# [R] lme unequal random-effects variances varIdent pdMat Pinheiro Bates nlme

Jacob Wegelin jawegelin at ucdavis.edu
Mon Jul 12 09:50:58 CEST 2004

```How does one implement a likelihood-ratio test, to test whether the
variances of the random effects differ between two groups of subjects?

Suppose your data consist of repeated measures on subjects belonging to
two groups, say boys and girls, and you are fitting a linear mixed-effects
model for the response as a function of time.  The within-subject errors
(residuals) have the same variance in both groups. But the dispersion of
the random effects differs between the groups.  The boys' random effects
-- say, the intercepts -- have greater variance than the girls'.  One can
see this by partitioning the data by sex and fitting two separate models.

The model for the girls,

library("nlme")
mylmeF0 <- lme( y ~ time, data=DAT, random=~time | id, subset=sex=="F")

yields a variance of about one for the random intercepts:

StdDev    Corr
(Intercept) 0.9765052 (Intr)
time        0.1121913 -0.254
Residual    0.1806528

whereas in the model for the boys, the corresponding variance is ten times
that amount:

mylmeM0 <- lme( y ~ time, data=DAT, random=~time | id, subset=sex=="M")

StdDev     Corr
(Intercept) 10.1537946 (Intr)
time         0.1230063 -0.744
Residual     0.1298910

I would like to use a likelihood ratio to test this difference.  The
smaller ("null") model would be

mylme0 <- lme( y ~ time, data=DAT, random=~time | id )  .

This model forces the random intercepts for both boys and girls to come
from a single normal distribution.

The larger model would allow the boys' and girls' random intercepts (or
more generally their random effects) to come from separate normal
distributions with possibly unequal variances.

There must be some straightforward obvious way to fit the larger model,
but I do not see it.

Pinheiro and Bates, chapter 5.2, show how to model unequal *residual*
("within-group") variances for the two groups using varIdent.  They also
tantalizingly say, "The single-level variance function model (5.10) can be
generalized to multilevel models" (page 206), which seems to suggest that
a solution to the current problem might exist.

The pdMat classes provide a way to *constrain* the dispersion matrix of
the random effects, not make it more general.

Of course, one way to test for unequal variances is to apply an F-test for
equal variances to the random intercepts. If the data are as shown at the
bottom of this email, the test can be implemented as follows:

stuff<-as.data.frame(summary(mylme0)\$coefficients\$random\$id)
stuff\$sex<-factor(substring(row.names(stuff), 1,1))
mysplit<-split(stuff[,"(Intercept)"], stuff[,"sex"])
ns<-sapply(mysplit, length)
vars<-sapply(mysplit, var)
p<- 1-pf( vars["M"]/vars["F"], ns["M"]-1, ns["F"]-1)

Alternatively, one could implement a permutation test for the ratio of the
variances of the random intercepts--these variances derived from the two
halves of the partitioned data.

But surely there's a direct, model-based way to do this?

Thanks for any suggestions

Jake

P.S. Here is the code by which the "data" were generated.

nb<-1
ntimepts<-3
girls<-data.frame(
y= rep(-nb:nb , each=ntimepts)
,
id=rep( paste("F", 1:(2*nb+1), sep=""), each=ntimepts)
,
time=rep(1:(2*nb+1), length=ntimepts)
)
boys <-data.frame(
y= rep(10*(-nb:nb) , each=ntimepts)
,
id=rep( paste("M", 1:(2*nb+1), sep=""), each=ntimepts)
,
time=rep(1:(2*nb+1), length=ntimepts)
)
DAT<-rbind(girls,boys)
DAT\$y<-DAT\$y + rnorm(nrow(DAT))/5
DAT\$sex<-factor(substring( as.character(DAT[,"id"]), 1,1))
row.names(DAT)<-paste( DAT[,"id"], DAT[,"time"], sep=".")

Jacob A. Wegelin
Assistant Professor
Division of Biostatistics, School of Medicine
University of California, Davis
One Shields Ave, TB-168
Davis CA 95616-8638 USA
http://wegelin.ucdavis.edu/
jawegelin at ucdavis.edu

```