Tests for random-effects covariance structures in the growth curve model with covariates
Let an observed random matrix on \(N\) individuals \(X(N\times p)\) have a growth curve model structure plus \(r\) additional covariates given by \(X=A\Xi B+Z\Theta+U\), where \(A(N\times k)\) and \(B(q\times p)\) are between and within individuals' design matrices, respectively, with unknown parameters \(\Xi(k\times q)\); \(Z(N\times r)\) is a given matrix of \(r\) covariates with unknown parameters \(\Theta(r\times p)\); \(U(N\times p)\) is the matrix of random errors, rows of which have independently and identically normal distribution with mean vector zero and covariance matrix \(\Sigma\). Assume that \(\text{rank}(A)=k\), \(\text{rank}(B)=q\) and \(\text{rank}(A,Z)=k+r\leq N-p\). The likelihood ratio test is considered for testing the covariance structure \[ H_{0c}: \Sigma=B_ c'\Delta_ c B_ c+\sigma^ 2_ c I_ p,\quad 0\leq c\leq q, \] where \(B_ c\) is the partitioned matrix of \(B\), composed of the first \(c\) rows and \(\Delta_ c\) is an arbitrary positive semi-definite matrix, against all alternatives. This covariance structure was derived by a random effects growth curve model by \textit{N. Lange} and \textit{N. M. Laird} [J. Am. Stat. Assoc. 84, No. 405, 241-247 (1989; Zbl 0678.62070)]. Reducing the problem to the canonical form, the authors derived an asymptotic distribution of the likelihood ratio statistic for large \(N\) for testing \(H_{00}\) \((\Sigma=\sigma^ 2 I_ p)\) and suggested an approximate likelihood ratio test for general hypotheses \(H_{0c}\) with an asymptotic distribution, due to the difficulty of maximizing the exact likelihood in the general case. The asymptotic distributions obtained are given by chi-squares.
scientific article; zbMATH DE number 93612