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.

This study investigated the possibility of increasing the reliability of direct genomic values (DGV) by combining reference populations. The data were from 3,735 bulls from Danish, Swedish, and Finnish Red dairy cattle populations. Single nucleotide polymorphism markers were fitted as random variables in a Bayesian model, using published estimated breeding values as response variables. In total, 17 index traits were analyzed. Reliabilities were estimated using a 5-fold cross validation, and calculated as the within-year squared correlation between estimated breeding values and DGV. Marker effects were estimated using reference populations from individual countries, as well as using a combined reference population from all 3 countries. Single-country reference populations gave mean reliabilities across 17 traits of 0.19 to 0.23, whereas the combined reference gave mean reliabilities of 0.26 for all populations. Using marker effects from 1 population to predict the other 2 gave a loss in mean reliability of 0.14 to 0.21 when predicting Swedish or Finnish animals with Danish marker effects, or vice versa. Using Swedish or Finnish marker effects to predict each other only showed a loss in mean reliability of 0.03 to 0.05. A combined Swedish-Finnish reference population led to an average reliability as high as that from the 3-country reference population, but somewhat different for individual traits. The results from this study show that it is possible to increase the reliability of DGV by combining reference populations from related populations.

BackgroundGenetic variation for environmental sensitivity indicates that animals are genetically different in their response to environmental factors. Environmental factors are either identifiable (e.g. temperature) and called macro-environmental or unknown and called micro-environmental. The objectives of this study were to develop a statistical method to estimate genetic parameters for macro- and micro-environmental sensitivities simultaneously, to investigate bias and precision of resulting estimates of genetic parameters and to develop and evaluate use of Akaike’s information criterion using h-likelihood to select the best fitting model.MethodsWe assumed that genetic variation in macro- and micro-environmental sensitivities is expressed as genetic variance in the slope of a linear reaction norm and environmental variance, respectively. A reaction norm model to estimate genetic variance for macro-environmental sensitivity was combined with a structural model for residual variance to estimate genetic variance for micro-environmental sensitivity using a double hierarchical generalized linear model in ASReml. Akaike’s information criterion was constructed as model selection criterion using approximated h-likelihood. Populations of sires with large half-sib offspring groups were simulated to investigate bias and precision of estimated genetic parameters.ResultsDesigns with 100 sires, each with at least 100 offspring, are required to have standard deviations of estimated variances lower than 50% of the true value. When the number of offspring increased, standard deviations of estimates across replicates decreased substantially, especially for genetic variances of macro- and micro-environmental sensitivities. Standard deviations of estimated genetic correlations across replicates were quite large (between 0.1 and 0.4), especially when sires had few offspring. Practically, no bias was observed for estimates of any of the parameters. Using Akaike’s information criterion the true genetic model was selected as the best statistical model in at least 90% of 100 replicates when the number of offspring per sire was 100. Application of the model to lactation milk yield in dairy cattle showed that genetic variance for micro- and macro-environmental sensitivities existed.ConclusionThe algorithm and model selection criterion presented here can contribute to better understand genetic control of macro- and micro-environmental sensitivities. Designs or datasets should have at least 100 sires each with 100 offspring.

