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Recently, I got my hands on modelling proportions [0,1]. Due to data type many of my variables are 0 and 1 inflated. Some of them are delicately affected by the bound values and some are heavily. I performed three types of simple beta regression:

  1. using betareg, fixed dispersion using y ~ x, logit link
  2. using betareg, variable dispersion using y ~ x | x, logit link
  3. using gamlss, BEINF family, logit link

Nevertheless, the results are substantially different between these three methods (i.e. they switch signs or shape). How we know which model for each variable in this case? The second quesiton is how we can compare the magnitude of several variables modeled by simple beta regressions if scalling is not applicable in this case?

@EDIT Example of predictions: enter image description here

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I would recommend to compare model predictions between these three specifications, in-sample and/or out-of-sample. You can compare overall fit, e.g., using information criteria (AIC, BIC) or other scoring rules (log-likelihood, CRPS, etc.). Or you can visualize predicted means or predicted probabilities across x. In some of these models rather different coefficients (in terms of signs or size) may in fact lead to more similar predictions than expected. Finally, nested model specifications can also be compared by likelihood ratio tests, for example.

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  • $\begingroup$ Thank you. I added prediction example for an outlook. As u can see curves are way different between each others and that happens in many cases. Model 1) becomes flat, model 2) goes up, model 3) goes down. What could be the reason and how to deal with it? $\endgroup$
    – Tom
    May 14, 2020 at 8:56
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    $\begingroup$ I would guess that the slopes of Model 1 and 3 do not differ significantly. Thus, these differ mainly in the way the upper boundary is handled. Model 2 is different because it captures the changes in precision which appear to be appropriate here. $\endgroup$
    – Achim Zeileis
    May 14, 2020 at 9:38
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    $\begingroup$ As yet another model you could also use a heteroscedastic censored Gaussian model. This is available in the crch package via crch(y ~ x, left = 0, right = 1, ...). $\endgroup$
    – Achim Zeileis
    May 14, 2020 at 9:39
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    $\begingroup$ Re: How can I tell that Model 2 captures the changes in precision appropriately. Maybe I need to phrase that more carefully: From the visualization it appears to be obvious that the precision changes from low on the left (where the points have high variance) to high on the right (where the variance is rather low). So trying to capture this effect is appropriate. Whether the model does this sufficiently well is, of course, not visible from the plot alone. $\endgroup$
    – Achim Zeileis
    May 14, 2020 at 21:43
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    $\begingroup$ Yes, exactly. But as a first quick check you could look at AIC/BIC. I wouldn't be surprised if the homoscedastic models (1, 3, and first crch model) are noticeably worse than the heteroscedastic models (2, and second crch model). $\endgroup$
    – Achim Zeileis
    May 14, 2020 at 22:55

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