[R] random effects in logistic regression (lmer)-- identification question

[Paul Johnson]

Hello R users!

I've been experimenting with lmer to estimate a  mixed model with a
dichotomous dependent variable.  The goal is to fit a hierarchical
model in which we compare the effect of individual and city-level
variables.  I've run up against a conceptual problem that I expect one
of you can clear up for me.

The question is about random effects in the context of a model fit
with a binomial family and logit link.  Unlike an ordinary linear
regression, there is no way to estimate an individual level random
error in the linear predictor

z_i = a + b*x_i + e_i

because the variance of e_i is unidentified.  The standard deviation
of the logistic is pi*s/3, and we assume s=1, so the standard
deviation is assumed to be pi/3 (just a bit bigger than 1, if you are
comparing against the Standard Normal).  The logistic fitting process
sets the variance of the error and the parameters a and b are
"rescaled" accordingly.

As a result, there is an implicit individual-level random effect
within a logistic model.  There is a good explanation of this issue in
Tony Lancaster's textbook An Introduction to Modern Bayesian
Econometrics.

So we usually end up thinking about a linear predictor in a logistic
regression like so

z_i = a + b*x_i

Random effects can be estimated for "groups" or "clusters" of
observations.  If j is a grouping variable, then we estimate

z_i = a + b*x_i + u_j

The variance component here is, as far as I understand, measured on
the same scale as the logistic distribution's standard deviation.

Currently, I'm working on a project in which there are observations
collected in many cities, represented by a variable PLACE.  We are
comparing the effect of several variables on a response for each of
the values of a RACE variable.  RACE is dichotomized into "White" and
"Nonwhite" by the people who collect the data.

For Nonwhites only, we can estimate the effect of individual level
predictors (x) on the output (y).

fm1 <- lmer( y ~ x + (1 | PLACE), data=dat, , family=binomial, subset=
Race %in% c("Nonwhite"))

The random effect in this model indicates the variance caused by a
Nonwhite's location on the response variable.

Random effects:
 Groups  Name        Variance Std.Dev.
 PLACE (Intercept) 0.047326 0.21754

Suppose I estimate models for the 2 races in a combined model like this:

fm1 <- lmer( y ~ -1 + Race / (x) + (-1 + Race | PLACE), data=dat,
family=binomial)

This gives fixed effects estimates that are pretty easy for
nonstatisticians to understand.  One can look and see the effect of a
variable x on people of different races.

But the random effect is a bit hard to understand.  Since the Race
variable is dichotomous, My aim was to see if the variance of the
random effect is different for the 2 racial categories. Here are the
estimates:

Random effects:
 Groups  Name                      Variance  Std.Dev. Corr
 PLACE RACE_ALLWhite      0.0095429 0.097688
             RACE_ALLNonwhite 0.1286597 0.358692 1.000

I can't quite grasp what the correlation means.  I BELIEVE the
variance values indicate that the experiences of whites are
homogeneous across cities, because the variance is negligible for
them, while the experiences of Nonwhites are much more city-dependent.

What does it mean when the correlation is 1.0?  The correlation takes
on that value when there are no city-level variables in this model, so
I GUESS that it means that all city-level variation is attributed to
the random effect. What do you think?

If i put in some city level predictors, then the estimates of the
variance components change--they essentially disappear to the minimum
values--and the correlation is not 1.0 anymore.

Random effects:
 Groups  Name          Variance Std.Dev.   Corr
 PLACE  RACEWhite    5e-10    2.2361e-05
          RACENonwhite   5e-10    2.2361e-05 0.001

This indicates that, after adding in the city level variables, the
unaccounted for city-level  variation is very small.  Correct?

Now, back to the idea that there is always an implicit individual
level random effect in a logistic regression. Is it meaningful to ask
"is that individual level random effect different for people of
different races?"  If so, How can I estimate that?  If e_i is the
implicit random error, can I ask for another random effect for
Nonwhites only, say u_iN, in a model like so:

z_i = a + b*x_i + e_i + u_iN

Suppose the unique respondent number is ID and we create a new variable

NonwhiteID = 0 for Nonwhites
                 = ID for Nonwhites

Here's my idea about how to check to see if the individual level
variance component for Nonwhites is different from the "baseline" of
Whites by fitting this:

fm1 <- lmer( y ~ -1 + Race / (x) + (-1 + Race | PLACE) + (1 |
NonwhiteID), data=dat, family=binomial)

Here, again, I leave out the city-level variables.

Random effects:
 Groups  Name          Variance   Std.Dev.   Corr
 NonwhiteID (Intercept)   5.0000e-10 2.2361e-05
 PLACE RACEWhite       1.0575e-02 1.0283e-01
         RACENonwhite        1.2880e-01 3.5889e-01 1.000
number of obs: 6201, groups: NonwhiteID, 1736; PLACE, 33

The variance component estimated for NonwhiteID means that the
variance observed among Nonwhite respondents is not substantially
different from the implicit, unestimated individual level random
error.  However, it still appears that there is a substantial place
effect, for Nonwhites only.

Do I understand that right?

Well, thanks in advance, as usual.

-- 
Paul E. Johnson
Professor, Political Science
1541 Lilac Lane, Room 504
University of Kansas