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There are many articles explaining the difference between regression and ordinal classification, most of them mentioned that regression is for continuous response while ordinal classification is for discrete response. However, I think it in another way, and that's my question:

As more and more discrete values added to the response set, it is more and more approximate a continuous response. Shall we really separate regression and ordinal classification as two different worlds? What models lie between regression and ordinal classification? What if i have a large number of discrete values in response variable (but still no rigorous continuous), integer 1 to 10000 for instance, what kinds of model can handles this issue? enter image description here

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  • $\begingroup$ You could search for limited dependent variable (e.g. Tobit), count data, ordered or multinomial models. $\endgroup$ – Christoph Hanck Mar 18 '19 at 15:13
  • $\begingroup$ ordered or multi-nominal models are good when the available values for response variable are "not to many", etc. 1~5, 1~10 is fine. What if the available values are 0~1000? $\endgroup$ – Master Shi Mar 18 '19 at 15:31
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As more and more discrete values added to the response set, it is more and more approximate a continuous response

No, you're wrong about this.

We use integers as group labels out of convenience. We could just as easily use strings as labels. Our problem is actually about predicting to which set an observation belongs, not a number per se.

Adding more and more categories does not make the problem any closer to regression, and so the two are still rightfully different.

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  • $\begingroup$ Sorry i correct my question: not regression vs. classification; it is regression vs. ordinal classification (ordinal regression). I understand what you say, $\endgroup$ – Master Shi Mar 18 '19 at 15:42
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So if I understood you correctly, what you are thinking is in a problem in which the response variable is formed by integer numbers, for example, as you said, numbers from 1 to 1000. For dealing with this kind of situations there is an specific kind of generalized regression models, called binomial models. In those models, you assume that the distribution of your response variable is a binomial distribution, which is the one that best fits this integer response situation.

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