[R-sig-ME] interpretation of categorical crossed effect in lme4

[Andrew McAleavey]

Hi,

I have a lmer model of the form:
y ~ x1 + x2 + (1 | group) + (0 +x2 | group) ;
where x1 is continuous, x2 is dichotomous and dummy-coded, and group has
about 250 levels (each with minimum 3 observations in each x2 level, but
the average is more like 7 per x2 level, and over 15 observations per group
on average, ignoring x2). My understanding is that this model separately
estimates variance components for each level of x2 across groups, and does
not model any correlation between them.

This was a better fit to the data than  the structure:
y ~ x1 + x2 + (x2 | group) ;
and I came to this model based on a series of threads on this list. Note
that under this model the correlation between random effects for x2 and the
intercept was .67, and as far as I can tell convergence was not a problem
in either model as it might be in some cases with smaller group numbers.

However, I would like to interpret, at least tentatively, the random
effects, and especially the relationship between them. My central
substantive question is whether groups vary with respect to differential
effectiveness with x2 levels (e.g., some groups were effective with x2=0
but not x2=1 while others were highly effective with both). Extracting the
random effects and plotting them suggests that even though the model does
not explicitly include correlations, the two random effects are correlated
at about r = .56.

My questions are these:
a) is a significant correlation like r = .56 common under conditions of my
model in which these effects were not modeled?
b) to interpret the random effects, I think I may need to treat them as
additive and correlate u1 with (u1 + u2), which leads to an even higher
correlation (r > .8). Am I correct in this? My thinking is that u2, as a
dummy coded variable, represents the deviation for x2 = 1 from x2 = 0, but
is that incorrect?

Thanks very much,
Andrew

-- 
Andrew McAleavey, M.S.
Department of Psychology
The Pennsylvania State University
346 Moore Building
University Park, PA 16802
aam239 at psu.edu

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