[R-sig-ME] Comparing Gaussian and bêta regression

[Ben Bolker]

On 2018-09-19 11:02 AM, Emmanuel Curis wrote:
> Thank you very much for the hints.
> 
> The « not very satisfactory » is from a "theoretical" point of view:
> I'm not very comfortable with modeling with a Gaussian a value
> constrained between 0 and 10, with the extremes obtained not so
> rarely.  From a practical point of view, it does not seem to produce
> unexpected results.  Of course, there are some effects that are
> borderline significant, that also makes the question uprise: what is
> the part of true signal and basically inadequate model in these
> effects? Still finding them with a more sounded model would make them a
> little bit more "trustable"...

  Fair enough.

> For the ordinal outcome: I have wrongly selected my example value, it
> induced in error, sorry ; the step is 0.1 and not 0.5 ; in practice,
> 46 different values were observed. Integer values and, to a less
> extent, half-integer values are clearly over-represented, I guess
> because of inconscient rounding during scoring.  I don't know how to
> handle this in a model, however, but that's another problem, and may
> be there is no need for that.  But, for the ordinal aspect, I fear
> that would make too much parameters in the model... 

    I agree.

> Just thinking... Would it be imaginable to make inferences on the
> beta-distribution model, since it seems to much better describe the
> data, but use the linear model on the raw scale just to have
> point-estimates of the changes in an easiest-to-interpret way?
> [despite it is problematic close to the boundaries...]

   It's certainly "imaginable", but my preference would be for taking
the time to understand logistic-scale parameters -- something like this
will be required any time you're dealing with predictions on a bounded
scale where the data go anywhere close to the boundaries ... if you're
going to go out of your way to use a model that's preferable on
theoretical grounds, why not go all the way?


> Is the Gelmann & Hill book you're thinking about this one: ?
> 
> Data Analysis Using Regression and Multilevel/Hierarchical Models
>  Cambridge University Press
> ISBN-10: 052168689X

 Yes, that's right. It's "Gelman" with one 'n', not to be confused with
the physicist Murray Gell-Mann -- although according to

https://www.edge.org/conversation/murray_gell_mann-the-making-of-a-physicist

> [M. Gell-Mann's brother] reformed the spelling of our surname and made
it just Gelman. He got tired of telling people about the double L, the
double N, the hyphen, and the capital M.

  FWIW there are probably lots and lots of explanations of logit-scale
parameters lying around on the internet, e.g.
https://stats.idre.ucla.edu/other/mult-pkg/faq/general/faq-how-do-i-interpret-odds-ratios-in-logistic-regression/


 Some rules of thumb from my GLM notes (partly taken from G & H):

– logit: log-odds change.

* for β∆x small, as for log (proportional)
* for intermediate values, linear change in probability with
slope ≈ β/4
* for large values, as for log ( 1 − x )



  cheers
    Ben Bolker



> 
> On Wed, Sep 19, 2018 at 09:49:51AM -0400, Ben Bolker wrote:
> « 
> « 
> « On 2018-09-19 03:30 AM, Emmanuel Curis wrote:
> « > Hello,
> « > 
> « > I'm doing my first try on bêta regression, with mixed effects model,
> « > and was wondering if my reasonning is correct...
> « > 
> « > The context is a clinical study where the outcome is a score variable,
> « > with continuous values between 0 and 10 (both included) and, in
> « > practice, values with only one decimal digit (eg. 1.5) There is
> « > about 400 patients. Random effect is the clinician who does the
> « > examination and afterthat collects the score that evaluates its
> « > intervention.
> « > 
> « > As a quick-and-dirty analysis, I did a linear mixed effect model on
> « > the raw data, with lmer. Residuals and random effects are not so bad,
> « > and results consistent & easy to interpret, but assuming a Gaussian
> « > distribution is not very satisfactory.
> « 
> « Can you expand on why "not very satisfactory"?  Do you get unrealistic
> « predictions etc.?
> « 
> «   This sounds like it could also be treated as an ordinal response (with
> « 21 values {0, 0.5, 1, ... 9.5, 10}).
> « > 
> « > Hence, I tried a bêta regression on the data after the transformation
> « > (y/10 * (n-1) + 0.5) / n, and used glmmTMB for that. And of course I
> « > wondered if the fit was better.
> « > 
> « > 1) Is it right that ln-likelihood of the model on the raw data
> « >    (Gaussian) and on the transformed data (bêta) cannot be compared,
> « >    because they involve probability densities and not probabilities,
> « >    hence depend on the data scale ?
> « 
> «   You can compare log-likelihoods (actually technically they're
> « log-likelihood *densities*, which is where the problem comes from) if
> « you account for the scaling.  In this case since you're doing a linear
> « transformation the scaling should be pretty easy.
> « > 
> « > 2) Is it right that the lmer model done on the raw data and the same
> « >    one done on the transformed data are conceptually the same, since
> « >    the transformation is linear — so that the ln-likelihood it gives
> « >    is « the same » expressed in the two different scales? (of course,
> « >    coefficients and so on will be different because of the scale
> « >    change)
> « 
> «    Should be. (You could do a simple test of this ...)
> « > 
> « > 3) And so, is it correct to compare the ln-likelihood (using logLik)
> « >    or the AIC given by glmmTMB with the bêta model and by lmer on
> « >    transformed data to compare the two models (raw data Gaussian vs
> « >    bêta)?
> « 
> «   I would think so.
> « > 
> « >    If so, the bêta model seems better than the Gaussian one. But now
> « >    comes the interpretation problem, other than « are coefficients
> « >    significantly different from 0? ».
> « > 
> « > 4) Since the default link is the logit for the mean, interpretation is
> « >    not quite clear for me.  For the Gaussian model on raw data,
> « >    interpretation is clear, for instance « men score 1 point lower
> « >    than women in average ».  But how can the coefficients of the
> « >    bêta-model be back-converted in a similar fashion ?
> « 
> «    You probably need to go read stuff about interpretation of
> « logit/log-odds  parameters: Gelman and Hill's book is good.
> « 
> « Quick rules of thumb:
> « 
> « * for β∆x small, as for log (proportional)
> « * for intermediate values, linear change in probability with
> « slope ≈ β/4
> « * for large values, as for log ( 1 − x )
> « > 
> « >    Would it be easier to use a log link and expression changes in the
> « >    scale as percent changes on the mean?
> « 
> «   This will work fine for low score values, but will run into trouble at
> « the upper end of the score range.
> « 
> « > 
> « > Thanks in advance,
> « >
> « 
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