Efficient tests for mean structure in random effects models (Q1378310)

A multivariate one-way classification model with random effects is defined by \[ y_{ij}=\mu+ b_i+e_{ij},\quad i=1,\dots,n,\;j=1, \dots, k, \] where \(y_{ij}\) is a \(p\)-component vector of the \(j\) th repeated observation of the \(i\) th individual, \(\mu\) is a total mean, \(b_i\) is a random effect of the \(i\) th individual, \(e_{ij}\) is a noise. Assume that \(b_i\)'s and \(e_{ij}\)'s are mutually independent and have \(p\)-variate normal distributions with mean vector 0 and covariance matrices \(\Gamma\) and \(\Sigma\), respectively. The problem is to test the linear hypothesis \(H:C\mu=0\) against all alternatives, where \(C\) is a known design matrix. It is shown that a simplified test based on the sample mean \(\sum^k_{j=1} y_{ij}/k\) has uniformly higher power than Wald-type tests and likelihood ratio tests. The methods are applied to other related models including a random coefficient growth curve model.