[R-sig-ME] Low intercept estimate in a binomial glmm
John Maindonald
john.maindonald at anu.edu.au
Fri Apr 5 08:24:48 CEST 2013
Surely it is an issue of how you define multi-collinearity.
Centering is a simple re-parameterisation that, like any
other re-parameterisation, makes no difference to the
predicted values and their standard errors (well, it will
make some small difference to the numerical computational
error, but with modern software that should be of scant
consequence). Re-parameterisation may however give
parameters that are much more interpretable, with much
reduced correlations and standard errors That is the
primary reason, if there is one, for doing it.
John Maindonald email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473 fax : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.
http://www.maths.anu.edu.au/~johnm
On 05/04/2013, at 4:40 PM, Paul Johnson <pauljohn32 at gmail.com> wrote:
> On Wed, Apr 3, 2013 at 2:58 PM, Zack Steel <zacksteel at 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 at gmail.com
>>
>> [[alternative HTML version deleted]]
>>
>>
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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
>
> [[alternative HTML version deleted]]
>
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