[R-sig-ME] deterministic correlation of intercept and slope random effects

[David Atkins]

Martin--

It sounds like your design has 2 and only 2 replicates per participants 
(thus, 181 subjects, 362 observations).

In essence, your larger model is trying to estimate 2 variances and a 
covariance - for which there are subject-specific realizations (i.e., 
the random-effects).  Thus, your model seems too complex for your data, 
which is (I think) what your AIC statistic is telling you, as you note 
that AIC prefers the random-intercept model.  (I often work with dyadic 
data, where we see similar problems; typically, we restrict models to 
random intercepts.)

Note that it is still possible to get perfect correlations even when 
there are larger group sizes.  The subject-specific effects (often 
called "empirical Bayes estimates") are weighted combinations of 
information about the individual as well as the sample as a whole. 
Relative to fitting data to *only* an individual's data, the empirical 
Bayes estimates are shrunken toward the sample mean.

The degree of shrinkage relates to how much variability there is between 
and within groups, and how much data there is.  If the variance of a 
random-effects is close to zero, the empirical Bayes estimates get 
shrunken to the sample average, which leads to perfect correlations 
between random-effects.

Doug Bates has slides from his workshops that discuss this in part - 
showing the difference between estimates from an individual OLS fit vs. 
random-effects estimate from mixed model.  Some of his slides can be found:

http://lme4.r-forge.r-project.org/

Hope that helps.

cheers, Dave
-- 
Dave Atkins, PhD
Research Associate Professor
Department of Psychiatry and Behavioral Science
University of Washington
datkins at u.washington.edu

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Dear group members,



Using lme4 for a poisson regression I found an odd effect: the correlation
between random intercept and random slope is exactly -1.



Currently, I analyze a data set from an experiment on cultural differences
in describing navigational routes. The outcome variables are counts of
verbal descriptors (like left-right descriptors, e.g. "turn right"). There
are three dichotomous predictors: Culture,  Instruction and Perspective.
Each subject was tested once in each of both Perspective conditions.



I started by determining whether a slope random effect was needed in
addition to the random intercept. This was done by comparing the AIC of the
following two models:



mr1<-lmer(LR~Culture*Instruction*Perspective+(1|Subj),data=cardin,
family=poisson)
mr2<-lmer(LR~Culture*Instruction*Perspective+(1+Perspective|Subj),data=cardi
n, family=poisson)



It turned out that the random intercept improves model fit (lower AIC).



Then I computed all possible combinations of the three fixed effects and
interactions with random slope and intercept and by lowest AIC arrived at:



m4c<-lmer(LR~Culture+Instruction+Perspective+Perspective:Culture+(1+Perspect
ive|Subj),data=cardin, family=poisson)



Now, what strikes me is the correlation of exactly -1 between random
intercept and slope. Indeed, the negative correlation has a straight forward
interpretation: Subjects already using left-right terms a lot are less
stimulated by the Perspective condition. But how can there be a complete
determination between intercept and slope?



Thanks in advance,

Martin.



Generalized linear mixed model fit by the Laplace approximation

Formula: LR ~ Culture + Instruction + Perspective + Perspective:Culture +
(1 + Perspective | Subj)

    Data: cardin

    AIC   BIC logLik deviance

709.3 740.5 -346.7    693.3

Random effects:

Groups Name             Variance Std.Dev. Corr

  Subj   (Intercept)      0.44923  0.67024

         PerspectiveRoute 0.41462  0.64391  -1.000 Number of obs: 362,
groups: Subj, 181



Fixed effects:

                               Estimate Std. Error z value Pr(>|z|)

(Intercept)                    1.78390    0.07249  24.608  < 2e-16 ***

CultureDutch                   0.49138    0.12595   3.901 9.56e-05 ***

InstructionCompass Rose       -0.01677    0.04247  -0.395  0.69283

PerspectiveRoute               0.63388    0.07205   8.797  < 2e-16 ***

CultureDutch:PerspectiveRoute -0.40283    0.13118  -3.071  0.00213 **

---

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1



Correlation of Fixed Effects:

             (Intr) CltrDt InstCR PrspcR

CultureDtch -0.534

InstrctnCmR -0.303  0.038

PerspectvRt -0.889  0.511  0.005

CltrDtch:PR  0.488 -0.929 -0.001 -0.549








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