BackgroundThe sensitivity to microenvironmental changes varies among animals and may be under genetic control. It is essential to take this element into account when aiming at breeding robust farm animals. Here, linear mixed models with genetic effects in the residual variance part of the model can be used. Such models have previously been fitted using EM and MCMC algorithms.ResultsWe propose the use of double hierarchical generalized linear models (DHGLM), where the squared residuals are assumed to be gamma distributed and the residual variance is fitted using a generalized linear model. The algorithm iterates between two sets of mixed model equations, one on the level of observations and one on the level of variances. The method was validated using simulations and also by re-analyzing a data set on pig litter size that was previously analyzed using a Bayesian approach. The pig litter size data contained 10,060 records from 4,149 sows. The DHGLM was implemented using the ASReml software and the algorithm converged within three minutes on a Linux server. The estimates were similar to those previously obtained using Bayesian methodology, especially the variance components in the residual variance part of the model.ConclusionsWe have shown that variance components in the residual variance part of a linear mixed model can be estimated using a DHGLM approach. The method enables analyses of animal models with large numbers of observations. An important future development of the DHGLM methodology is to include the genetic correlation between the random effects in the mean and residual variance parts of the model as a parameter of the DHGLM.

The possibility of breeding for uniform individuals by selecting animals expressing a small response to environment has been studied extensively in animal breeding. Bayesian methods for fitting models with genetic components in the residual variance have been developed for this purpose, but have limitations due to the computational demands. We use the hierarchical (h)-likelihood from the theory of double hierarchical generalized linear models (DHGLM) to derive an estimation algorithm that is computationally feasible for large datasets. Random effects for both the mean and residual variance parts of the model are estimated together with their variance/covariance components. An important feature of the algorithm is that it can fit a correlation between the random effects for mean and variance. An h-likelihood estimator is implemented in the R software and an iterative reweighted least square (IRWLS) approximation of the h-likelihood is implemented using ASReml. The difference in variance component estimates between the two implementations is investigated, as well as the potential bias of the methods, using simulations. IRWLS gives the same results as h-likelihood in simple cases with no severe indication of bias. For more complex cases, only IRWLS could be used, and bias did appear. The IRWLS is applied on the pig litter size data previously analysed by Sorensen & Waagepetersen (2003) using Bayesian methodology. The estimates we obtained by using IRWLS are similar to theirs, with the estimated correlation between the random genetic effects being -0·52 for IRWLS and -0·62 in Sorensen & Waagepetersen (2003). © Cambridge University Press 2013. Estimation of breeding values for mean and dispersion, their variance and correlation using double hierarchical generalized linear models SummaryThe possibility of breeding for uniform individuals by selecting animals expressing a small response to environment has been studied extensively in animal breeding. Bayesian methods for fitting models with genetic components in the residual variance have been developed for this purpose, but have limitations due to the computational demands. We use the hierarchical (h)-likelihood from the theory of double hierarchical generalized linear models (DHGLM) to derive an estimation algorithm that is computationally feasible for large datasets. Random effects for both the mean and residual variance parts of the model are estimated together with their variance/covariance components. An important feature of the algorithm is that it can fit a correlation between the random effects for mean and variance. An h-likelihood estimator is implemented in the R software and an iterative reweighted least square (IRWLS) approximation of the h-likelihood is implemented using ASReml. The difference in variance component estimates between the two implementations is investigated, as well as the potential bias of the methods, using simulations. IRWLS gives the same results as h-likelihood in simple cases with no severe indication of bias. For more comple...

Trait uniformity, or micro-environmental sensitivity, may be studied through individual differences in residual variance. These differences appear to be heritable, and the need exists, therefore, to fit models to predict breeding values explaining differences in residual variance. The aim of this paper is to estimate breeding values for micro-environmental sensitivity (vEBV) in milk yield and somatic cell score, and their associated variance components, on a large dairy cattle data set having more than 1.6 million records. Estimation of variance components, ordinary breeding values, and vEBV was performed using standard variance component estimation software (ASReml), applying the methodology for double hierarchical generalized linear models. Estimation using ASReml took less than 7 d on a Linux server. The genetic standard deviations for residual variance were 0.21 and 0.22 for somatic cell score and milk yield, respectively, which indicate moderate genetic variance for residual variance and imply that a standard deviation change in vEBV for one of these traits would alter the residual variance by 20%. This study shows that estimation of variance components, estimated breeding values and vEBV, is feasible for large dairy cattle data sets using standard variance component estimation software. The possibility to select for uniformity in Holstein dairy cattle based on these estimates is discussed.

The purpose of this study was to: 1) establish the optimal body-mass exponent for maximal oxygen uptake (VO(2)max) to indicate performance in elite-standard men cross-country skiers; and 2) evaluate the influence of course inclination on the body-mass exponent. Twelve elite-standard men skiers completed an incremental treadmill roller-skiing test to determine VO(2)max and performance data came from the 2008 Swedish National Championship 15-km classic-technique race. Log-transformation of power-function models was used to predict skiing speeds. The optimal models were found to be: Race speed = 7.86 · VO(2)max · m(-0.48) and Section speed = 5.96 · [VO(2)max · m(-(0.38 + 0.03 · α)) · e(-0.003 · Δ) (where m is body mass, α is the section's inclination and Δ is the altitude difference of the previous section), that explained 68% and 84% of the variance in skiing speed, respectively. A body-mass exponent of 0.48 (95% confidence interval: 0.19 to 0.77) best described VO(2)max as an indicator of performance in elite-standard men skiers. The confidence interval did not support the use of either "1" (simple ratio-standard scaled) or "0" (absolute expression) as body-mass exponents for expressing VO(2)max as an indicator of performance. Moreover, results suggest that course inclination increases the body-mass exponent for VO(2)max.

The genetic improvement in pig litter size has been substantial. The number of teats on the sow must thus increase as well to meet the needs of the piglets, because each piglet needs access to its own teat. We applied a genetic heterogeneity model to teat counts in pigs, and estimated a medium heritability for teat counts (0.35), but found a low heritability for residual variance (0.06), indicating that selection for reduced residual variance might have a limited effect. A numerically positive correlation (0.8) was estimated between the breeding values for the mean and the residual variance. However, because of the low heritability of the residual variance, the residual variance will probably increase very slowly with the mean.

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The present study explored the possibility of selection for uniformity of days from calving to first service (DFS) in dairy cattle. A double hierarchical generalised linear model with an iterative reweighted least-squares algorithm was used to estimate covariance components for the mean and dispersion of DFS. Data included the records of 27 113 Iranian Holstein cows (parity, 1–6) in 15 herds from 1981 to 2007. The estimated additive genetic variance for the mean and dispersion were 32.25 and 0.0139; both of these values had low standard errors. The genetic standard deviation for dispersion of DFS was 0.117, indicating that decreasing the estimated breeding value of dispersion by one genetic standard deviation can increase the uniformity by 12%. A strong positive genetic correlation (0.689) was obtained between the mean and dispersion of DFS. This genetic correlation is favourable since one of the aims of breeding is to simultaneously decrease the mean and increase the uniformity of DFS. The Spearman rank correlations between estimated breeding values in the mean and dispersion for sires with a different number of daughter observations were 0.907. In the studied population, the genetic trend in the mean of DFS was significant and favourable (–0.063 days/year), but the genetic trend in the dispersion of DFS was not significantly different from zero. The results obtained in the present study indicated that the mean and uniformity of DFS can simultaneously be improved in dairy cows.