The HPLMIXED Procedure

Matrix Notation

Suppose that you observe n data points y 1 comma ellipsis comma y Subscript n Baseline and that you want to explain them by using n values for each of p explanatory variables x 11 comma ellipsis comma x Subscript 1 p Baseline, x 21 comma ellipsis comma x Subscript 2 p Baseline, ellipsis comma x Subscript n Baseline 1 Baseline comma ellipsis comma x Subscript n p Baseline. The x Subscript i j values can be either regression-type continuous variables or dummy variables that indicate class membership. The standard linear model for this setup is

y Subscript i Baseline equals sigma-summation Underscript j equals 1 Overscript p Endscripts x Subscript i j Baseline beta Subscript j Baseline plus epsilon Subscript i Baseline i equals 1 comma ellipsis comma n

where beta 1 comma ellipsis comma beta Subscript p Baseline are unknown fixed-effects parameters to be estimated and epsilon 1 comma ellipsis comma epsilon Subscript n Baseline are unknown independent and identically distributed normal (Gaussian) random variables with mean 0 and variance sigma squared.

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

Start 4 By 1 Matrix 1st Row y 1 2nd Row y 2 3rd Row vertical-ellipsis 4th Row y Subscript n Baseline EndMatrix equals Start 4 By 4 Matrix 1st Row 1st Column x 11 2nd Column x 12 3rd Column ellipsis 4th Column x Subscript 1 p Baseline 2nd Row 1st Column x 21 2nd Column x 22 3rd Column ellipsis 4th Column x Subscript 2 p Baseline 3rd Row 1st Column vertical-ellipsis 2nd Column vertical-ellipsis 3rd Column Blank 4th Column vertical-ellipsis 4th Row 1st Column x Subscript n Baseline 1 Baseline 2nd Column x Subscript n Baseline 2 Baseline 3rd Column ellipsis 4th Column x Subscript n p Baseline EndMatrix Start 4 By 1 Matrix 1st Row beta 1 2nd Row beta 2 3rd Row vertical-ellipsis 4th Row beta Subscript p Baseline EndMatrix plus Start 4 By 1 Matrix 1st Row epsilon 1 2nd Row epsilon 2 3rd Row vertical-ellipsis 4th Row epsilon Subscript n EndMatrix

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

bold y equals bold upper X bold-italic beta plus bold-italic epsilon

where bold y denotes the vector of observed y Subscript i’s, bold upper X is the known matrix of x Subscript i j’s, bold-italic beta is the unknown fixed-effects parameter vector, and bold-italic epsilon 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 (prime), inverse (Superscript negative 1), generalized inverse (Superscript minus), determinant (StartAbsoluteValue dot EndAbsoluteValue), and matrix multiplication. See Searle (1982) for details about these and other matrix techniques.

Last updated: December 09, 2022