Research On The Equivalence Of Predictor/Estimator Of Random Effect Model

Random effects model is an important statistical model,commonly used in statistical analysis and inference in the fields of biostatistics,public health,psychometrics,education and sociology,and is one of the more widely used models in modern statistics.This article mainly combines the relevant knowledge of linear algebra and matrix theory.With regard to linear random effect models,linear random effect model with equality constraints and multivariate random effect model and their corresponding over-parameterized model,the optimal parameter functions under the pairwise models the equivalence between best linear unbiased prediction(BUP)/ best linear unbiased estimation(BLUE)is as follows:First,for the linear random effect model and its corresponding over-parameterized model,that is,a new linear random effect model with some more variables than the original model,by solving the extreme value of the constrained matrix function,the analytical expressions and some properties of the two model parameter functions BLUP / BLUE are obtained.At the same time,in the sense of absolute equality and equal probability 1,the conditions for the equivalence of the two model parameter functions BLUP / BLUE are discussed.Furthermore,for the linear random effect model with equality constraints and its over-parameterized model,by solving the extreme value of the matrix function with constraints,the analytical expressions and some properties of the two model parameter functions BLUP / BLUE are obtained.Similar to the study of unconstrained problems,the relevant knowledge of matrix rank and block matrix theory is used to discuss and give the condition that the two model parameter functions BLUP / BLUE are absolutely equal.Finally,consider the linear random effect model with multiple dependent variables corresponding to multiple independent variables,that is,the multivariate random effect model.For the multivariate random effect model and its correspondingover-parameter model,the model is transformed into a dependent variable correspondence through the straightening operation of the matrix.The linear random effect model with multiple independent variables,and then by solving the extreme value of the constrained matrix function,gives the analytical expression and properties of the two model parameter function BLUP and the equivalent conditions of the two model parameter function BLUP.