![]() Chapter Contents |
![]() Previous |
![]() Next |
The MIXED Procedure |
A statistical model is a mathematical description of how data are generated. The standard linear model, as used by the GLM procedure, is one of the most common statistical models:
In this expression, y represents a vector of observed data,
is an unknown vector of fixed-effects parameters with known
design matrix X, and
is an unknown random error
vector modeling the statistical noise around
. The focus
of the standard linear model is to model the mean of y by using
the fixed-effects parameters
. The residual errors
are assumed to be independent and identically
distributed Gaussian random variables with mean 0 and
variance
.
The mixed model generalizes the standard linear model as follows:
Here, is an unknown vector of random-effects parameters
with known design matrix Z, and
is an unknown
random error vector whose elements are no longer required to be
independent and homogeneous.
To further develop this notion of variance modeling, assume that
and
are Gaussian random variables
that are uncorrelated and have expectations 0 and variances G and R, respectively. The variance of
y is thus
Note that, when and Z = 0, the
mixed model reduces to the standard linear model.
You can model the variance of the data, y, by specifying the
structure (or form) of Z, G, and R. The model matrix
Z is set up in the same fashion as X, the model matrix for
the fixed-effects parameters. For G and R, you must
select some covariance structure. Possible
covariance structures include
By appropriately defining the model matrices X and Z, as well as the covariance structure matrices G and R, you can perform numerous mixed model analyses.
![]() Chapter Contents |
![]() Previous |
![]() Next |
![]() Top |
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.