50 th
Annual Meeting of the European Association for Animal Production, Zürich, August 22-26, 1999
RANDOM REGRESSION MODELS TO DESCRIBE PHENOTYPIC VARIATION IN
WEIGHTS OF BEEF COWS WHEN AGE AND SEASON EFFECTS ARE CONFOUNDED
Karin Meyer
Animal Genetics and Breeding Unit, University of New England, Armidale, NSW 2351
SUMMARY
Weights of beef cows between 19 and 84 months of age, recorded on a monthly basis during a selection experiment, were analysed using random regression models. Only phenotypic animal effect were considered.
Curves fitted included orthogonal (Legendre) polynomials of age, fitted up to an order of 22, segmented quadratic polynomials and Fourier series approximations. There were strong seasonal effects on weights of cows, and due to short calving seasons, a high degree of confounding between age and month of recording.
Data exhibited cyclic seasonal fluctuations in variances which could only partially be attributed to scale effects. Random regression models proved capable of modelling such pattern of variation adequately, but required high orders of fit and thus a large number of parameters.
Keywords : Random regression, growth curve, mature weights, beef cattle.
INTRODUCTION
Growth of animals is a prime example of a 'trait', measured repeatedly per individual along a continuous scale (time), which changes gradually and continually. Analyses fitting a random regression (RR) model account for the non-constant nature of such traits. Moreover, the resulting estimates of covariance functions provide estimates of covariances for the complete range of ages considered, and allow a great deal of flexibility in modelling variances changing over time. Records for weights of beef cows from a selection experiment at the Wokalup research station in Western Australia have been analysed earlier using this approach (Meyer 1999). Although cows were weighed monthly, except during the calving season, only single annual records taken in January were considered. At this point in time, cows were typically at their top weight, strong seasonal effects causing substantial fluctuations in weights of cows over the year. This paper presents RR analyses of these data on a phenotypic level including records throughout the year, examining the scope of RR to model a cyclic pattern of variation.
MATERIAL AND METHODS
Data . Data consisted of weights of cows from a selection experiment carried out at the Wokalup research station in the South of Western Australia. The experiment comprised two herds of about 300 cows, a purebred Polled Hereford (PH) herd and a herd of a four-breed synthetic formed by mating Charolais x
Brahman bulls to Friesian x Angus or Friesian x Hereford cows, the so-called 'Wokalups' (WOK), see Meyer et al . (1993) for details. Management of the herds was characterised by short (2 months) mating periods and correspondingly short calving seasons. Most calves were born during April and May each year. This resulted in a high degree of confounding between age and month of weighing. Throughout the experiment all animals were weighed on a monthly basis, except for the calving season. After basic edits, this yielded a total of
87,516 weights taken between 1977 and 1990. Records considered were weights of cows taken between 19 and 84 months of age (inclusive). Earlier and later records were excluded to minimise the influence of maternal effects and problems associated with small numbers per age and subclass, respectively. Similarly, small numbers of weights taken in April and May were ignored. This gave 27,728 records for 922 PH cows and 29,033 records for 1020 WOK cows with weights ranging from 193 to 836 kg and 226 to 946 kg, respectively, and a maximum of 62 records per cow.
Analyses . Data were analysed fitting a RR model with age at weighing as the independent covariable. Only phenotypic animal effects, comprising animals' genetic and permanent environmental effects, were fitted and relationships between animals were ignored. The same fixed effects were fitted for all analyses, comprising contemporary groups (CG), defined as paddock-year-week of weighing subclasses, and a fixed regression on orthogonal polynomials of age at weighing to model the mean growth curve. With partial confounding of CG and age, a cubic regression sufficed. A range of RR models was investigated. These included RR on orthogonal (Legendre) polynomials (LP) of age, up to an order of fit of 22 (denoted as LP k for an order of fit of k ), and RR on segmented quadratic polynomials (denoted as SQ q for a polynomials with q -3 knots)
(Fuller 1969) for 6- and 12-month segments, resulting in 14 and 9 RR coefficients, respectively. In addition,
RR on first (F2) or second order (F4) Fourier series approximations with a periodicity of 12 months,
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Annual Meeting of the European Association for Animal Production, Zürich, August 22-26, 1999 superimposed on LPs to model age trends were examined (denoted as LP k F f for f =2,4). Temporary environmental effects were considered to be independently distributed throughout with 'measurement error'
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Annual Meeting of the European Association for Animal Production, Zürich, August 22-26, 1999
2500 600
2000
500
1500
1000
WOK
PH
PH
WOK
400
500
300
0
25 37 49
61 73 85
200
Figure 1. Numbers of records and mean weights for individual ages. variances  i
2 , for i =1, m . Alternatives considered were homogeneous variances ( m =1),  i
2 grouped for 6 months periods with separate components at the extremes ( m =15 : 19, 20-21, 22-24, 25-30, 31-36, …, 73-78,
79-81, 82-83, 84 months) and individual 

