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Two mixed model equations ( MME ) for best linear unbiased prediction ( BLUP ) of breeding values and for restricted  BLUP  of breeding values were derived by maximum likelihood from the joint normal probability distribution of the observations and breeding values. As a result,  MME  is actually more general than maximum likelihood because we can prove that each set of solutions of  MME  are identical to  BLUP  and restricted  BLUP  of breeding values and then it does not depend on normality. In the present study, the author shows deriving directly each  MME  from  BLUP  and restricted  BLUP  equations for breeding values without assuming the joint normal distribution of the data and random effects. However, if we cannot assume the multivariate normal density distribution of the estimated aggregate breeding value and each breeding value for selected traits, the response to selection by restricted  BLUP  may deviate from the expected values.

animal science journal

An alternative derivation method of mixed model equations from best linear unbiased prediction (BLUP) and restricted BLUP of breeding values not using maximum likelihood