What problems, if any, would exist if I were to treat a dependent variable with relatively low cardinality (e.g. 10 distinct values) as continuous versus binary (the latter requiring that I create some threshold to "binarize" the column)?

For example, I have a dependent variable consisting of points assigned to answers on a survey. There are several thousand respondents in my survey and for this particular variable the points are distributed somewhat uniformly, if not somewhat right-skewed.

Does this warrant a different type of regression, e.g. Poisson?

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    $\begingroup$ There are many problems with using linear models in that context, as discussed at length elsewhere on this site. Use a methods that is dedicated to ordinal response data, e.g. the proportional odds ordinal logistic model, which is a generalization of the Wilcoxon test. See my RMS book and course notes for more. $\endgroup$ Jan 27 '20 at 21:28

The obvious problem behind using a Gaussian to model count (or ordinal data) is that we might get negative values. In addition to that, we might have the issues of predicting decimal numbers instead of integers and of the linear model having a "fixed tolerance" as the estimated mean gets larger while with a Poisson-like regression model we would be more flexible because the estimated variance and the mean are proportional to each other. In short, both the statistical inference as well as the predictions from that model are problematic.

Also, without going too much detail, binning a continuous variable is a plainly bad because it adds subjectivity to our analysis as well as lowers the statistical power of our models. I will not expand on that because CV.SE already has a couple of great threads on that. I would recommend checking the threads on: What is the effect of dichotomising variables? and When should we discretize/bin continuous independent variables/features and when should not?. They outline the strong consensus for avoiding binarisation/dichotomising continuous variables.

That said, Poisson-like regression models (e.g. Poisson regression, Quasi-Poisson, Negative Binomial, etc.) are not a panacea themselves. In a situation like the one you describe they are suboptimal because they can still give out of range predictions and most importantly do not account for additive nature of the ratings. (Off the bat, we might say that a model like a beta-binomial might fix the "range issue" but that's a stop-gap measure really.)

As a rating is an ordinal response, (i.e. an ordered categorical response variable) the function MASS::polr should be more appropriate; it implements the proportional odds logistic regression routine. Simplifying things a bit: through a proportional odds model instead of modelling the probability of response in a particular category (as we would do if we simply assumed a multinomial response without any ordering), we model the cumulative probability that the response is not greater than a chosen category. This is also the core point behind the proportional odds assumption that this model relies one; i.e. that the estimated "rate of change" across two response levels is the same regardless of which pair of outcomes we consider. Particular to the case mentioned, the "right-skewness" of the data is not a problem either. Ananth & Kleinbaum (1997) Regression models for ordinal responses: a review of methods and applications is a very accessible paper on the matter. A&K point to some example in SAS but a very comprehensive tutorial on the analysis of ordinal response variables using R can be found in the UCLA Stats consulting pages: here; I strongly recommend it as it offers a step-by-step explanation with code examples.

So to recap, for the data at hand, a proportional odds logistic regression is probably the most appropriate. It allows to avoid unnecessary dichotomisation of our data as well as encapsulate well the nature of ordinal response variable.


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