1 181.7 SQ13.E1 92 299 931.2 224.2 12 91 SQ14.E1 106 291 885.4 181.4 13 105 SQ14.E1 106 2100 112.3 228.8 13 105 SQ9.E66 111 294 621.6 SQ14.E66 171 299 223.8 SQ14.E66 171 291 295.4 SQ13F2.E1 121 299 109.4 198.7 SC9.E1 46 294 972.0 268.4 LP12F2.E1 106 299 470.5 206.6 SC14.E1 106 295 045.0 274.0 12 103 LP121F2.E1 82 299 675.5 207.9 LP4F2.E1 22 297 928.6 363.1 LP201M F4.E1 221 298 424.5 171.0 1814 218 ] LP4F4.E1 37 297 895.7 361.8 6 34 LP201M LP10.E1 266 298 358.4 168.0 1817 257 ] LP6F2.E1 37 295 601.9 276.1 SQ131M F2.E1 95 299 268.0 200.6 12 94 ] LP6F4.E1 56 295 555.6 275.0 SQ141M F2.E1 109 299 136.7 197.3 13 108 ] LP6F6.E1 79 295 534.7 273.7 SQ141M F4.E1 116 299 038.2 190.4 13 115 ] a Number of parameters full rank. b Reduced rank of estimated covariance matrix; left blank if matrix was not constrained. c Number of parameters if reduced rank covariance matrix has been estimated. K . Meyer Livestock Production Science 65 2000 19 –38 29 Table 3 Maximum log likelihood values log + for analyses of data transformed to logarithmic scale, together with estimates of the measurement 2 error variance s for models fitting a single component e a 2 b c a 2 b c Model p log + s r p Model p log + s r p e e Polled Hereford SQ131M F2.E1 95 252 091.4 7.8 ] LP20.E1 211 250 379.0 6.9 SQ131M LP2.E12C 106 251 395.7 10 100 ] LP20.E12C 222 249 814.9 Wokalup LP61M LP6.E1 43 254 788.4 12.5 LP20.E1 211 254 079.5 7.3 ] LP101M LP6.E1 77 251 911.2 8.8 LP20.E12C 222 253 395.6 ] LP121M LP6.E1 100 251 316.7 7.9 LP121M LP6.E1 100 255 127.1 8.4 ] ] LP121M F2.E1 82 251 427.4 8.1 LP121M F4.E1 89 255 088.0 8.3 ] ] LP121M F4.E1 89 251 313.0 7.9 SQ131M F2.E1 95 256 164.4 8.3 ] ] a Number of parameters full rank. b Reduced rank of estimated covariance matrix; left blank if matrix was not constrained. c Number of parameters if reduced rank covariance matrix has been estimated. Fig. 5. Estimates of phenotypic standard deviations for Polled Herefords, from analyses fitting random regressions on Legendre polynomials of order k 5 4 to k 5 22, assuming homogeneous measurement error variances. described by a quadratic function in age. This parameters to be estimated in individual analyses, suggests that for a sufficient order of fit, i.e. an order thus making model comparisons easier. of fit which does not produce gross over- or under- Comparison with univariate estimates. When 2 estimates of phenotypic variances at individual ages, fitting individual s for all ages m 5 66, estimates e different regression curves to model between animal of s exhibited a distinct cyclic pattern with peaks e variation can be examined under the assumption of every 12 months. As shown in Fig. 8 for model homogeneous measurement error variances. This LP20.E66, estimates of s from RR analyses for both e reduces computational requirements and potential breeds agreed closely with estimates of residual convergence problems due to large numbers of standard deviations from univariate analyses of 30 K . Meyer Livestock Production Science 65 2000 19 –38 Fig. 6. Maximum log likelihood for analyses of Polled Hereford data, fitting random regressions on Legendre polynomials of age, allowing for single m 5 1, j, grouped m 5 15, d and individual m 5 66, m measurement error variances. Fig. 7. Estimates of the between animal top row and measurement error bottom row standard deviations kg for Polled Herefords, from analyses fitting random regressions on Legendre polynomials of order k 5 10 first column, k 5 12 second column, k 5 14 third column and k 5 20 last column, and fitting different numbers of measurement error variances dark grey: single variance, light grey: 15 variances, black: 66 variances. K . Meyer Livestock Production Science 65 2000 19 –38 31 Fig. 8. Estimates of between animal squares and measurement error circles standard deviations in kg from univariate analyses open symbols and analyses fitting random regressions on Legendre polynomials of order k 5 20 and allowing for separate measurement error variances for individual ages m 5 66 closed symbols, for Polled Herefords left and Wokalups right. individual ages. Corresponding estimates of between models .E1 and .E12C were found for other RR animal components showed reasonable agreement for models for PH LP141M F2, SQ131M F2 and ] ] earlier ages, in particular for WOK. Estimates from analyses on the logarithmic scale Table 3. This both types of analyses followed a similar, cyclic suggests that the cyclic measurement error variances pattern, indicating that the RR model fitted modelled indeed represent a reasonable compromise between the existing variation in the data adequately. How- parsimony and adequateness of modelling temporary ever, values from RR analyses were consistently environmental variation. Conversely, it indicates that higher for later ages. Presumably, this reflected, in there were additional, age-specific and non-seasonal part at least, a reduction in variation between animals differences in variation which were only fully mod- due