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In a comment on another question, clarification was asked if the topic under discussion was "count proportions" or "continuous proportions", and a followup indicated that the difference was critical information (to the topic of logistic/binomial vs. beta regression).

What is the distinction between the two, and where does the distinction matter? What are important things to keep in mind when working with "count proportions" versus when working with "continuous proportions"?

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Perhaps an example would help. Suppose you observe a number of people and count how many of them are women. The resulting proportion is what is called a count proportion and takes on values between zero and one but only $n+1$ of them where $n$ is the total number you observed. Suppose you buy a sausage from your local supermarket and observe on the label that it is 80% pork that is an example of a continuous proportion and could take on any value between 0 and 100.

The distinction in modelling is that in the first case it is meaningful to predict the probability of a random person being a woman (logistic regression) but in the second case that is not a sensible question and something else (often beta regression) would be preferred.

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    $\begingroup$ Agreed, but the difference can be less than implied. Men and women are counted, in principle and in practice. Clay, silt and sand particles are, whatever the principle, not counted in practice. The probability of being (e.g.) clay (rather than silt or sand) then refers to notional small amounts of sediment or soil. Whether land is (say) rural or urban is a problem in principle in measuring areas (but in practice this may still reduce to some kind of counting of small unit areas!). But the principle is that count proportions are discrete and continuous proportions continuous. $\endgroup$ – Nick Cox Oct 10 '16 at 15:18
  • $\begingroup$ @Nick even if the constituents are particles, we can't be merely counting them unless you introduce the unrealistic assumption that particles of silt sand or clay must all be the same (the same mass if we're measuring proportion by mass, say) both within and across types. As such continuous proportions may often be fundamentally different from count proportions, were "1" typically doesn't alter much in size. Certainly some properties may be shared (not least because both are on the unit interval) but in some ways they will be or will often be actually different in important ways. $\endgroup$ – Glen_b Oct 10 '16 at 23:40
  • $\begingroup$ Indeed the example is muddy, because although particles may be discrete, their mass not their number is what we want to measure. I wanted an example where entities are discrete, but we measure in practice, to make the point that the distinction between counting and measuring is a little fuzzy. A better example would be welcome. $\endgroup$ – Nick Cox Oct 11 '16 at 8:10
  • $\begingroup$ I would have to say @NickCox that I struggled to think of an absolutely cast-iron, 22 carat, bullet-proof example and as for a definition I gave up. $\endgroup$ – mdewey Oct 11 '16 at 11:00
  • $\begingroup$ @mdewey Sympathies. As you know, in statistics something is considered discrete if in someone's judgement the discreteness is evident and important for the immediate purpose, and continuous otherwise, and vice versa. In quite another context a pedologist offered the definition that a soil is anything so called by a competent authority and so blew a raspberry (British English that you and I will understand, don't know how far it will translate) at anguish over definitions. $\endgroup$ – Nick Cox Oct 11 '16 at 11:18

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