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After attempting to produce a linear mixed model I was left with a great deal of heterogeneity.

lme1 <- lme(Average.payoff ~ Game + Type + Others.Type + Game:Type + 
  Game:Others.Type + Type:Others.Type, random=~1|Subjects, method="REML", 
  data=Subjectsm1)

The response term Average.payoff is continuous, whilst all explanatory variables are binary.

When I look to validate, I can clearly see that the spread of the residuals decreases with the larger fitted values. So, I decided to look at using the weights argument. I came up with the following model which seems to be the best fit (AIC and loglik).

vf1 <- varComb(varIdent(form=~1|Others.Type), varIdent(form=~1|Game),     
  varIdent(form=~1|Type), varIdent(form=~1|Male))

lme1 <- lme(Average.payoff ~ Game + Type + Others.Type + Game:Type + 
  Game:Others.Type + Type:Others.Type, random=~1|Subjects, method="ML", 
  weights=vf1, data=Subjectsm1)
  1. Can I justify using the varIdent function alongside random effects?
  2. Can I justify combining varIdent functions so the variance of all explanatory variables are accounted for?
  3. If I remove a fixed effect explanatory variable for being insignificant should the same variable be removed from the variance structure?
  4. If I get a better fit (using AIC and loglik) without including the random effects by re-modelling with gls(), can I justify using the model without the random effects?

The experiment involved $20$ subjects, for each subject I have two samples, one where Others.Type=0 and one where Others.Type=1.

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This smells like a Kiefer & Wolfowitz (1956) inconsistency of the MLE problem with many variance parameters (and, really, many location parameters, which was considered before them, but they mention it, too). I don't think that your variance estimates are worth much with 40 observations; and AIC/BIC are not worth anything at all in your situation, as they explicitly assume large samples (40 could be reasonably large for a mean of i.i.d. data with symmetric distribution, but not large enough for anything else) and i.i.d. data (which you obviously don't have). So to me, you cannot justify much, except switching to the GLS. Modeling the variance usually brings more problems than it solves (first and foremost, specification errors lead to disasters in the variance components and the standard errors of your main model), unless your variability changes by a factor of a couple of order of magnitude, and cannot be reasonably ignored.

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  • $\begingroup$ Thanks, in this case do you think a transformation would be a better approach? $\endgroup$ – Jonathan Bone Aug 28 '12 at 10:52

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