to culling: RR analyses incorporating records at elled by fitting individual measurement error vari- all ages are expected to account for this while their ances for all ages. univariate counterparts are not. Regression line. Finally, it was attempted to 2 Cyclic measurement error variances. The pat- model changes in s over time through a polynomial e tern in estimates of s observed above Fig. 8 regression equation. This involved estimating a e suggested that a model with 12 distinct components single measurement error variance at the mean age repeated cyclically .E12C might fit the temporary 0 on the standardised scale and the coefficients of a 2 environmental variation in the data without increas- regression line to predict s at individual ages. A e ing the number of parameters to be estimated similar approach, in retrospect though rather than dramatically over the assumption of homogeneous integrated in the analysis, has been taken by Jam- residuals. As shown in Table 2 for k 5 20 for PH, rozik and Schaeffer 1997 who estimated individual 2 adding 11 parameters to go from model .E1 to model s for test day records in dairy cattle for 29 days-in- e i .E12C increased log + by 318.1 while adding the milk classes and then fitted a regression to the additional 55 parameters to fit individual measure- estimates. Fitting a regression to model trends in ment error variances yielded another, again signifi- measurement error variances increased log + sig- cant increase of 361.6. Similar differences between nificantly over a model assuming homogeneity of 32 K . Meyer Livestock Production Science 65 2000 19 –38 variance results not shown. However, it failed to RR coefficients to be fitted for each animal. This model the cyclic variation identified in the data provided a better fit to the data than a RR on LPs of adequately, even for high orders of polynomial fit. At age, involving the same number of parameters at equal numbers of parameters to be estimated, models k 5 8 log + 5 2 94, 478.3 for model SQ8.E1 and .E15 resulted in higher log +, i.e. better fit to the log + 5 2 94, 566.6 for LP8.E1 for PH; see Table data. 2. Shifting knots by one month, however, made a quadratic approximation for intervals of this length 3.4. Alternative curves clearly inappropriate. At z 5 6 and 9 RR coefficients, likelihoods for models SQ9.E1 and SQ9.E66 were While RR on orthogonal polynomials proved considerably lower than for SQ8.E1. capable of modelling the cyclic, seasonal variation in Choice of knots was less critical when considering variance encountered in the data, high orders of quadratic approximation in 6 months segments. For polynomial fit were required. This resulted in a large PH, model SQ14.E1 gave a significantly better fit to number of parameters to be estimated and corre- the data than SQ13.E1, while for WOK SQ13.E1, spondingly large computational requirements. Other fitting 14 parameters less than SQ14.E1, produced a curves might be more suitable for this kind of higher log +. The latter indicates convergence pattern, i.e. might provide similar results requiring problems for that analysis SQ14.E1 in WOK. Both less RR to be fitted for each animal. models SQ13 and SQ14 yielded considerably higher Segmented polynomials. The first alternative maximum log + values than LP14, at the same considered were segmented polynomials. Such func- number SQ14 or less SQ13 parameters fitted, in tions have found occasional use in animal breeding both breeds Table 2. As shown in Fig. 9, this was applications, e.g in modelling growth curves of goats clearly due to the greater scope of the SQ to follow Gipson, 1997 or in describing lactation curves El the cyclic changes of growth and its variation. Faro et al., 1998. More recently, there has been Differences in log + between SQ14 and LP14 were 2 interest in cubic spline functions to smooth lactation considerably less when fitting individual s .E66 e 2 curves White et al., 1999; see also Verbyla et al. than when assuming homogeneous s , being 136.6 e 1999 for a detailed exposition on the use of and 558.0, respectively, for PH and 126.6 and 443.9, smoothing splines for the analysis of longitudinal respectively, for WOK Table 2. This suggested that data in a linear model framework. Allowing for part of the advantage of SQ over LP was due to different polynomials in individual segments, these being able to better account for differences in functions were expected to be more flexible in temporary environmental variation. approximating the seasonally fluctuating growth Convergence of the SQ analyses was markedly curves in our data than polynomials fitted to the slower than that of corresponding LP analyses. whole range of ages. There has been concern about Presumably this was due to the fact that ordinary the numerical stability of covariance function esti- polynomials 5quadratics were used which, by mates involving high orders of fit and thus high default, were highly correlated. Similarly, analyses powers of age