The HPMIXED Procedure

Assumptions and Notation

The linear mixed models fit by the HPMIXED procedure can be represented as linear statistical models in the following form:

$\displaystyle \mb {y}$	$\displaystyle = \bX \bbeta + \bZ \bgamma + \bepsilon$
$\displaystyle \bgamma$	$\displaystyle \sim N(\mb {0},\bG )$
$\displaystyle \bepsilon$	$\displaystyle \sim N(\mb {0},\sigma ^2\bI )$
$\displaystyle \mr {Cov}[\bgamma ,\bepsilon ]$	$\displaystyle = \mb {0}$

The symbols in these expressions denote the following:

$\mb {y}$: the $(n \times 1)$ vector of responses
$\bX$: the $(n \times k)$ design matrix for the fixed effects
$\bbeta$: the $(k \times 1)$ vector of fixed-effects parameters
$\bZ$: the $(n \times q)$ design matrix for the random effects
$\bgamma$: the $(q \times 1)$ vector of random effects
$\bepsilon$: the $(n \times 1)$ vector of unobservable residual errors

As is customary for statistical models in the linear mixed model family, the random effects are assumed normally distributed. The same holds for the residual errors and these are furthermore distributed independently of the random effects. As a consequence, these assumptions imply that the response vector $\mb {y}$ has a multivariate normal distribution.

Further assumptions, implicit in the preceding expression, are as follows:

The conditional mean of the data—given the random effects—is linear in the fixed effects and the random effects.
The marginal mean of the data is linear in the fixed-effects parameters.