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

Emmanuel Curis emm@nuel@curi@ @ending from p@ri@de@c@rte@@fr
Wed Sep 19 09:30:14 CEST 2018

```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.

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 ?

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)

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)?

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 ?

Would it be easier to use a log link and expression changes in the
scale as percent changes on the mean?