Estimation of Genetic and Phenotypic Covariance Functions for Body Weight as Longitudinal Data of SD-II Swine Line

: Growth records over six generations of 686 pigs in SD-II Swine Line were used to estimate the genetic and phenotypic covariance functions for body weight as longitudinal data. A random regression model with Legendre polynomials of age as independent variables was used to estimate the (co)variances among the regression coefficients, thus the coefficients of genetic and permanent environmental covariance functions by restricted maximum likelihood employing the average information algorithm. The results showed that, using litter effect as additional random effect, a reduced order of fit did not describe the data adequately. For all five orders of fit, however, the change trends of genetic and phenotypic (co)variances were very similar from κ =3 onwards. (Asian-Aust. J.


INTRODUCTION
An animal's phenotype changes with age. A trait that changes with age can be represented as a trajectory, that is, a function of time. Often, such traits are measured on the same individual(s) at various times or ages. Such records are commonly referred to as longitudinal data and such trajectories as infinite-dimensional characters. The infinitedimensional model offers several advantages over earlier attempts to adapt quantitative genetics to growth trajectories (Kirkpatrick and Heckman, 1989;Kirkpatrick et al., 1990).
Covariance functions (CFs) have been recognized as a suitable alternative to the conventional multivariate mixed model to describe genetic and phenotypic variation for longitudinal data (Meyer, 1998a). In essence, CFs are the infinite-dimensional equivalents to covariance matrices in a traditional, finite-multivariate analysis (Kirkpatrick et al., 1990).
As the name indicates, a CF describes the covariance between records taken at certain ages as a function of these ages. A suitable class of functions is the family of orthogonal polynomials (Kirkpatrick et al., 1990). Kirkpatrick et al. (1990Kirkpatrick et al. ( , 1994 modeled CFs using orthogonal polynomials of age, choosing Legendre polynomials, and described a generalized least-squares procedure to determine the coefficients of a CF from an estimated covariance matrix. However, Meyer and Hill (1997) thought that this was often not available or computationally expensive to obtain. These authors showed that the coefficients of CFs could be estimated directly from the data by restricted maximum likelihood (REML) through a simple reparameterisation of existing, finite-dimensional multivariate REML algorithms. In the general case, however, this approach required a multivariate mixed model matrix proportional to the number of ages in the data to be set up and factored, even for a reduced order of fit. This severely limited practical applications, especially for data records at all ages. Meyer (1998a,b) described another alternative procedure for the estimation of CFs. It was shown that the CF model was equivalent to a random regression model (RRM) with polynomials of age as independent variables, and that REML estimates of the coefficients of the CF could be obtained as (co)variances among the regression coefficients.
SD-II Line is a synthesized specialized dam line of Duroc and Shanxi Black Swine using a closed nucleus breeding system. This paper estimates the genetic and phenotypic CFs for body weight of the SD-II Line using a RRM so as to reflect the changes of genetic and phenotypic (co)variances in continuous time scale. The resultant genetic parameters from estimated CFs can be further used in genetic evaluation.

Data
Data on body weight collected from the SD-II Line for six generations were used for this study. The data set comprised 3430 records from 686 pigs, each weighed at birth, 35, 70, 120 and 180 days of age. (table 1). Detailed descriptions of the selection and management procedures followed in this line were given by Zhou et al. (1999).

Random regression model
Let *

Covariance structure
The covariance between two records for the same animal is then Generally, measurement errors are assumed to be identically independent distributed with variance 2 ε σ , so that

REML estimation
Considering all animals, equation (1) can be written in matrix form as with y the vector of N observations measured on N D animals, b the vector of fixed effects, α the vector of k A ×N A additive genetic random regression coefficients (N A ≥N D ) denoting the total number of animals in the analysis, including parents without records), γ the vector of k R ×N D permanent environmental random regression coefficients, ε the vector of N measurement errors, and X, Z * and * D Z denoting the corresponding 'design' matrices. The superscript ' * ' marks matrices incorporating orthogonal polynomial coefficients.
Analyses were carried out using program DXMRR (Meyer, 1998c), which employed the average information REML of Johnson and Thompson (1995) for parameter estimation. A separate measurement error variance component for each days of age (five variances) was fitted. Fixed effects fitted were generation (6 levels), sex (2 levels) and litter size (13 levels). Additional random effect fitted was litter with 220 levels. Additive genetic and permanent environmental CFs were fitted to the same order throughout ). Orders of fit considered ranged from 1 to 5.
A likelihood ratio test such that -2(logL 1 -logL 2 ) has a x 2 distribution with k 2 -k 1 degrees of freedom, where i k is the different order of fit and i L is the maximum value of the restricted likelihood function, was carried out in order to select the appropriate order of fit. Alternatively, Akaike's information criterion (Akaike, 1973) that is not dependent on the number of free parameters in the whole model was also used.

Likelihoods and information criterion
Log restricted likelihoods (logL), Akaike's information criterion (AIC) together with estimates of measurement errors ( 2 i ε σ , i =1, 5), are summarized in table 2. It can be seen that increasing k from 1 to 5 yields significant increases in logL and decreases in measurement error. Both likelihood ratio tests and Akaike's information criteria indicate that the appropriate order of fit is 5 and that a reduced order of fit cannot describe the data adequately.

Coefficients of CFs
Estimates of the coefficients of the genetic CF, A and the permanent environmental CF, R are listed in table 3. Thus, under the suitable order of fit, estimated CFs for SD-II Line, with i t denoting the i th standardized age, can be

Genetic and phenotypic (co)variances
The additive genetic and phenotypic (co)variances from the estimated CFs for different orders of polynomial fit are shown in figures 1 and 2. With the increase of age, both genetic and phenotypic covariance increases. For k=1 (not shown), all (co)variances are equal. Graphically, that is a plane parallel to the base. For k=2, (co)variances are linear functions of the ages, resulting in a tilted plane. Including quadratic ages for k=3 then gives a semi-parabolic surface. Considering cubic (k=4) or quartic (k=5) terms does not change its shape dramatically. For all orders of fit, the change trend of genetic and phenotypic covariance was very similar from k=3 onwards.

DISCUSSION
Often, a reduced order of fit involves fewer parameters and smoothes out differences in estimates of covariance (Kirkpatrick et al., 1990;Meyer and Hill, 1997), but from this study, a reduced order of fit did not describe the data adequately. This may be due to the selection of additional random effect. Using the same data set, Liu  studied influences of different additional random effects on estimating CFs. The results showed that a reduced order of fit (k=3) was feasible using animal rather than litter effect as additional random effect. All those results indicate that although litter effect is an important permanent environmental effect at early stage of growth, it cannot reflect all the permanent environmental effects in the entire growth period.
Because growth, especially early growth traits, were much influenced by maternal effects, it may be necessary to include the maternal effects in the model. Meyer (1999Meyer ( , 2000, Albuquerque and Meyer (2001) and Liu   genetic and permanent environmental effects. For data used in this paper, a further study is needed for estimation of CFs modeling maternal effects, besides direct genetic effect. RRM and CFs model observations (namely means) and (co)variances over time respectively. Their combination can effectively explain the genetic and environmental variations in continuous scale. For this reason, CFs have been used in beef cattle (eg. Meyer, 1999;, dairy cattle (eg. Van der Werf et al., 1998), Zebu cattle (eg. Albuquerque and Meyer, 2001), swine , and goats (Liu Wenzhong, 2001). Because of the availability of readymade software, extensive application of CFs to longitudinal data in other species is expected.