# Models for a discrete numerical response with only 5 distinct values

I have a response variable that takes on only 5 distinct values: -15, -5, 0, 5, 15. These values are obtained in a clearly categorical manner, meaning: if event A happens, then the outcome is 5, if event B happens, then the outcome is -15, etc... So the eventual response value for a certain observation isn't a result of some cumulative counting, but rather a pure assignment of a numerical value to a categorical outcome. With that being said, there's a clear ordering and a clearly defined distance between the numerical values being assigned to different categories, and that distance properly reflects the difference in their inherent values (unlike in some other cases of ordinal data, where only the order is known, but not necessarily the distance between the categories). Last note is that the marginal distribution of the response values is heavily asymmetrical here: the negative values (-5, -15) are much rarer than the non-negative ones, with 0 being the most frequent value by far, and 15 is the 2nd, and 5 is a distant 3rd:

My question is: what kind of modeling approaches exist out there for such a response type? Would it be the generalized ordered logit model, where I'd focus on modeling the probabilities of each distinct value, rather than modeling the value directly? But if so, could it account for the known distances between the categories? Or maybe there might be a Bayesian approach that allows modeling an expected value of an arbitrary discrete variable with 5 numerical values (based on each category's probability and designated value)?

PS: I have already tried a variety of classic linear regression techniques with a continuous response assumption, and the fit diagnostics are quite terrible, as expected. I'm attaching diagnostic plots from fitting a thin-plate spline regression with multiple predictors.

• It depends on what you're trying to answer. In general, I'd use a multinomial model with such data because there are only 5 possible values for the outcome. Even though their distance is known and there is ordering, I would treat them as categorical anyways unless there is a good reason for including the distance information (that's why I said it depends). Commented Apr 14, 2023 at 21:47
• My primary goal (or at least "my hope") is really to study the impacts of explanatory variables on the !expected value! of the response, so that it was possible to eventually say "as variable x increases by this amount, the expected value of the response will change by that amount". All the logit models (including multinomial, I believe) mostly focus on impacts to odds of one category vs the other, but I don't know if it'd be directly transferable to the interpretations as the impact on the expected value.. It could still yield point predictions, yes, but effect interpretation is trickier. Commented Apr 14, 2023 at 21:58
• You should model the data generating process and then see if there could be such an interpretation associated with it. You're dealing with a weird data generating process and chances are there is a latent continuous process that led to such data. For example, why other values are not possible? If you are dealing with an instrument that can't measure other values then the underlying truth is continuous and should be modelled as such. Then you should connect the latent process to the observed discrete data. Commented Apr 14, 2023 at 22:22

This seems like a situation where ordinal regression would be appropriate. See Chapter 13 of Frank Harrell's Regression Modeling Strategies or this UCLA web page for the commonly used ordinal logistic regression.

Ordinal regression ignores the "known distance between the categories," but it's not immediately clear how important those known distances are, except for their ordering. It's not clear (to me, at least) what the "generalized" ordered logit model would add to that, and so far as I can tell that type of model also doesn't take the "known distance between the categories" into account.

First, you aren't limited to a proportional-odds model. The R ordinal package provides several alternatives for ordinal regression.

Second, the latent-variable interpretation of a cumulative-link ordinal regression provides something very close to "as x increases, the expected value of the response will change by ...". The idea is that there is some continuous underlying (latent) unobserved variable associated with the predictors, and that when you cross a threshold you jump into the next observed category. See Section 2.4 of the vignette on cumulative link models for the ordinal package, and illustrations as in Figure 1 of the vignette.

In that interpretation, a regression coefficient represents the change in the expected value of the latent variable for a change in the associated predictor. The intercepts are related to the thresholds. With your data, analysis might benefit from the ability to specify a "symmetric" structure of thresholds in the ordinal package (although I don't have experience with that myself).

This presumably can be handled by a Bayesian approach under a specific probability model, but that's outside my expertise.

• I just highly doubt that the proportional odds assumption will be satisfied, in no small part due to the fact of a clear non-equidistance between the response "categories" as far as their designated numerical value, hence mentioning the generalized one. Granted, I haven't run either just yet. My biggest concern about running the logistic approaches here is that they end up solely focusing on probabilities of categories, disregarding the numerical values designated to those categories. My main hope is to be able to interpret the effect of predictors on the !expected value! of the response Commented Apr 15, 2023 at 3:00
• By that last part, I mean I'd really like something along the lines of "as x increases, the expected value of the response will change by ...". I don't think logit models will grant me that capability. They could help with a point prediction, yes - I could just estimate point probability predictions and then do a weighted sum with the designated values to get the expected value - but not with the interpretation of the direct effect on the expected value.. Commented Apr 15, 2023 at 3:09
• @UsDAnDreS I've edited the question to address some of your concerns. You aren't restricted to a proportional-odds assumption, and the latent-variable interpretation of an ordinal model seems very close to the way you want to interpret regression coefficients.
– EdM
Commented Apr 15, 2023 at 9:15
• Thanks so much, really appreciate your help with this! I've been studying the materials and, although it doesn't amount to that pure "change in x -> change in expected response value" interpretation (latent variable is somewhat little comfort), I think it could still be turned into something I wanted. I will still wait another day or two just in case someone could suggest something else, and then will mark it as solved/most helpful, giving you credit. Commented Apr 16, 2023 at 20:04