This separation is clearly done by the Animal Model. As this factor is random, we have distributional assumptions, namely that ~ where the is the numerator relationship matrix that describes the additive genetic relationship between individuals, and is the additive variance.Īs the error or residual term,, is also a random effect, we have their distributional assumptions of ~, with an identity matrix and the residual variance (or mean square error).Īs you can see, the central problem in predicting breeding values from observed phenotypic data is separating the genetic and environmental effects. However, what is more interesting to us is, corresponding to the breeding values. The above model is the basis for most genetic analyses with what is known as the Animal Model, that once fitted provides us with an estimation of the breeding values for each of the ‘animals’ (individuals) considered in the model.Ĭonsidering the Animal Model on its matrix notation of above, we have corresponding to a series of nuisance fixed effects, for example contemporary group, age or replicate. Where, ,, and are vectors of observations, fixed effects, random effects, and random residuals, respectively and and are design matrices connecting the observations to the effects. For this, we will consider the model written in matrix notation: Let’s have a look at how solutions are obtained from a linear mixed model. He developed the mixed model equations (MME, see equation ) for LMMs in order to calculate BLUPs of breeding values (or any random effect) and BLUEs of fixed effects.
The use of BLUPs to predict random effects was first described by C. Solutions are formed from a linear combination of the observationsĮxpectations of these solutions are equal to their true values But what does Best Linear Unbiased mean? BestĪmong all possible unbiased linear estimators these solutions have minimum variance But what are these?īest Linear Unbiased Estimates (BLUEs) are the solutions (or estimates) associated with the fixed effects and Best Linear Unbiased Predictions (BLUPs) are the solutions (but identified as predictions) associated with the random effects of a model. This is where we need to differentiate between BLUEs and BLUPs. Once these variance components are estimated, we then go onto the second step, where these variances are used to obtain estimates of the fixed effects and predictions of random effects. In ASReml and Genstat this is done using restricted (or residual) maximum likelihood based on a complex algorithm that uses the average information (AI) algorithm and sparse matrix operations. The first one is the estimation of the variance components. The fitting of a linear mixed model (LMM) can be divided into two steps.
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