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Reducing the bias of estimates of genotype by environment interactions in random regression sire modelsAbstract: Random regression models are widely used to describe effects that change gradually over a continuous scale, for instance in genotype by environment interaction studies, where the genotype effect is modeled as a function of the environment [1]. A common measurement of the interaction is the variance in the slope of the sire reaction norms, i.e. sire breeding values regressed on an environmental variable. The interaction is regarded as significant if the slope variance is significant [e.g. [2,3,1]].For the estimation of genotype by environment interactions, both sire models or animal models are used, however sire models are computationally less demanding. Thus the sire model is preferred when the model is complex, the amount of data is large, or the analysis has to be repeated many times, as in QTL analyses in which testing many positions is necessary.Performing genetic analyses with a sire model gives an estimate of the "sire-variance", which is one fourth of the genetic variance. The remaining genetic variance (3/4) is modeled through the residual term together with the environmental variance. When the genetic variance is heterogeneous because of genotype by environment interactions, the residual variance will also be heterogeneous since part of it is genetic. Therefore, a random regression model that also accounts for heterogeneous residual variance is preferred [4,1].One way to account for heterogeneous residual variance over environments is to divide the environment into classes and to assume homogeneous variance within each environmental class, but with different residual variances across classes [1]. The drawbacks of this method are that classes have to be arbitrarily defined and that the number of classes increases with the number of parameters that need to be estimated [5]. A more advantageous approach would be to model the residual variance as a function of the environment in the mixed model, but commonly used software does not facilitate this option [6]. An
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