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random effects correlation in lmer

2 messages · Kurt Smith, Daniel Malter

Original
Kurt Smith
# 
I am having an issue with lmer that I wonder if someone could explain.

I am trying to fit a mixed effects model to a set of longitudinal data
over a set of individual subjects:
(fm1 <- lmer(x ~ time + (time|ID),aa))


I quite often find that the correlation between the random effects is 1.0:
Linear mixed model fit by REML
Formula: x ~ time + (time | ID)
   Data: aa
   AIC   BIC logLik deviance REMLdev
 28574 28611 -14281    28561   28562
Random effects:
 Groups   Name        Variance Std.Dev. Corr
 ID       (Intercept)  77.035   8.7770
          time         10.817   3.2889  1.000
 Residual             112.151  10.5901
Number of obs: 3539, groups: ID, 1000

Fixed effects:
            Estimate Std. Error t value
(Intercept)  98.7601     0.3894  253.64
time          1.3671     0.2001    6.83

Correlation of Fixed Effects:
     (Intr)
time -0.045


All other parameters seem to converge as I increase the size of the
data set, or have a reasonable distribution over several bootstrap
samples. This suggests to me there is a singularity or something in
solving for the random effects correlation. Does anyone have any
insight?
Thanks,
Kurt Smith
Daniel Malter
# 
It implies that the random intercept is perfectly collinear with the random
slope, as you suggested. I attach an example. 

The data generating process of y1 has a random intercept, but no random
slope. When you fit a model with random intercept and random slope, the
correlation between the two is estimated at -1. However, note that the
variance of the random slope is almost zero. Thus, we fit the wrong model. A
random intercept only model would have sufficed. 

The data generating process of y2 includes a random slope, but those with
the higher intercepts also have the greater slopes. Random intercept and
random slope estimates are perfectly collinear. This leads to the problem
you encounter. Models 2a and 2b provide random intercept only and random
slope only estimates for comparison. It would suffice in this case to fit a
random intercept model.

Finally, the data generating process for y3 has random intercept and slope,
but both are independent, so that the problem does not occur.

library(lme4)

tim=rep(10:19,10)
id=rep(1:10,each=10)
rand.int=rep(rnorm(10,0,1),each=10)
rand.slop=rep(rnorm(10,0,1),each=10)
e=rnorm(100,0,0.5)

y1=10+rand.int+tim+e
y2=10+rand.int+tim+e
y3=10+rand.int+tim+rand.slop*tim+e

reg1=lmer(y1~tim+(tim|id))
summary(reg1)

reg2=lmer(y2~tim+(tim|id))
summary(reg2)

reg2a=lmer(y2~tim+(1|id))
summary(reg2a)

reg2b=lmer(y2~tim+(-1+tim|id))
summary(reg2b)

reg3=lmer(y3~tim+(tim|id))
summary(reg3)

HTH,
Daniel
Kurt Smith-3 wrote: