The HPLMIXED Procedure

Matrix Notation

Suppose that you observe n data points $y_1, \ldots , y_ n$ and that you want to explain them by using n values for each of p explanatory variables $x_{11}, \ldots , x_{1p}$, $x_{21}, \ldots , x_{2p}$, $\ldots , x_{n1}, \ldots , x_{np}$. The $x_{ij}$ values can be either regression-type continuous variables or dummy variables that indicate class membership. The standard linear model for this setup is

\[  y_ i = \sum _{j=1}^ p x_{ij} \beta _ j + \epsilon _ i \quad i=1,\ldots ,n  \]

where $\beta _1, \ldots , \beta _ p$ are unknown fixed-effects parameters to be estimated and $\epsilon _1, \ldots , \epsilon _ n$ are unknown independent and identically distributed normal (Gaussian) random variables with mean 0 and variance $\sigma ^2$.

The preceding equations can be written simultaneously by using vectors and a matrix, as follows:

\[  \left[\begin{array}{c} y_1 \\ y_2 \\ \vdots \\ y_ n \end{array} \right] = \left[\begin{array}{cccc} x_{11} &  x_{12} &  \ldots &  x_{1p} \\ x_{21} &  x_{22} &  \ldots &  x_{2p} \\ \vdots &  \vdots & &  \vdots \\ x_{n1} &  x_{n2} &  \ldots &  x_{np} \end{array} \right] \left[\begin{array}{c} \beta _1 \\ \beta _2 \\ \vdots \\ \beta _ p \end{array} \right] + \left[\begin{array}{c} \epsilon _1 \\ \epsilon _2 \\ \vdots \\ \epsilon _ n \end{array} \right]  \]

For convenience, simplicity, and extendability, this entire system is written as

\[  \mb{y} = \mb{X}\bbeta + \bepsilon  \]

where $\mb{y}$ denotes the vector of observed $y_ i$’s, $\mb{X}$ is the known matrix of $x_{ij}$’s, $\bbeta $ is the unknown fixed-effects parameter vector, and $\bepsilon $ is the unobserved vector of independent and identically distributed Gaussian random errors.

In addition to denoting data, random variables, and explanatory variables in the preceding fashion, the subsequent development makes use of basic matrix operators such as transpose ($’$), inverse ($^{-1}$), generalized inverse ($^{-}$), determinant ($|\cdot |$), and matrix multiplication. See Searle (1982) for details about these and other matrix techniques.