Should I keep or remove random effects?

Question

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.

I decided alternatively to see what would happen if I just fit using gls().

gls1 <- gls(Average.payoff ~ Game + Type + Others.Type + Game:Type + 
  Game:Others.Type + Type:Others.Type, method="REML", data=Subjectsm1)
anova(lme1, gls1)
#
#         Model df      AIC      BIC    logLik   Test   L.Ratio p-value
#    lme1     1  9 67.81805 81.28662 -24.90902                         
#    gls1     2  8 66.14661 78.11867 -25.07330 1 vs 2 0.3285588  0.5665

As you can see the gls() model gives a better fit than the lme() model. Even though I know from the experimental design that I have random effects, am I justified to remove them from the model if the fit is better without them?

Answer 1

An initial issue: when estimating the final parameters of a ME model, the restricted maximum likelihood (ReML) is used; as you have already done. ReML tries practically to find linear combinations of the responses, $k$, such that $k^TX= 0$ and thus to exclude any fixed terms parameters from the likelihood function ($X$ : $n\times p$ model matrix). However, ML should be used for the model selection procedure as the theory for model comparisons is based on ML estimation. As ReML will try to transform the fixed effect responses in the manner described above, this would lead to a series of different transformations for each model setting, making them incomparable. Therefore one should use ML estimators if likelihood ratio tests are to be employed (AIC etc.).

(Chapt.4 in Applied Longitudinal Analysis by Fitzmaurice et al. (towards the end of section 4.5 - I have the book but I just checked it and that section seems available in Google books also.))