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I am building a GLS model following protocol in "Zuur, 2009. Mixed effects models..." on p.90.

I have 5 continuous predictors. VarConstPower variance structure works best for me. At first the fixed part of the model includes all covariates. At that point I get the lowest AIC when variance part also includes all 5 covariates. But with model selection my fixed part shrinks to only 2 covariates. Can I still retain all 5 covariates in my variance part?

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  • $\begingroup$ Removing covariates in this way biases $\sigma^2$ towards zero, which will invalidate $P$-values and confidence limits. $\endgroup$
    – Frank Harrell
    Commented Feb 7, 2014 at 13:24

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You don't have to treat the variance structures like you would treat random effects in a non-gls model. With random effects you usually assume that they have zero mean, hence you include the corresponding fixed effect covariate to ensure this zero mean.

Working with variances, all you want to achieve is predicting the heteroscedasticity as well a possible, so there's no problem in including a variable in the variance prediction that is not part of your fixed effects structure. However, you should keep an eye on other information criteria like the Bayesian/Schwarz information criterion. They punish the addition of parameters more than the AIC does and hence prevent you from overfitting.

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