How to prove variable is ordinal

Question

I want to run a model that determines the probability of self curing for loan customers who entered arrears. The thing is that when a customer enters arrears there are 3 ways he could go:

-not cure

-cure not by himself

-self cure

I was thinking of running either an ordinal logistic regression or a multinomial logistic, but I figured maybe I had to prove that my variable of interest is ordinal (let's say in the order: not cure, cure not by himself and self cure). Is there any test I can do to prove this?

Answer 1

Ordinality (here with "cure not by himself" in the middle) exists in the mind of the beholder and is established by consensus of experts. Then, given data on predictors and outcome you move on to examination of assumptions of various ordinal regression models, i.e., the proportional odds assumption. Or you allow for violations of their assumptions (e.g., partial proportional odds model). The only statistical assessment of ordinality comes from assessing whether $Y$ is ordinal with respect to a given $X$. This is all discussed in detail in Regression Modeling Strategies.

Answer 2

When we say that a variable $X \in \mathscr{X}$ is (at least) "ordinal" we mean that there is some binary relation $\succeq$ on that set of values that is a total order. So if you want to show that a variable is (at least) ordinal, you just have to construct a total order on the set of possible values for that variable and then explain why that ordering is a useful ordering for the problem. In the present case, if you set some reasonable criterion for ordering the possible outcomes, and show that this ordering is useful for analysis, that would be sufficient to "prove" that you have an ordinal variable.

As a slight complication to this, it is important to note that when we refer to a variable as "ordinal", we also often mean that it is not more than ordinal ---i.e., it is not an "interval" or "ratio" variable (see information on measurement scales). If this latter sense is intended then proof that a variable is "ordinal" would require you to prove the absence of any meaningful measurements of difference or ratio on the variable. You are not really going to be able to "prove" this in the strict mathematical sense, but you can certainly give a compelling argument for why difference and ratio operations on the variable are not meaningful.

Answer 3

I don't have enough reputation to comment. But let me give you my view.

Ordinal data is related to information organized in a particular order without indicating a specific relationship between each item. Items may be greater than or less than other items. The order of items is often defined by assigning numbers to them to show their relative position.

Nominal and ordinal data are qualitative data. But, let me provide you an example with rankings and ratings. A ranking establishes an order relationship for a set of items known as a weak order of elements (a sequence of ordinal numbers); the first is better than the second, the second is better than the fourth, and so on. A list of items ranked by its numerical score (real numbers), that list is called a rating. Therefore, every rating list generates a ranking list, but no vice-versa. So, from quantitative data (rating), you can obtain qualitative data, a ranking (ordinal data).