On Wed, Apr 3, 2013 at 2:58 PM, Zack Steel <zacksteel@gmail.com> wrote:

> Hello all,
>
> I am running a glmer using the lme4 package and the binomial family and am
> getting somewhat unexpected results, which I'm hoping someone can help me
> make sense of. My data look something like the following:
>
> id        group      successes   total         fe1_center     fe2_center
> 1713    A              0                  11          -0.0911
> -17.2868
> 1717    A              0                  155        -0.0911       -17.2886
> 2272    B              49                 49          -0.0911      -32.2868
> 2289    B              7                   22          -0.2416
>  -32.2868
> 1487    B              0                   20          0.0537        2.7132
> 8199    C              10                127        -0.2416       -59.2868
> .....
>
> Where my response variable is the proportional of successes. I have
> centered the two fixed effects variables to alleviate some problems of
> multicollinearity and am also interested in their interaction.


I'm just interrupting to say that is incorrect. Centering does not
alleviate multicollinearity. It just shifts the y axis. It alters the
location at which the point estimates are provided, sometimes making people
think they are "better" because the ratio of estimate to std.error
changes.  But there really is no difference.

I got angry about it a couple of years ago when I was told I needed to
teach mean centering in a regression class. That advice is fairly widely
distributed, but it is just wrong.  I wrote functions meanCenter and
residualCenter in the rockchalk package so this would be easier for people
to see.  The evidence is in the vignette

http://pj.freefaculty.org/R/rockchalk.pdf

scroll down to page 22. There's a very clear publication of this.

Echambadi, R., & Hess, J. D. (2007). Mean-Centering Does Not Alleviate
Collinearity Problems in
Moderated Multiple Regression Models. Marketing Science, 26(3), 438–445.
doi: 10.1287/mksc.1060.0263

I'm absolutely completely positive their argument is correct for linear
models. How might the GLM's transformation via the link affect that?  Well,
it doesn't affect multicollinearity on the right hand side at all. And
that's the main point. It may be that the QR decomposition that insulates
OLS from numerical roundoff is not relevant in IRLS algorithms for GLM. But
I bet it is.

Now, coming back to your question about why, when your fixed effects are
set to the mean, is your estimated probability value so much lower than the
observed proportion of successes.

Can I suggest the culprit is your random effect? You may be forgetting the
model is nonlinear.  In your story, the predicted value is low, you are on
the bottom of the S shaped curve.  Now add a symmetric amount of noise on
either side of the linear predictor.  The noise on the left has a very
small effect on predicted probability, it is pushing you further down the
slope you've already slid down.  But the other half of the noise is
positive, and it is pushing you up the steep part of the S shaped curve.

I think this is the point at which people start talking about "average
marginal effect"...

pj

The data are
> clustered spatially within groups so I am using group as a random/grouping
> variable. When running the glmm, the coefficients of the fixed effects and
> their interaction seem reasonable (see below). However, when plotting the
> predictions vs. the response the curve is consistently lower than i would
> expect. E.g., the predicted proportion is lower than the mean proportion of
> the data across the full range of data.
>
> #running the model
> resp = cbind(data$successes, (data$total - data$successes))
> model = glmer( resp ~ fe1_c * fe2_c + (1|group) ,
>              data=data, family = binomial, REML=F)
> summary(model)
>
>
> Random effects:
>  Groups Name            Variance   Std.Dev.
>  group  (Intercept)       4.6432      2.1548
> Number of obs: 12271, groups: group, 392
>
> Fixed effects:
>                                       Estimate Std. Error              z
> value    Pr(>|z|)
> (Intercept)                      -2.3776295    0.1112830     -21.37
> <2e-16 ***
> fe1_c                             -0.8771395    0.0362946     -24.17
> <2e-16 ***
> fe2_c                              0.0109161    0.0001074      101.65
> <2e-16 ***
> fe1_c:fe2_c                   -0.0528655    0.0010090     -52.39     <2e-16
> ***
> ---
> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
> Correlation of Fixed Effects:
>                      (Intr)     fe1_c    fe2_c
> fe1_c            -0.012
> fe2_c             0.000  -0.687
> fe1_c:fe2_c  -0.022   0.411   -0.071
>
> I suspect my problem has something to do with how the "average" intercept
> is estimated (-2.378). Since I have centered my predictor variables I would
> expect the intercept to be equal to the grand mean (is this a correct
> assumption?), but in fact it is quite a bit lower.
>
> mean(data$successes/data$total)   # equal to 0.2008
> logistic (-2.3776295)                        # equal to 0.0849
>
> Perhaps the model is weighting the unique group intercepts differently
> leading to something other than a true average intercept? My group sizes
> vary greatly (data comes from messy observations, not experiments) so could
> this be affecting the estimate?
>
> Any incite you could give me would be much appreciated. Thank you for the
> help.
> Zack
>
>
> --
> Zack Steel
> Landscape Ecologist
> University of California, Davis
> zacksteel@gmail.com
>
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>
>
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>
>


-- 
Paul E. Johnson
Professor, Political Science      Assoc. Director
1541 Lilac Lane, Room 504      Center for Research Methods
University of Kansas                 University of Kansas
http://pj.freefaculty.org               http://quant.ku.edu

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