Meaning of Corr of random-effects with a cross-level interaction

Hi Thierry and list

This was actually cross-posted at CrossValidated yesterday:

https://stats.stackexchange.com/questions/488984/corr-of-random-effects-when-a-cross-level-interaction-in-lme4

I have the impression that Simon is experimenting with a toy dataset,
rather than analysing their own study, which is a great way to learn, in
my opinion.

As you can see from my answer and the comments to it, the real question
(actually, two questions) is this:

Suppose we have two groups of schools, with a single explanatory variable
at the student level. Suppose further that the correlation between the
random slopes for that variable and the random intercepts in the two groups
is very different.  The first question is what the overall correlation
represents ? I thought that it would probably be some kind of average of
the two.  I did some simulations that indicate that this seems to be the
case. The followup question (see the last comment to my answer) asks how to
uncover the correlations in the two groups ? From my simulations so far the
only way I can see of doing this is by splitting the data by group and
fitting two models.

Best regards
Robert Long



On Fri, Sep 25, 2020 at 9:04 AM Thierry Onkelinx via R-sig-mixed-models <
r-sig-mixed-models at r-project.org> wrote:

Dear Simon,

A perfect correlation between random effect parameters indicates a
problem.
Note the failed convergence warning.
Standardising ses makes things even worse: it yields a singular fit error.

Removing the random slope of ses or the sector interaction solves the
problem. i.e. the model runs and yields sensible output.

Looking at the data, it seems like both math and ses have bounds. Ses
even seems to have some data above its upper bound.
The model assumes no bounds in the response variable. Maybe this is the
cause of the problem.

ggplot(hsb, aes(x = ses, y = math, colour = factor(sector))) +
  geom_point()

Best regards,

Thierry


ir. Thierry Onkelinx
Statisticus / Statistician

Vlaamse Overheid / Government of Flanders
INSTITUUT VOOR NATUUR- EN BOSONDERZOEK / RESEARCH INSTITUTE FOR NATURE AND
FOREST
Team Biometrie & Kwaliteitszorg / Team Biometrics & Quality Assurance
thierry.onkelinx at inbo.be
Havenlaan 88 bus 73, 1000 Brussel
www.inbo.be


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Op do 24 sep. 2020 om 18:39 schreef Simon Harmel <sim.harmel at gmail.com>:

Dear All,

I had a quick question. I have a cross-level interaction in my model

below

(ses*sector). My cluster-level predictor "sector" is a binary variable
(0=Public, 1=Private). My level-1 predictor is numeric.

QUESTION:  The `Corr = 1` is indicating the correlation between
intercepts and slopes across BOTH public & private sectors (like their
average) OR something else?

hsb <- read.csv('
https://raw.githubusercontent.com/rnorouzian/e/master/hsb.csv')

summary(lmer(math ~ ses*sector + (ses|sch.id), data = hsb))


Random effects:
 Groups   Name        Variance Std.Dev.     Corr
 sch.id   (Intercept)  3.82107    1.9548
          ses                0.07587     0.2754        1.00
 Residual             36.78760 6.0653

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