Comparing Gaussian and bêta regression

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"...

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... 

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...]

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

? 
?

? > Hello,
? > 
? > I'm doing my first try on b?ta 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 b?ta 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 (b?ta) 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 b?ta model and by lmer on
? >    transformed data to compare the two models (raw data Gaussian vs
? >    b?ta)?
? 
?   I would think so.
? > 
? >    If so, the b?ta 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
? >    b?ta-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 ??x small, as for log (proportional)
? * for intermediate values, linear change in probability with
? slope ? ?/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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Emmanuel CURIS
                                emmanuel.curis at parisdescartes.fr

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