Correlation of -1: is it a problem?

On Fri, Mar 26, 2010 at 1:18 PM, Eric Castet

Dear all,

I would be grateful if you could help me with the following question
using lmer()
I want to test the effect of a categorical factor with two levels
(called 'couleurs')
The only random factor is 'nom'.

I first start with all random effects (I only report the lines for the
random effects):
Linear mixed model fit by maximum likelihood
Formula: lRT ~ couleurs + (1 + couleurs | nom)
Random effects:
?Groups ? ? ? ? ?Name ? ? ? ? ? ?Variance ?Std.Dev. Corr
?nom ? ? ? ? ? ? ?(Intercept) ? ? 0.1376693 0.371038
? ? ? ? ? ? ? ? ? ? ?couleurs1 ? ? ? 0.0030358 0.055098 *-1.000 *
?Residual ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0.5118424 0.715432
Number of obs: 7927, groups: nom, 10


Then, I remove the random effect of 'couleurs' ?with the following result:

Linear mixed model fit by maximum likelihood
Formula: lRT ~ couleurs + (1 | nom)
? Data: jb

Random effects:
?Groups ? ? ? Name ? ? ? ? ? ?Variance Std.Dev.
?nom ? ? ? ? ?(Intercept) ? ? 0.11768 ?0.34304
?Residual ? ? ? ? ? ? ? ? ? ? ? ? 0.51263 ?0.71598
Number of obs: 7927, groups: nom, 10

I then compare the two models and see that I should go with the first
model Df=6:
?> anova (jb.lmer1, jb.lmer2)
Data: jb
Models:
jb.lmer2: lRT ~ couleurs + (1 | nom)
jb.lmer1: lRT ~ couleurs + (1 + couleurs | nom)
? ? ? ? Df ? AIC ? BIC ?logLik ?Chisq Chi Df Pr(>Chisq)
jb.lmer2 ?4 17259 17287 -8625.4
jb.lmer1 ?6 17251 17293 -8619.4 12.078 ? ? ?2 ? 0.002384 **

My questions are the following:

a/ is it really a statistical (or numerical) problem to have a -1
correlation in the model that I should keep?

Yes, it is.  The fitted model is has a singular variance-covariance
matrix for the random effects and that is not good.  In fact, it is no
longer a linear mixed model.

b/ is it possible to remove the correlation between Intercept and
Couleurs, as I would do if Couleurs were not a categorical factor?

I would fit another model of

IRT ~ couleurs + (1|nom:couleurs) + (1|nom)

and see how that works.  This model is, in some sense, intermediate to
the models that you have fit above.

Thanks in advance,

Eric Castet




--

Eric Castet

Institut de Neurosciences Cognitives de la M?diterran?e -- INCM CNRS

31 chemin Joseph Aiguier

13402 Marseille cedex 20 (France)

tel : (+33)(0)4-91-16-43-34

fax : (+33) (0)4-91-16-44-98

UMR 6193 du CNRS

Universit? Aix-Marseille II

http://www.incm.cnrs-mrs.fr/equipedyva.php

http://www.incm.cnrs-mrs.fr/pperso/ecastet.php




? ? ? ?[[alternative HTML version deleted]]

_______________________________________________
R-sig-mixed-models at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models

An embedded and charset-unspecified text was scrubbed...
Name: not available
URL: <https://stat.ethz.ch/pipermail/r-sig-mixed-models/attachments/20100329/53e9f909/attachment.pl>

-----Mensaje original-----
De: r-sig-mixed-models-bounces at r-project.org [mailto:r-sig-mixed-models-bounces at r-project.org] En nombre de Eric Castet
Enviado el: lunes, 29 de marzo de 2010 19:45
Para: Douglas Bates
CC: r-sig-mixed-models at r-project.org
Asunto: Re: [R-sig-ME] Correlation of -1: is it a problem?

Dear Doug,

Thanks for your reply.

I've done as you suggested (your point b/), i.e. I've fit another model that considers 'couleurs' within 'nom'
However, after running anova() (likelihood ratio test), I find that I should keep the initial model that contains the -1 correlation:

 > anova (jb.lmer1, jb.lmer2)
Data: jb
Models:
jb.lmer2: lRT ~ couleurs + (1 | nom) + (1 | nom:couleurs)
jb.lmer1: lRT ~ couleurs + (1 + couleurs | nom)
          Df   AIC   BIC  logLik  Chisq Chi Df Pr(>Chisq)
jb.lmer2  5 17260 17295 -8625.2
jb.lmer1  6 17251 17293 -8619.4 11.584      1  0.0006654 ***

So, the question is now:

a/ How can I justify to refuse the initial model that contains the -1 correlation?

b/ And a parallel question is: what is wrong with the -1 correlation? Is it because it is exactly -1 ? Would it still be a problem if it were -0.99 ?

c/ Ultimately, how can I report what appears to me as an important
result: namely the high correlation between the intercept and the 'couleurs' effect for each subject?

--------

When I see a correlation very close to 1 or -1 between parameter estimates (mostly in nonlinear models, because that is my main area) I start to think of ways to re-parameterize the model, i.e. some change in the algebraic structure that introduces different parameters, because a nearly-perfect correlation is telling me that one of the highly correlated parameter estimates is redundant: the sample does not have any more information about one parameter than it has about the other.
Probably the perfect correlation is telling you that you don't need a statistical test to tell what you want to tell.

HTH

Rub?n


____________________________________________________________________________________ 

Dr. Rub?n Roa-Ureta
AZTI - Tecnalia / Marine Research Unit
Txatxarramendi Ugartea z/g
48395 Sukarrieta (Bizkaia)
SPAIN