As the molecular marker density grows, there is a strong need in both genome-wide association studies and genomic selection to fit models with a large number of parameters. Here we present a computationally efficient generalized ridge regression (RR) algorithm for situations in which the number of parameters largely exceeds the number of observations. The computationally demanding parts of the method depend mainly on the number of observations and not the number of parameters. The algorithm was implemented in the R package bigRR based on the previously developed package hglm. Using such an approach, a heteroscedastic effects model (HEM) was also developed, implemented, and tested. The efficiency for different data sizes were evaluated via simulation. The method was tested for a bacteria-hypersensitive trait in a publicly available Arabidopsis data set including 84 inbred lines and 216,130 SNPs. The computation of all the SNP effects required ,10 sec using a single 2.7-GHz core. The advantage in run time makes permutation test feasible for such a whole-genome model, so that a genome-wide significance threshold can be obtained. HEM was found to be more robust than ordinary RR (a.k.a. SNP-best linear unbiased prediction) in terms of QTL mapping, because SNPspecific shrinkage was applied instead of a common shrinkage. The proposed algorithm was also assessed for genomic evaluation and was shown to give better predictions than ordinary RR. HIGH-dimensional problems are increasing in importance in genetics, computational biology, and other fields of research where technological developments have greatly facilitated the collection of data (Hastie et al. 2009). In genome-wide association studies (GWAS) and genomic selection (GS), the number of observations n is generally in the order of hundreds/thousands whereas the number of marker effects to be fitted p is in the order of hundreds of thousands. This is a rather extreme p ) n problem, and the methods developed for analyses of the data need to be computationally feasible. At the same time the models fitted should be flexible enough to capture the important genetic effects that are often quite small .Methodologies regarding high-dimensional genomic data focus on both detection and prediction purposes. There is currently a trend that GWAS and GS could potentially apply the same framework of models. Such models fit the whole genome based on penalized likelihood or Bayesian shrinkage estimation (see the review by de los Campos et al. 2013). Ordinary GWAS usually avoids high-dimensional models and turns the problem into multiple testing instead (e.g., the review by Kingsmore et al. 2008). The tests of all the SNPs (single nucleotide polymorphisms) are dismembered. Such routine sacrifices both detective and predictive power. Using detected QTL (quantitative trait loci that are genome-wide significant), the prediction can be rather poor, which led to the insignificant application of marker-assisted selection (MAS) (Dekkers 2004). GS, however, has been practically use...

Residual feed intake (RFI) is a candidate trait for feed efficiency in dairy cattle. We investigated the influence of lactation stage on the effect of energy sinks in defining RFI and the genetic parameters for RFI across lactation stages for primiparous dairy cattle. Our analysis included 747 primiparous Holstein cows, each with recordings on dry matter intake (DMI), milk yield, milk composition, and body weight (BW) over 44 lactation weeks. For each individual cow, energy-corrected milk (ECM), metabolic BW (MBW), and change in BW (ΔBW) were calculated in each week of lactation and were taken as energy sinks when defining RFI. Two RFI models were considered in the analyses; RFI model [1] was a 1-step RFI model with constant partial regression coefficients of DMI on energy sinks (ECM, MBW, and ΔBW) over lactation. In RFI model [2], data from 44 lactation weeks were divided into 11 consecutive lactation periods of 4 wk in length. The RFI model [2] was identical to model [1] except that period-specific partial regressions of DMI on ECM, MBW, and ΔBW in each lactation period were allowed across lactation. We estimated genetic parameters for RFI across lactation by both models using a random regression method. Using RFI model [2], we estimated the period-specific effects of ECM, MBW, and ΔBW on DMI in all lactation periods. Based on results from RFI model [2], the partial regression coefficients of DMI on ECM, MBW, and ΔBW differed across lactation in RFI. Constant partial regression coefficients of DMI on energy sinks over lactation was not always sufficient to account for the effects across lactation and tended to give roughly average information from all period-specific effects. Heritability for RFI over 44 lactation weeks ranged from 0.10 to 0.29 in model [1] and from 0.10 to 0.23 in model [2]. Genetic variance and heritability estimates for RFI from model [2] tended to be slightly lower and more stable across lactation than those from model [1]. In both models, RFI was genetically different over lactation, especially between early and later lactation stages. Genetic correlation estimates for RFI between early and later lactation tended to be higher when using model [2] compared with model [1]. In conclusion, partial regression coefficients of DMI on energy sinks differed across lactation when modeling RFI. Neglect of lactation stage when defining RFI could affect the assessment of RFI and the estimation of genetic parameters for RFI across lactation.