2 for each age ( m =66). A number of analyses transformed records to logarithmic scale to eliminate scale effects. Covariances of RR coefficients, phenotypic covariance functions (P) and 

2 were estimated by restricted maximum likelihood using program D X M RR (Meyer
1998). Up to 276 parameters were estimated simultaneously. Covariance matrices for the ages in the data were computed from estimates of P. In addition, corresponding 'standard' univariate analyses fitting a repeatability model were carried out for individual ages, considering records for ages i -1, i and i +1 for an analysis for the i -th age. This provided estimates of the between animal and temporary environmental variance for each age.
RESULTS AND DISCUSSION
Numbers of records for each age and corresponding means are given in Figure 1. Climate at the research station was of Mediterranean type, with winter rains and annual summer (Southern hemisphere) droughts from November to April. This resulted in summer dearth and mean cow weights decreased by almost 100 kg from January to June each year. With close confounding between age and month of recording, this annual cycle is reflected in the changes in means over ages. Lowest weights during winter were accompanied by lowest phenotypic variances. Coefficients of variation, however, were highest in winter and lowest in
January (ranging from 10.4% to 9.1% for PH and 11.1% to 9.7% for WOK), indicating that seasonal differences in variances could only partially be attributed to scale effects.
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50 th
Annual Meeting of the European Association for Animal Production, Zürich, August 22-26, 1999
80
60
40 k=4 log L = -98,496 p=11
 2
=401.1
k=6 log L = -96,452 p=22
 2
=316.1
k=8 log L = -94,567 p=37
 2
=260.7
k=10 log L = -93,540 p=56 s2=220.1
70
60
50
40 k=12 p=79 log L= -92,944
 2
=208.1
k=14 log L = -92,443 p=106
 2
=191.2
k=20 log L = -90,748 p=211
 2
=148.4
k=22 log L = -90,575 p=254
 2
=142.0
25 37 49 61 73 85 25 37 49 61 73 85 25 37 49 61 73 85 25 37 49 61 73 85
Age (mon)
Figure 2. Estimates of phenotypic standard deviations for different orders of fit (k); Polled Hereford, assuming homogeneous measurement error variances (

2 ). k=10 k=12 k=14 k=20
70
50
30
30
20
10
25 37 49 61 73 25 37 49 61 73 25 37 49 61 73
Age (mon)
25 37 49 61 73
Figure 3. Estimates of between animal (top row) and measurement error (bottom row) standard deviations for Legendre polynomials of different orders of fit (k), fitting a single (black), 15 (light grey) and individual (dark grey) measurement error variances (Polled Hereford).
Orthogonal polynomials . Estimates of phenotypic standard deviations ( 
P
) for PH from RR analyses, fitting
LP with k ranging from 4 to 22 and assuming homogeneous measurement error variances, are shown in
Figure 2 together with corresponding values for the maximum log likelihood (log L), the number of parameters estimated (p) and the estimate of  2 . Values of k >22 were not examined due to computational requirements. For both breeds log L increased consistently and significantly with k . For example, increasing k from 19 to 20, added 20 parameters while augmenting log L for PH by 289.4. For low orders of fit ( k <10),

P
for later ages (>70 months) were dramatically overestimated. From k =12 onwards, there was little difference in the average trend in estimates of 
P
. However, as k increased plots showed more oscillations, i.e. RR modelled seasonal differences in variation more explicitly. Transforming data to logarithmic scale only slightly dampened the wave pattern.
Estimates of between animal ( 
A
) and error standard (