Kirkpatrick et al., 1994. Typically, fitting segmented cubic polynomials SC9 and SC14 this created ‘wiggly’ surfaces when plotting esti- were unsuccessful, resulting in log + which were mated covariances in the data, especially at the considerably lower than those of corresponding extremes of the ages considered. Fitting segments of analyses fitting quadratic functions SQ8 and SQ13. low order polynomials was anticipated to have the This may well have been a convergence and parame- additional benefit of being less susceptible to such terisation problem. Perhaps an alternative representa- problems. tion would aid convergence. For instance, cubic Knots, i.e. joining points between segments were splines are seldom presented in the form of 5 but chosen after inspecting results from univariate analy- are often specified in a ‘value and second derivative’ ses and RR analyses involving Legendre polyno- at each knot form. mials. Choosing segments of 12 months duration Fourier series approximation. The cyclic pattern yielded 5 knots and, for a quadratic approximation, 8 of growth and its variances encountered cf. Fig. 8 K . Meyer Livestock Production Science 65 2000 19 –38 33 Fig. 9. Estimates of phenotypic standard deviations in kg for Polled Herefords d and Wokalups s from analyses fitting segmented quadratic polynomials and Legendre polynomials superimposed by sinusoid and cosinoid curves, assuming homogeneous measurement error variances. made an approximation by sinusoidal curves with a those for LP at very high orders of fit k 5 20, 22; periodicity of 12 months an obvious choice. Allow- see Figs. 5 and 8 for comparison. Correlations ing for covariances between all regression coeffi- between RR coefficients on LPs and RR on sine or cients, a model superimposing a Fourier series cosine curves were generally weak and on average approximation consisting of f curves onto Legendre close to zero. For analysis LP14F2.E1 in PH, for polynomials fitted to order k, LPkFf.Em, involved the instance, the 28 correlations ranged from 2 0.39 to same number of parameters as a model LPk 1 f .Em 0.38 with an average of 0.08. Assuming the two at equal m. As shown in Table 2, a minimum order types of regression coefficients were uncorrelated of fit of k 5 10 for LPs was required, sufficient to models Pp 1Ff resulted in dramatically increased describe the average trend in growth without distor- log + over values for the base polynomial function tion, before models LPkFf.E1 fitted the data better Pp while adding only f f 1 1 2 covariances to be than their counterparts LPk 1 f .E1. Most of the estimated. For LP121F2.E1 and LP141F2.E1 in improvement in fit was achieved by fitting only one PH, for example, increases in log + over LP12.E1 sine and one cosine curve f 5 2. While increasing f and LP14.E1 were as large as 1233.3 and 1270.4, to 4 or 6 at constant LPk increased log + sig- respectively, for 3 additional parameters Table 2. nificantly in all cases examined, at equal number of However, the resulting estimates of s exhibited P parameters models LPkF4 generally had substantially a very different pattern to those obtained from lower log + than LPk 1 2F2. corresponding analyses assuming covariances were As demonstrated in Fig. 9, allowing for covar- non-zero. This is illustrated in Fig. 9 for model iances between coefficients of the Fourier series LP121F2.E1, compared to LP12F2.E1. Generally, approximation and the underlying polynomial curve estimates of s from models Pp 1F2.E1 were very P enforced a cyclic pattern with amplitudes similar to similar to those from Pp.E1 P standing for LP or 34 K . Meyer Livestock Production Science 65 2000 19 –38 SQ, with only a small proportion of variance instance, used covariance functions to model chang- between animals contributed by the covariance func- ing variation in test day of dairy cows records across tion pertaining to the Ff part of the model. This the lactation and production level parsimoniously, suggested that a large proportion of the benefit with days in milk and herd level as independent achieved in adding an uncorrelated set of regression variables. While month did not quite represent a coefficients on sine and cosine functions, was totally continuous, infinite-dimensional scale, regres- through better modelling of the temporary environ- sions on month of recording were fitted for each mental variation. Alas, fitting cyclic error variances animal in an attempt to separate seasonal and age- for analysis LP141F2 in PH, increased log + by determined variation in weight. As shown in Table 2, 389.2 LP141F2.E12C vs. LP141F2.E1; Table 2, this was done using Legendre polynomials and indicating that this was only a partial explanation and Fourier series approximations. In all instances, co- that a better fit could be achieved by modelling variances with RR coefficients on age were assumed heterogeneous error variances explicitly. to be zero. Again, fitting any additional function increased log 3.5. Random regression