A matrix handling of predictions of new observations under a general random-effects model

Linear regression models that include random effects are commonly used to analyze longitudinal and correlated data. Assume that a general linear random-effects model y = Xβ + ε with β = Aα+γ is given, and new observations in the future follow the linear model yf = Xfβ +εf . This paper shows how to establish a group of matrix equations and analytical formulas for calculating the best linear unbiased predictor (BLUP) of the vector φ = Fα +Gγ +Hε +Hfεf of all unknown parameters in the two models under a general assumption on the covariance matrix among the random vectors γ , ε and εf via solving a constrained quadratic matrix-valued function optimization problem. Many consequences on the BLUPs of φ and their covariance matrices, as well as additive decomposition equalities of the BLUPs with respect to its components are established under various assumptions.