An Unbalanced Nested Model with Random Effects Having Unequal Variances

Consider the model Y ijk =μ + a i + b ij + e ijk (i=1, 2,…, t; j=1,2,…, B i ; k=1,2…,n ij ), where μ is a constant and a 1 ,b ij and e ijk are distributed independently and normally with zero means and variances σ 2 a d ij and σ 2 , respectively, where it is assumed that the d i 's and d ij 's are known. In this paper procedures for estimating the variance components (σ 2 , σ 2 a and σ 2 b ) and for testing the hypothesis σ 2 b = 0 and σ 2 a = 0 are presented. In the last section the mixed model y ijk , where x ijkkm are known constants and β m 's are unknown fixed effects ( m = 1, 2,…, p ), is transformed to a fixed effect model with equal variances so that least squares theory can be used to draw inferences about the β m 's.