on month of recording + significantly throughout. Effects of the additional set of regression coefficients were most pronounced While most RR model analyses in the literature for models involving low orders of fit for RR on age consider a single continuous, independent variable Table 2. This is illustrated in Fig. 10 for models only, the approach readily accommodates multiple LP121M Pp.E1 versus models LP20PM Pp.E1. ] ] ‘meta-meters’. Veerkamp and Goddard 1998, for There was little difference in estimates of s be- P Fig. 10. Estimates of phenotypic standard deviations in kg for Polled Herefords from analyses fitting random regressions on month of recording m: Legendre polynomials with order of fit 6 Model LPk 1M LP6.E1, \: Legendre polynomials with order of fit 10 Model ] LPk 1M LP10.E1, d: Fourier series approximation with f 5 2 curves Model LPk 1F2.E1, and + : no regression for month Model ] LPk.E1 in addition to regressions on Legendre polynomials of age, for order of fit of k 5 12 left graph and k 5 20 right graph, assuming homogeneous measurement error variances. K . Meyer Livestock Production Science 65 2000 19 –38 35 tween orders of fit for LPs of months, k 5 6 and coefficients was to a large part due to better model- k 5 10, respectively. As shown in Table 2, when ling of temporary environmental variation. fitting models LPk 1M LP10.E1, the matrix of co- ] variances between RR coefficients for months con- 3.6. Analyses on the logarithmic scale sistently had a reduced rank of only 7, indicating that a full order fit for months represented an over- Results from analyses of data transformed to parameterisation. logarithmic scale for selected models are given in Estimates of s fitting Legendre polynomials for Table 3. On the whole, differences between analyses P RR on months of recording 1M LPk were lower corresponded to those of the untransformed data. ] than those obtained fitting a Fourier series approxi- Existence of a scale effect on variances had been mation for months 1M Ff , substantially so for noticed above Fig. 2. Estimates of s for model P ] k 5 12 and to a lesser extent for k 5 20; see Fig. 10. LP20.E12C are contrasted in Fig. 11 to corre- Values for log + from analyses fitting a regression sponding values from univariate analyses. Again on sine and cosine functions of age 1Ff and those there was close agreement between analyses for fitting a corresponding regression on month estimates of s , the logarithmic transformation hav- e 1M Ff as a second, independent set of RR ing levelled out univariate estimates, especially for ] coefficients were very similar Table 2. Further- PH. Estimates of the between animal components more, resulting estimates of s for equal f were showed less differences than for the untransformed P virtually indistinguishable not shown. This empha- data; see Fig. 8 for comparison. sizes the close confounding between age and month While estimates of between animal standard devia- of recording in these data. The difference between tions still exhibited a cyclic pattern, this was less models 1M LPk and models 1M Ff also supports distinct and following a clear annual cycle than ] ] the hypothesis above that the improvement in log + observed previously. Peaks became particularly and fit of the model due to these additional RR prominent for older ages. On the whole, there was a Fig. 11. Estimates of between animal squares and measurement error circles standard deviations in kg from univariate analyses open symbols and analyses fitting random regressions on Legendre polynomials of order k 5 20 and fitting 12 cyclic, measurement error variances closed symbols, for Polled Herefords left and Wokalups right for data transformed to logarithmic scale 3 100. 36 K . Meyer Livestock Production Science 65 2000 19 –38 slight downward trend in variance for later ages, stone et al. 2000 found that parametric curves fitted indicating that the upwards trend in estimates on the the data best, but produced negative estimates of untransformed scale was associated with the increase genetic correlations between early and late lactation in average weight. As the cyclic pattern on the while RR on orthogonal polynomials did not. The logarithmic scale shows, seasonal variation in vari- rationale for concentrating on variances or standard ances clearly could only partially attributed to scale deviations in this study was that it seemed futile to effects. Moreover, Fig. 11 suggests that the non-size examine covariances, unless variances had been related, seasonal differences became more important modelled correctly. at later ages, presumably because variation in the RR models proved well capable of describing the ability of cows to cope with dearth increased with existing, complex pattern of variation. Not requiring age. any prior knowledge about the shape of curves to be modelled, RR on orthogonal polynomials of age performed admirably. While they required more

4. Discussion parameters to achieve the same degree of fit mea-