negative variances

lm2<-lmer(ppd~month+(month|id)) 

lm2<-lmer(ppd~month+(month|id),control = list(msVerbose = TRUE) ) 

lm3<-lmer2(ppd~month+(month|id),control = list(msVerbose = TRUE) ) 

From: "Douglas Bates" <bates at stat.wisc.edu>
To: "Tu Yu-Kang" <yukangtu at hotmail.com>
CC: r-sig-mixed-models at r-project.org
Subject: Re: [R] negative variances
Date: Thu, 12 Apr 2007 08:07:57 -0500

On 4/12/07, Tu Yu-Kang <yukangtu at hotmail.com> wrote:

Dear Prof Bates,

Many thanks for your email. I tried lmer() and received the 
following
messages:

 lm2<-lmer(ppd~month+(month|id))

Warning message:
Estimated variance-covariance for factor 'id' is singular
 in: `LMEoptimize<-`(`*tmp*`, value = list(maxIter = 200, 
tolerance =
1.49011611938477e-08,

I then tried lmer2():

 lm3<-lmer2(ppd~month+(month|id))
summary(lm3)

Linear mixed-effects model fit by REML
  AIC  BIC logLik MLdeviance REMLdeviance
 1146 1166 -567.9       1126         1136
Random effects:
 Groups   Name        Variance  Std.Dev. Corr
 id       (Intercept) 0.1566631 0.395807
          month       0.0022331 0.047255 1.000
 Residual             0.6391120 0.799445
Number of obs: 420, groups: id, 140

Fixed effects:
            Estimate Std. Error t value
(Intercept)  6.43595    0.07017   91.72
month       -0.36619    0.01642  -22.30

Correlation of Fixed Effects:
      (Intr)
month -0.544

I'm a bit pressed for time right now so this will be brief.  I 
suggest
that you add the optional argument

control = list(msVerbose = TRUE)

to the calls to lmer and to lmer2 and look at the progress of the
iterations and also at the value of the deviance.  I have seen
situations where lmer2 converges to a fit with a significantly 
smaller
deviance than can lmer because it progresses through the 
near-singular
region of the parameter space.

What was the deviance (or the log-likelihood) from the MLWin fit 
that
gave negative variance components?

Did you plot the data as ppd versus month by id using xyplot from 
the
lattice package?  That should always be the first step in the 
analysis
of longitudinal data.

However, I am not sure about the results, because MLwiN showed both 
random
effects were negative values (-0.196 and -0.023).

I start to notice this problems of negative variances when I am 
learning
how to use structural equation modeling software to run multilevel 
models
for longitudinal data. To my great surprise, it occurs quite 
frequently. In
SEM, this problem sometimes may be overcome by estimating  a 
nonlinear
model by freeing the factor loadings. For example, in this data, 
PPD
(probing pocket depth) was measured three times at month 0, 3 and 
6. I only
fixed the first and last factor loadings to be 0 and 6 to get a 
non-linear
relation, and I also allow the level-1 residuals to be different on 
each
occasion. However, in some data, I failed to get a satifactory 
model no
matter how I modified my models.

I looked for the discussion in several multilevel modeling 
textbooks but
only found one short discussion in the book by Brown and Prescott. 
SEM
literature usually suggest fixing the negative variances to 0. 
However, I
wander whether this is the only way to get around this problem or 
the
sensible way because if the random effects are fixed to 0 the model 
is no
longer a random effects model.

With best regards,

Yu-Kang