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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.
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