Sequential Transformation for Multiple Traits for Estimation of (Co)variance Components with a Derivative-Free Algorithm for Restricted Maximum Likelihood

Transformation of multiple-trait records that undergo sequential selection can be used with derivative-free algorithms to maximize the restricted likelihood in estimation of covariance matrices as with derivative methods. Data transformation with appropriate parts of the Choleski decomposition of the current estimate of the residual covariance matrix results in mixed-model equations that are easily modified from round to round for calculation of the logarithm of the likelihood. The residual sum of squares is the same for transformed and untransformed analyses. Most importantly, the logarithm of the determinant of the untransformed coefficient matrix is an easily determined function of the Choleski decomposition of the residual covariance matrix and the determinant of the transformed coefficient matrix. Thus, the logarithm of the likelihood for any combination of covariance matrices can be determined from the transformed equations. Advantages of transformation are 1) the multiple-trait mixed-model equations are easy to set up, 2) the least squares part of the equations does not change from round to round, 3) right-hand sides change from round to round by constant multipliers, and 4) less memory is required. An example showed only a slight advantage of the transformation compared with no transformation in terms of solution time for each round (1 to 5%)

Estimates of Genetic Parameters for 320-Day Pelvic Measurements of Males and Females and Calving Ease of 2-Year-Old Females

Records from 12 breed groups collected from 1983 to 1991, included in the Germ Plasm Utilization project at the U.S. Meat Animal Research Center, were analyzed separately by breed group and combined to estimate heritabilities and genetic correlations for 320-d male and female pelvic width, height, and area, and for 320-d male pelvic and female 2-yr-old calving ease. Calving ease was analyzed as a trait of the dam using 1) actual and 2) binary scale calving ease scores with a covariate of calf birth weight. A bivariate animal model and derivative-free REML incorporating sparse matrix techniques were used. When breed groups were analyzed separately, heritability estimates of male and female 320-d pelvic traits varied by breed group and sex. Average genetic correlations between male and female 320-d pelvic width, pelvic height, and pelvic area were large and positive. When breed groups were combined (n = 26,071), heritability estimates for 320-d pelvic traits were moderate in size. Genetic correlations of .68, .48, and .61, between male and female 320-d pelvic width, height, and area, respectively, suggest male and female pelvic traits are largely under the same genetic control but are correlated traits rather than the same trait. Heritability estimates for actual calving ease in 2-yr-olds ranged from .00 to .49 in separate breed group analyses, and from .00 to .37 for binary measures. When breed groups were combined, heritability was .ll for actual calving ease and was .09 on the binary scale. Genetic correlations by breed groups between 320-d male pelvic traits and calving ease of 2-yr-old females were variable. When breed groups were combined, genetic correlation estimates between 320-d male pelvic traits and actual calving ease of 2-yr-old females (on a 6-point scale) were negative and moderate as were genetic correlations between male 320-d pelvic traits and binary calving ease of 2-yr-old females. A bull one phenotypic SD above the mean in pelvic area would be expected to increase his daughters\u27 average pelvic area by 1.30 cm2 and improve its calving ease score by .03 of a score compared with an average breed bull