[R-sig-ME] using lme for cov structures?

[Ben Pelzer]

Dear list,

With longitudinal data, the package nlme offers the possibility to 
specify particular covariance structures for the residuals. In the 
examples below I used data from 35 individuals measured at 4 time 
points, so we have 4 observations nested in each of 35 individuals. 
These data are discussed in Singer and Willett, Applied Longitudinal 
Data Analysis, chapter 7.

In several sources I found that e.g. compound symmetry can easily be 
obtained with gls from package nlme, by using the correlation structure 
corCompSymm, as in:

     comsym <- gls(opp~1,opposites,
                   correlation=corCompSymm(form = ~ 1 |id), method="REML")

where id is subject-identifier for the individual. With gls however one 
cannot specify random effects, as opposed to lme. To have lme estimate 
compound symmetry is simple, one would not even need have to specify a 
particular correlation structure, it would suffice to say 
"random=~1|id". However, for more complex covariance structures, e.g. 
heterogeneous compound symmetry, I was only able to find syntax for gls, 
but not for lme.

Then I thought that tricking lme might be an option by having a kind of 
"fake" random effect. That is, I constructed a variable "one" which 
takes value 1 for all cases, and let the intercept be random across "all 
one groups". This led to the following lme model:


     comsym.lme <- lme(opp~1,opposites, random= ~1|one,
                       correlation=corCompSymm(form = ~ 1 |one/id), 
method="REML")

And actually to my surprise, the results of this lme are highly similar 
to those of the gls above.
The output of both is attached below. The loglik's are identical, the 
AIC and BIC are not, which I can understand, as the lme has an extra 
variance to be estimated, compared to the gls. Also, the fixed 
intercept's standard error is different, the one of the lme being about 
twice as large.

I added some independent variables (not shown below) but the 
similarities between gls and lme remain, with only the AIC and BIC and 
the standard error of the fixed intercept being different for the two 
models.

My question is threefold.
1) Suppose the individuals (say pupils) would be nested in schools, then 
I assume that with lme I could add school as a random effect, and run a 
"usual" model, with pupils nested in schools and observations in pupils, 
and then use any of the possible residual covariance structures for the 
observations in pupils. Would you agree with this? (with gls one cannot 
use an additional random effect, like e.g. school)
2) Are the lme results indeed to be trusted when using this "fake" 
random effect, apart from the differences with gls mentioned? Could you 
imagine a situation where lme with this trick would produce wrong results?
3) I don't understand the variance of the intercept in the lme output. 
Even when I specify a very simple model: lme(opp ~ 1, random = ~1|one, 
opposites, method="REML")
lme is able to estimate the intercept variance...but what is this 
variance estimate expressing?

Thanks a lot for any help!!!

Ben.





*----------------- gls ---------------------------------------------------

 > comsym <- gls(opp~1,opposites,
                 correlation=corCompSymm(form = ~ 1 |id), method="REML")
 > summary(comsym)

Generalized least squares fit by REML
Model: opp ~ 1
Data: opposites
      AIC      BIC    logLik
1460.954 1469.757 -727.4769

Correlation Structure: Compound symmetry
Formula: ~1 | id
Parameter estimate(s):
   Rho
0.2757052

Coefficients:
                Value Std.Error  t-value p-value
(Intercept) 204.8143  5.341965 38.34063       0

Standardized residuals:
   Min          Q1         Med          Q3         Max
-2.75474868 -0.71244027  0.00397158  0.56533908  2.24944156

Residual standard error: 46.76081
Degrees of freedom: 140 total; 139 residual


#------------------ lme ---------------------------------------------------

 > comsym.lme <- lme(opp~1,opposites, random= ~1|one,
                 correlation=corCompSymm(form = ~ 1 |one/id), method="REML")
 > summary(comsym.lme)

Linear mixed-effects model fit by REML
Data: opposites
      AIC      BIC    logLik
1462.954 1474.692 -727.4769

Random effects:
   Formula: ~1 | one
         (Intercept) Residual
StdDev:    10.53875 46.76081

Correlation Structure: Compound symmetry
Formula: ~1 | one/id
Parameter estimate(s):
   Rho
0.2757052
Fixed effects: opp ~ 1
                Value Std.Error  DF  t-value p-value
(Intercept) 204.8143  11.81532 139 17.33463       0

Standardized Within-Group Residuals:
   Min          Q1         Med          Q3         Max
-2.75474868 -0.71244027  0.00397158  0.56533908  2.24944156

Number of Observations: 140
Number of Groups: 1


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