[R-sig-ME] Interpreting Estimates from GLMM following a beta-distribution (glmmTMB)

[Ben Bolker]

On Mon, Nov 9, 2020 at 1:12 PM Jan Velimsky via R-sig-mixed-models
<r-sig-mixed-models using r-project.org> wrote:
>
> Dear R project mixed models users,
>
> I am struggling to interpret the estimates from a GLMM following a
> beta-distribution with a logit link. There is not much literature
> regarding the interpretation of this special case.
>
> We have been estimating a GLMM model following a beta-distribution (with
> a logit link) with the glmmTMB-package. The model consists of factors
> influencing referendum turnout in German municipalities. The primary
> units of investigation are city districts nested in referendums nested
> in cities. The dependent variable (0-100) has been transformed to the
> unit interval 0-1.

  Is there a reason you can't get information on the voter base for
each municipality? That would make the model into a more standard
binomial model, which would have two advantages: (1) municipalities of
different sizes would be appropriately weighted; (2) it would be a
more familiar model type (less convincing of reviewers, wider range of
available software, etc.)

>
> Here an example model:
>
>   glmmTMB (ref_turnout ~ unemployment + contestation+ (1|
> city/referendum), family=list(family="beta", link ='logit'), data = ml)
>
>
> Results example model:
>
> Estimate      Std. Error    zvalue     Pr(>|z|)
>
> (Intercept) -0.583             0.131      -4.455     8.4e-06***
>
> unemployment rate (in%) -0.067              0.002      -30.397    <
> 2e-16 ***
>
> contestation 0.008              0.003        2.398     0.0165 *
>
> All explanatory variables are grand mean centered.
>
> Taking into account the link function and the parameterization of beta
> regression, my interpretation for the effect of unemployment would be:
> One percent increase from the average unemployment-rate in districts of
> German municipalities (grand-mean) is associated with a  0.067 unit
> decrease from the overall mean of the participation rate in referendums
> (log odds)
>
> 1) Is this interpretation correct?

  Yes

> 2) Are there more intuitive options for interpretation (e.g. with odd
> ratios or marginal effects )

  You could exponentiate the coefficient and conclude that there was a
proportional change of 0.064 (1-exp(-0.067)), or 6.4%, in the *odds*
of turnout per 1% increase in the unemployment rate (the numbers
(0.067, 0.064) are similar because the coefficient is <<1).

  Because your baseline rate is intermediate (exp(-0.583) is 0.56, in
the range 0.3-0.7 where the following rule of thumb applies) you could
say there is *approximately* a decrease of beta/4 = 1.7 percentage
points, in turnout per percent increase in unemployment (this is an
*absolute*, linear change rather than a proportional change). (See
e.g. Gelman and Hill's book.)

   You should be a little careful about the effect of transformation
and variation: see the last ("Bias adjustment") section of the
"transformations" vignette from the emmeans package.



> Thanks a lot for your help!
>
> Jan
>
> --
> Jan Velimsky, M.A.
> Wissenschaftlicher Mitarbeiter
> Lehrstuhl für Politische Systeme und Europäische Integration
> Geschwister-Scholl-Institut für Politikwissenschaft
> Ludwig-Maximilians-Universität München
>
> D-80538 München
> Oettingenstrasse 67
> Tel. 0176 73292389
>
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