Computing approximate standard errors for genetic parameters derived from random regression models fitted by average information REML

Troy M. F 1 2 Arthur R. G 0 1 Julius H.J. van der W 1 2 0 NSW Agriculture, Orange Agricultural Institute , Orange, NSW, 2800 , Australia 1 Australian Sheep Industry CRC 2 School of Rural Science and Agriculture, University of New England , Armidale, NSW, 2351 , Australia - Approximate standard errors (ASE) of variance components for random regression coefficients are calculated from the average information matrix obtained in a residual maximum likelihood procedure. Linear combinations of those coefficients define variance components for the additive genetic variance at given points of the trajectory. Therefore, ASE of these components and heritabilities derived from them can be calculated. In our example, the ASE were larger near the ends of the trajectory. 1. INTRODUCTION maximum likelihood (REML) methods. In contrast, Meyer [9] published confidence intervals of genetic parameter estimates derived from Bayesian analyses using Gibbs sampling. With REML estimation by the average information algorithm, approximate variances of variance components are obtained from the inverse of the information matrix. Variance components as well as heritabilities at given trajectory points can be calculated from variances of random regression coefficients and therefore approximate standard errors (ASE) of these derived parameters can also be obtained. The aims of this note are to describe how to calculate ASE for genetic parameter estimates derived from RR models and to apply the method to a field data set. 2. MATERIALS AND METHODS 2.1. Random regression model Consider a variance-covariance (VCV) matrix G0 of rank t for repeated measurements of weight at t given trajectory points (e.g. ages). Under the covariance function (CF) approach defined by Kirkpatrick et al. [5], G0 is modelled with a reduced number of parameters. The genetic CF of order k, where k t, can be estimated from G0 such that: where G is an approximation of G0. Meyer [8] showed that K can be estimated directly from data using RR. The matrix K of order k contains the variance components for the RR coefficients in the model. The matrix of order t k contains orthogonal polynomial coefficients evaluated at t standardised trajectory points (ages) with elements ij = j(xi), being the jth polynomial coefficient for the ith point xi [6]. The covariance structure for the environmental effects is fitted as an unstructured t t covariance matrix. This yields the model: where yi is the vector of ti observations measured on animal i, b is a vector of fixed effects, i a vector of additive genetic RR coefficients and ei a vector of residual errors pertaining to yi. Xi and Zi are design matrices relating b and i to yi, where Zi contains the elements pertaining to ages in the data. Extending the model to n individuals, the corresponding variances are defined as var() = K A, where K contains the additive genetic variances and covariances for the RR coefficients, A is the numerator relationship matrix among individuals and the symbol denotes direct product. The solution for K can be used as in equation (1) to calculate the variances and covariances among defined trajectory points. 2.2. Calculation of standard error of parameters derived from RR coefficients Consider a genetic variance covariance matrix, G , derived from equation (1), G = K , where has dimension t k, K has dimension k k and G is t t. We can write the elements of G in vector form, such that the variances and covariances of these parameters can be summarized in a matrix. Hence, equation (1) can also be written as where has dimension (t t) (k k), vec( K) is the vector form of K of dimensions (k k) 1 achieved by stacking the columns of K below one another, and similarly vec(G ) is the vector form of G of dimensions (t t) 1. It can be checked for a small example that equations (1) and (3) are equivalent, written in matrix and vector form respectively. The variance of estimates in G can be calculated in a similar manner whereby where var(vec(K )) has dimensions (k k) (k k) and var(vec(G )) is a (t t) by (t t) matrix. Var(vec(K )) can be approximated from the appropriate elements of the inverse of the average information matrix in a REML procedure (e.g. as given in the *.vvp file in ASReml) [3]. The same principles apply to other random effects in the RR model, and the covariance between variances of different random effects. Hence, this methodology can be extended to the matrices estimated for other random regression effects and the covariance between random effects. Subsequently these matrices are summed as in equation (5) to build a matrix containing estimates of variance of phenotypic (co)variance components, var(vec(P )), which also has dimension (t t) (t t). = var vec G + 2 cov vec G , vec E For functions of variance components (such as heritabilities) a Taylor series expansion can be used to approximate the variance of a variance ratio as detailed by Lynch and Walsh [7]. For the ratio of genetic to phenotypic variance, var gi,i/pi,i = var hi2 pi2,ivar gi,i + gi2,ivar pi,i 2gi,ipi,icov gi,i, pi,i /pi4,i where gi,i and pi,i are elements of G and P, var(gi,i), var(pi,i) and cov(gi,i, pi,i) represent variance and covariance of genetic and phenotypic variance at time i. The ASE for the heritability estimate at time i (for univariate and RR estimates) is obtained by taking the square root of equation (6). 2.3. Example of application of method to RR coefficients estimated from field data s2000 d r co1500 e R fo1000 r e b 500 m u N A VCV matrix for additive genetic and phenotypic effects for weight over a 450-day trajectory was derived based on the analysis performed by Fischer et al. [2]. Data for this analysis originated from the LAMBPLAN database and consisted of 16 826 weight records on 5 420 Poll Dorset sheep. The number of records at different ages is represented in Figure 1. Fischer et al. [2] used RR to estimate CF coefficients for direct and maternal genetic and environmental effects. The model also included heterogeneous residual variance across ages of measurement. ASReml [3] was used for this analysis. Based on a third order CF for additive genetic effects, a VCV matrix (G ) was constructed for weights at 10 equidistant ages (i.e. defining ). Similarly, VCV matrices were derived for the other random effects. Furthermore, adding the resultant variance matrices together resulted in a phenotypic VCV matrix (P ) with (co)variance components for weights at the 10 equidistant ages. We then obtained the variance of vec(G ) as in equation (4) and similarly for the two types of maternal effects, which in this case were all matrices of dimensions 100 100. Following equations (4), (5) and (6) we obtained the ASE of the heritability estimate for each age. Results from this example are shown in Figure 2. In addition, a series of piecewise estimates at specific ages were obtained using the equivalent univariate model (direct and matern (...truncated)