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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, 2019 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, 2019 at 15:31
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    $\begingroup$ Maybe you mean "ordinal regression", not "ordinal classification"? $\endgroup$
    – ttnphns
    Jul 17, 2021 at 11:02
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    $\begingroup$ @MasterShi nominal (polytomous; unordered categorical variables) run into trouble (require too many parameters to be estimated) when there are many categories of Y. Ordinal models do not have that problem. $\endgroup$
    – Frank Harrell
    Jul 17, 2021 at 11:17

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All continuous interval-scaled variables are ordinal, but not all ordinal variables are interval-scaled. An ordinal variable may have any number of distinct values, all the way to having no ties in the data. See the ordinal regression for continuous Y chapter in RMS. The R rms package orm function can computation-time-wise handle more than 6000 distinct ordinal values for Y. That is the number of intercepts in the model plus one. The number of slopes stays the same no matter how many intercepts you have, and the intercepts are forced to be in order, hence the model is not overparameterized.

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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, 2019 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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