# Evaluation of a GLS fit

My first post in this useful community. Ive been trying to use GLS methods in non-linear mixed effects models and have found this post, Prediction with GLS extremely informative and helpful. Some of my questions are probably more of a follow up to the above post which discusses the predictive capability of GLS. Below I will state my question and give a brief description of why I would like to know more.

• How does one evaluate a GLS fit for a real dataset as compared to a regular fit?

Using simulations I could show that GLS is robust to error model mis-specification, especially when the data is sparse or there is deviation from normality (heavy-tailed distribution or time-varying distribution of residuals). For a simulated model, there is decreased bias and improved precision of the model parameters in a gls fit compared to the regular nlme fit. However, I don't understand how would I evaluate my gls fit for a real dataset as compared to a regular fit. Standard errors/Goodness of Fit/etc..?

• At which stage of model building does one test a GLS fit?

Carrol and Ruppert state that when there is enough evidence of heteroschedasticity of your variance, then GLS estimators are very efficient. I obviously could see that in simulations studies, but how do I justify a better fit in a real dataset when the parameter estimates of the model from a gls fit are not very different from those obtained after a regular fit (even though the gls estimator is more efficient). This obviously comes back to the question, why (or actually why not) use GLS estimation at all.

• The estimation of residual variance parameters is independent of response prediction. T/F?

If the variance changes with your predictions (e.g. a constant CV model), is my understanding of the above statement correct that the estimation of the residual variance parameters is independent of model prediction of responses (or vice-versa). I believe this is the basic principle behind GLS.

• Can anyone suggest good references whether gls is design dependent ( both structure and density of observations)

• Any comments on gls performance/robustness for variable selection. Given bullet 3 above, can we expect gls to be more sensitive in picking up predictors for a response variable.