Prediction Of Breeding Values For Mean And Environmental Variance With An Iterative BLUP- Procedure H.A. Mulder * , L. Rönnegård †,‡ , R. F. Veerkamp* Introduction Animals may not only differ genetically in phenotypic trait values, but also in their response to environmental factors, which would lead to genetic variation in environmental variance. Several studies have shown the existence of genetic variation in environmental variance (e.g. Sorensen and Waagepetersen (2003); review Mulder et al. (2007); Wolc et al., (2009)). Genetic variation in environmental variance can be used to breed for increased robustness and uniformity (Mulder et al. (2007, 2008)). Mulder et al. (2007) proposed a selection index framework to predict breeding values for mean and environmental variance in the presence of genetic variation in environmental variance. In principle this idea can be easily extended to a mixed model framework using the phenotype and squared trait values adjusted for fixed effects. However, in the presence of genetic trend it is theoretically better to use squared residuals instead of squared trait values adjusted for fixed effects, which means that a bivariate iterative BLUP-procedure is required to update the squared residuals. Updating the squared residuals is important when the breeding value for mean and the breeding value for environmental variance are correlated and to account for heterogeneous residual variance in phenotype. Such an iterative BLUP- procedure has not been tested before, but has the advantage that it can be implemented in software for routine genetic evaluation, such as MiXBLUP (Lidauer and Stranden (1999)). The aim of this paper was to test an iterative BLUP-procedure to predict breeding values for mean and environmental variance. Bivariate models for phenotype and squared residuals with and without accounting for heterogeneous residual variance in phenotype were compared using simulation. In addition, a univariate analysis was studied where the bivariate mixed model was split into two independent univariate models. Material and methods The model. The bivariate mixed model used for the iterative BLUP-procedure was:         +       +         =       2 e p v m e p p e e a Z a Z Xb Xb e p 2 ˆ ˆ 2 , where p ( 2 p e ) is a vector with phenotypes (squared residuals of phenotypes), X and Z are incidence matrices for fixed effects and additive * Animal Breeding and Genomics Centre, Wageningen UR Livestock Research, PO Box 65, 8200 AB Lelystad, The Netherlands † Statistics Unit, Dalarna University, SE-781 70 Borlange, Sweden ‡ Department of Animal Breeding and Genetics, Swedish University of Agricultural Sciences, SE-750 07 Uppsala, Sweden