Mixture model equations for marker-assisted genetic evaluation

Marker‐assisted genetic evaluation needs to infer genotypes at quantitative trait loci (QTL) based on the information of linked markers. As the inference usually provides the probability distribution of QTL genotypes rather than a specific genotype, marker‐assisted genetic evaluation is characterized by the mixture model because of the uncertainty of QTL genotypes. It is, therefore, necessary to develop a statistical procedure useful for mixture model analyses. In this study, a set of mixture model equations was derived based on the normal mixture model and the EM algorithm for evaluating linear models with uncertain independent variables. The derived equations can be seen as an extension of Henderson's mixed model equations to mixture models and provide a general framework to deal with the issues of uncertain incidence matrices in linear models. The mixture model equations were applied to marker‐assisted genetic evaluation with different parameterizations of QTL effects. A sire‐QTL‐effect model and a founder‐QTL‐effect model were used to illustrate the application of the mixture model equations. The potential advantages of the mixture model equations for marker‐assisted genetic evaluation were discussed. The mixed‐effect mixture model equations are flexible in modelling QTL effects and show desirable properties in estimating QTL effects, compared with Henderson's mixed model equations.