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Let's say we have two continuous variables, $a$ and $b$, both being positive with $a\leq b$ and the outcome in a regression is their ratio. The typical solution is beta regression (see for example here). However, something started to bother me. If we divide them and apply a regression for the proportion, then $1/5$ will be the exact same as $1000/5000$... while my intuition is that they shouldn't be identical, as in the latter case we are much more sure that the proportion is $0.2$ than in the former (the sampling variability is much smaller).

Isn't it a problem for beta regression? Or I overlook something...?

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    $\begingroup$ You seem implicitly to be thinking of $a$ and $b$ as counts. In that case, there is an issue. Otherwise, our intuition about counts doesn't necessarily apply. $\endgroup$ – whuber Dec 1 '19 at 21:50
  • $\begingroup$ @whuber Hm. So you mean, that if they're integer, then there is a problem, but if they're continuous, then it's OK? (It is really hard to imagine: how can the answer depend on whether they're integer or not...??) $\endgroup$ – Tamas Ferenci Dec 1 '19 at 21:59
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    $\begingroup$ "Integer" covers much more territory than "counts." Count data typically have distributions closely related to Binomial or Poisson variables, whereas integer data could have any distribution whatsoever, depending on what kinds of measurement they are. $\endgroup$ – whuber Dec 1 '19 at 22:09
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    $\begingroup$ Take care with the logic! The first half of your restatement is correct, but not the second half. Count data are special. About any other kind of integer data I cannot tell you anything in general, so your "not otherwise" is unsupported. Regardless, adopting a conditional Beta distribution for a response is tantamount to supposing a continuous model will work reasonably well. In your 1/5 case that's an unrealistic assumption. Thus, the magnitude of $b$ provides some guidance about the advisability of using Beta regression, no matter what the data mean. $\endgroup$ – whuber Dec 1 '19 at 22:23
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    $\begingroup$ I would be wary of your intuition in this case; you can rescale your two continuous variables by dividing each by 1000 (converting, e.g., meters to kilometers) and the ratio will still be 1/5 - do you think the ratio is more accurate if the numerator and denominator are 1023 meters and 5214 meters respectively than if they are 1.023 km and 5.214 km respectively? $\endgroup$ – jbowman Dec 2 '19 at 2:41

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