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

[Andrew McAleavey]

Thanks for the reply,

I take from your message that my interpretation of the x2 random effect is
correct - it is the variance of the deviation from the group mean with x2=1
compared with x2=0. So the total effect of group when x2=1 would be the sum
of both random effects, yes?

The model you suggest:
y ~ x1 + x2 +  (1 +x2 | group);
is identical to:
y ~ x1 + x2 +  (x2 | group),
right? It seems to be based on my testing and understanding of lmer
defaults. In any case that model does not improve model fit according to
AIC/BIC and LRT, so I went with the one I described in my first email.

My problem is that these variances should not be correlated (because there
is no covariance between them, right?), though the estimates (pulled from
ranef() ) seem to be meaningfully correlated. Is this just a chance
occurrence or artifact, like when factor scores from uncorrelated factors
are highly correlated? Should I not interpret the high observed correlation
due to the lack of formal modeling and nonsignificant improvement? Am I
just capitalizing on chance?

Thanks!
Andrew

On Sat, Dec 6, 2014 at 3:23 PM, Ken Beath <ken.beath at mq.edu.au> wrote:

> The random effect for x2 is giving the variation in the effect of x2, that
> is the difference in levels (from x2=0 to x2=1), with id.
>
> I would first try the model, and see if it improves AIC.
>
> y ~ x1 + x2 +  (1 +x2 | group)
>
> This now allows for the random effects for the intercept and x2 to be
> correlated
>
> On 7 December 2014 at 02:12, Andrew McAleavey <andrew.mcaleavey at gmail.com>
> wrote:
>
>> 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
>>
>>         [[alternative HTML version deleted]]
>>
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>>
>
>
>
> --
>
> *Ken Beath*
> Lecturer
> Statistics Department
> MACQUARIE UNIVERSITY NSW 2109, Australia
>
> Phone: +61 (0)2 9850 8516
>
> Building E4A, room 526
> http://stat.mq.edu.au/our_staff/staff_-_alphabetical/staff/beath,_ken/
>
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