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.

Dispersion Effects

Modeling dispersion effects is not very widely covered in textbooks, with the exception of the Taguchi framework. In a Taguchi-style experiment, this is handled by taking multiple measurements across settings of an outer array, constructing a new response which measures the variability off-target across this outer array, and then fitting the model to find out the factors that produce minimum variability. This kind of modeling requires a specialized design that is a complete cartesian product of two designs. The method of this chapter models variances in a more flexible, model-based approach. The particular performance statistic that Taguchi recommends for variability modeling is STD = -log(s). In JMP’s methodology, the log(s2) is modeled and combined with a model that has a mean. The two are basically equivalent, since log(s2)=2 log(s).

Model Specification

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.

Notes

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.