Statistical inference for parallelism hypothesis in growth curve model

Abstract

Let $\mathit{y}=(y_1,\dots ,y_p{)}^{\prime }$ be a $p$-dimensional random vector measurable on the individuals drawn from each of $k$ $p$-dimensional normal populations ${\prod }_i:N_p({\mu }_i,\Sigma ),i=1,\dots ,k$. In this paper we consider the growth curve model which has a mean structure as follows: ${\mathit{\mu }}_i=X{\mathit{\theta }}_i,i=1,\dots ,k$, where $X$ is a $p\times q$ given matrix with rank $q$ and ${\mathit{\theta }}_i$’s are unknown parameter vectors. First we derive an $\text{LR}$ test for a parallelism hypothesis $H_{\text{1}}:X{\mathit{\theta }}_i-X{\mathit{\theta }}_k={\gamma }_i{\mathbf{1}}_p,i=1,\dots ,k-1$, where ${\gamma }_i$’s are unknown parameters, and ${\mathbf{1}}_p$ is the $p$-dimensional vector with all the elements 1. Next we obtain the MLE of $\mathit{\gamma }=({\gamma }_1,\dots ,{\gamma }_{k-1}{)}^{\prime }$ and its distribution, and propose a simultaneous confidence interval for linear combinations of $\mathit{\gamma }$.