Modeling and Multivariate Methods - SAS

Chapter 12 Fitting Dispersion Effects with the Loglinear Variance Model 305
Overview of the Loglinear Variance Model
Overview of the Loglinear Variance Model
The loglinear-variance model (Harvey 1976, Cook and Weisberg 1983, Aitken 1987, Carroll and Ruppert
1988) provides a way to model the variance simply through a linear model. In addition to having regressor
terms to model the mean response, there are regressor terms in a linear model to model the log of the
variance:
mean model: E(y) = Xβ
variance model: log(Variance(y)) = Z λ,
or equivalently
Variance(y) = exp(Z λ)
where the columns of X are the regressors for the mean of the response, and the columns of Z are the
regressors for the variance of the response. The regular linear model parameters are represented by β, and λ
represents the parameters of the variance model.
Log-linear variance models are estimated using REML.
A dispersion or log-variance effect can model changes in the variance of the response. This is implemented in
the Fit Model platform by a fitting personality called the Loglinear Variance personality.
Model Specification
Notes
Log-linear variance effects are specified in the Fit Model dialog by highlighting them and selecting
LogVariance Effect from the Attributes drop-down menu. &LogVariance appears at the end of the effect.
When you use this attribute, it also changes the fitting Personality at the top to LogLinear Variance. If you
want an effect to be used for both the mean and variance of the response, then you must specify it twice,
once with the LogVariance option.
The effects you specify with the log-variance attribute become the effects that generate the Z’s in the model,
and the other effects become the X’s in the model.
Every time another parameter is estimated for the mean model, at least one more observation is needed, and
preferably more. But with variance parameters, several more observations for each variance parameter are
needed to obtain reasonable estimates. It takes more data to estimate variances than it does means.
The log-linear variance model is a very flexible way to fit dispersion effects, and the method deserves much
more attention than it has received so far in the literature.