The HPLMIXED Procedure
Suppose that you observe 
data points 
and that you want to explain them by using values for each of 
explanatory variables 
, 
, 
. The 
values can be either regression-type continuous variables or dummy variables that indicate class membership. The standard
linear model for this setup is
![\[ y_ i = \sum _{j=1}^ p x_{ij} \beta _ j + \epsilon _ i \quad i=1,\ldots ,n \]](images/stathpug_hplmixed0160.png){kind=small}
where 
are unknown fixed-effects parameters to be estimated and 
are unknown independent and identically distributed normal (Gaussian) random variables with mean 0 and variance 
.
The preceding equations can be written simultaneously by using vectors and a matrix, as follows:
![\[ \left[\begin{array}{c} y_1 \\ y_2 \\ \vdots \\ y_ n \end{array} \right] = \left[\begin{array}{cccc} x_{11} & x_{12} & \ldots & x_{1p} \\ x_{21} & x_{22} & \ldots & x_{2p} \\ \vdots & \vdots & & \vdots \\ x_{n1} & x_{n2} & \ldots & x_{np} \end{array} \right] \left[\begin{array}{c} \beta _1 \\ \beta _2 \\ \vdots \\ \beta _ p \end{array} \right] + \left[\begin{array}{c} \epsilon _1 \\ \epsilon _2 \\ \vdots \\ \epsilon _ n \end{array} \right] \]](images/stathpug_hplmixed0163.png)
For convenience, simplicity, and extendability, this entire system is written as
![\[ \mb {y} = \mb {X}\bbeta + \bepsilon \]](images/stathpug_hplmixed0001.png){kind=small}
where 
denotes the vector of observed 
’s, 
is the known matrix of ’s, 
is the unknown fixed-effects parameter vector, and 
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 (
), generalized inverse (
), determinant (
), and matrix multiplication. See Searle (1982) for details about these and other matrix techniques.