Question regarding the estimation of the correlation between random intercepts and random slopes in the lme4 package

Dear list members,

I posted a question regarding the lme4 package in R on stackoverflow.com
(see
http://stackoverflow.com/questions/24778642/estimating-correlation-between-random-slopes-and-random-intercepts-using-the-lme)
and a stackoverflow-user kindly  referred me to this mailing list. I hope
that my question is appropriate for this mailing list. If it is not, please
let me know!

This is my question:

For answering my research question I am interested in the correlation
between the random slopes and random intercepts in a multilevel model,
estimated using the R library lme4.

The data I have is: Y (test-scores from students), SES (socio-economic
status for each student) and schoolid (ID for each school).

I am using the following syntax to estimate random intercepts and slopes
for the schools:

library(lme4)
model3 <- lmer(Y ~ SES + (1 + SES | schoolid))

 The reference I used for this syntax is this pdf:

http://www.bristol.ac.uk/cmm/learning/module-samples/5-concepts-sample.pdf

On page 19, a similar analysis is described. It is said that by defining
the random intercepts and slopes toghether, it is indirectly specified that
we want the random intercepts and slopes to covary. Therefore, also the
correlation between random slopes and random intercepts is estimated.
Basically, exactly what I need for answering my research hyptohesis.

However, when I look at the results:

 summary(model3)

 I am getting the following output:

Linear mixed model fit by REML ['lmerMod']
Formula: Y ~ SES + (1 + SES | schoolid)

REML criterion at convergence: 8256.4

Scaled residuals:
    Min      1Q  Median      3Q     Max -3.1054 -0.6633 -0.0028  0.6810  3.5606

Random effects:
 Groups   Name        Variance  Std.Dev. Corr
 schoolid (Intercept) 0.6427924 0.80174
      SES             0.0009143 0.03024  1.00
 Residual             0.3290902 0.57366
Number of obs: 4376, groups: schoolid, 179

Fixed effects:
             Estimate Std. Error t value(Intercept) -0.036532   0.060582  -0.603
SES          0.062491   0.009984   6.259

Correlation of Fixed Effects:
    (Intr)
SES 0.226

 As stated in the output, the correlation between the random slopes and
random intercepts equals 1.00. I find this hard to believe. When I call in
R:

VarCorr(model3)$schoolid

 I am getting the following output which gives the correlations and
covariance matrix:

                (Intercept)          SES(Intercept)  0.64279243 0.0242429680
SES          0.02424297 0.0009143255

attr(,"stddev")(Intercept)         SES
 0.80174337  0.03023782

attr(,"correlation")
        (Intercept) SES(Intercept)           1   1
SES                   1   1

 It seems as if the correlation between the slopes and intercepts was set
to 1.00 by R. I did not see this in the output of anyone else when I was
searching the internet on references on multilevel modelling.

Does anybody know what can be the cause of this correlation? Can it be that
the correlation is set to 1.00 because otherwise the model would not be
identified? Or is it because the variance of the random slopes is so small
(0.0009) that the correlation can not be estimated?

I have tried to simulate data in order provide the code for a small
reproducible dataset. I was however not yet able to reproduce this output
by means of simulated data. As far as I have the code I will eidt my post
and add the code.


Many thanks for you time and effort,
Kind regards,

Inga