) deviations for PH from analyses fitting LP and m =1,
15 and 66 are contrasted in Figure 3. While estimates of
 differed substantially and log L increased
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Annual Meeting of the European Association for Animal Production, Zürich, August 22-26, 1999 dramatically with increasing m , there was little difference in estimates of

A
for k >10. This suggests that for sufficient orders of fit, RR models for animal effects can be compared assuming homogeneous errors, thus reducing computational requirements. For m=66, estimates of

showed a distinct cyclic pattern with sharp peaks every 12 months at the ages corresponding to the lowest weights. As shown in Figure 4 for k =20, there
70 Polled Hereford Wokalup
50
30
10
25 37 49 61 73 85 25
Age (mon)
37 49 61 73 85
Figure 4. Estimates of between animal (circles) and measurement error (squares) standard deviations from univariate (grey symbols) and random regression model (black symbols) analyses fitting
Legendre polynomials to order k=20 and individual error variances (n=66).
70
60
50
40
30
60
LP14 log L = -92,443
LP12F2 log L =-91,524
LP14F2 log L=-91,065
25 37 49 61 73 85
Age (mon)
50
40
SQ14 log L = -91,885
LP12F4 log L=-91,421
30
25 37 49 61 73 85 25 37 49 61 73 85
Figure 5. Estimates of phenotypic standard deviations for Polled Hereford fitting random regressions on alternative curves (see text) and single measurement error variances. was excellent agreement with corresponding estimates from univariate analyses for both breeds, indicating that the RR analysis modelled the existing variation in the data adequately. While exhibiting the same pattern, estimates of

A
from RR analyses were consistently higher than their univariate counterparts, increasingly so with rising age. To some extent this might be explained by effects of culling of cows on variances.
Other curves . Plots suggested that alternative curves might be better suited to model the cyclic seasonal variation encountered in the data, requiring fewer parameters than LP. As shown in Figure 5, fitting a SQ involving 6-months segments, log L for SQ14 was higher than for LPs with the same number of RR coefficients (LP14). Clearly, this was due to a greater scope for the SQ to accommodate seasonal fluctuations. Model LP12F2 fitted LP to order 12 superimposed by a first order Fourier series approximation
(one sine and one cosine curve). Assuming RR coefficients for the two types of curves were correlated, this
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Annual Meeting of the European Association for Animal Production, Zürich, August 22-26, 1999 involved the same number of parameters ( p =106) as LP14 and SQ14 but provided a better fit, the plot for
LP12F2 showing similar oscillations to those for LP20 and LP22. Fitting an additional set of RR on sine and cosine curves, model LP12F4, increased log L significantly but not as much as increasing the number of LPs correspondingly, model LP14F2. With p =137 both provided a better fit than LP16 (not shown) with log
L=91,710.
CONCLUSIONS
Analyses identified a distinct cyclic pattern in variances of body weights of cows, with highest values at low weights after calving and a period of poor nutrition. Random regression models proved well suited to the analysis of growth data. Complex patterns of variation, such as encountered in this study, can be modelled adequately, although a large number of parameters can be required. Legendre polynomials do not require any assumptions about the shape of curves to be modelled and are thus useful as general purpose functions, but alternative curves, utilising the known periodicity of events, can provide more parsimonious models.
REFERENCES
Fuller, W.A. (1969). Grafted polynomials as approximating functions. Aust. J. Agric. Econ . 13 : 35-46.
Meyer, K. (1998). "D X MRR" a program to estimate covariance functions for longitudinal data using a random regression model. Proc. 6 th World Congr. Genet. Appl. Livest. Prod.
Vol. 27 : 465-466.
Meyer, K. (1999). Estimates of genetic and phenotypic covariance functions for postweaning growth and mature weight of beef cows. J. Anim. Breed. Genet . 116 : 181-205.
Meyer, K., Carrick, M.J. and Donnelly, B.P.J. (1993). Genetic parameters for growth traits of Australian beef cattle from a multi-breed selection experiment. J. Anim. Sci . 71 : 2614-2622.
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