Tests for random-effects covariance structures in the growth curve model with covariates
type: Item
instance of: scholarly article
title: Tests for random-effects covariance structures in the growth curve model with covariates
zbMATH Open document ID: 0759.62022
author: Takahisa Yokoyama
author: Yasunori Fujikoshi
published in: Hiroshima Mathematical Journal
publication date: 1993-01-16T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 62H15
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62H10
zbMATH DE Number: 93612
zbMATH Keywords: asymptotic chi-square
zbMATH Keywords: growth curve model
zbMATH Keywords: additional covariates
zbMATH Keywords: likelihood ratio test
zbMATH Keywords: covariance structure
zbMATH Keywords: canonical form
zbMATH Keywords: approximate likelihood ratio test
reviewed by: Nariaki Sugiura
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206128618
OpenAlex ID: W1543025915
DOI: 10.32917/HMJ/1206128618
description: scientific article; zbMATH DE number 93612