The Equalities of BLUPs for Linear Combinations Under Two General Linear Mixed Models.

For the general linear mixed model ℱ: {y,Xβ + Zu + e,Σ}, where the fixed effectsuand random effectseare correlated and the covariance matrixΣmay be singular. We establish a new explicit expression of the best linear unbiased predictors (BLUPs) and give the necessary and sufficient conditions for the BLUP of the predictable parameter functionLβ + Muunder the ℱ to remain BLUP under its misspecified model {y,X0β + Z0u + e,Σ0}. We also derive the necessary and sufficient conditions for the BLUPs under the multiple partitioned linear mixed model {y,X1β1 + X2β2+ … +Xkβk + Z1u1 + Z2u2+ … +Zkuk + e,Σ} to be the sum of BLUPs under its small models {y,Xiβi + Ziui + e,Σi},i = 1, 2,…,k. [ABSTRACT FROM PUBLISHER]