Hello,
Sorry to be a bit longwinded, but I've struggled quite a bit with the following over the last few days. I've many of the archive entries related to spatial autocorrelation and haven't found what I'm after. If it's okay, I'm going to first describe my general understanding of the process by which a mixed model can account for correlated errors. If possible, please briefly point out any misunderstanding I have to help my work overall (the literature I've found on this area does not go into extensive explanation).
I'm aware that mixed models are currently in use to fit fixed effects while controlling for correlation among residuals. I believe this is often done by specifying a theoretical variogram that one believes describes the spatial structure of the error correlation and which is then used to modify the variance-covariance error matrix that is used in model fitting (which I think in this case would be block diagonal with distance input into chosen variogram model determining matrix element value). So, as the fixed effects are adjusted algorithmically to maximize likelihood, simultaneously the parameters of the theoretical variogram (which enter as a random effect) are similarly adjusted which in turn influences the variance-covariance error matrix. The combined goal of these two parallel adjustments (I believe) would be to maximize overall model likelihood.
Although I plan to use Stat, here is an example of R code that uses a nonlinear mixed model in this way.
http://www.ats.ucla.edu/stat/r/faq/spatial_regression.htm
It seems that in the example given in this link, the incorporated correlation structure is not specifically on the error term but instead on the reponse itself. Therefore, it seems that the effect of the explanatory variable is diluted by this approach. For instance, if you had a 'true' model where temperature was only a function of elevation but elevation was strongly autocorrelated, the approach in the link would likely leave elevation as a nonsignificant part of the model. Versus, if the correlation structure was assigned to model error this would not happen. Is this true or am I speaking of 6 of one and half dozen of the other (that in practice it makes no difference to results)?
If the above example is not an example of modeling the correlation among model errors, is there a good example somewhere that does this that I can reference?
Thanks, Seth Myers
PS I plan to read all the excellent books suggested in other threads, but ask this now to help me digest this material more quickly.
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