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The textbooks I have differentiate between nominal, ordinal, interval and ratio scales. The ordinal scale is quite popular in the wild, used for basically all subjective data, and also for dividing interval and ratio scales into better digested bins.

But when dealing with the data itself, the need for different "exceptions" arises. Sometimes it is just different kinds of null flavors, which can be written off as a meta data issue. But there are also more interesting cases. I have encountered satisfaction measurements where the overall satisfaction with a feature is considered to be "excellent", "good" or "bad" if the distribution of answers peaks at the expected points in the scale, but "polarising" if the distribution is U-shaped, and "unclear" if there is a flattish or weirdly multimodal distribution. Obviously, the "excellent", "good" and "bad" part is ordinal, but the last two are nominal. Other examples include a "prefer not to say" box in self-reported measures, if the reluctance of giving information is meaningful under the employed model.

I am pretty sure that we don't yet have qualitative methods to analyze these mixed scales. But I hope that the situation is at least recognized in theoretical texts, even if the only recommendation is "treat them like nominal". Also, while a mixed statistical analysis is difficult, maybe there are at least visualization approaches capable of representing the true complexity of the situation. But I don't know how to look for such literature, since I have never encountered a name for this kind of scale.

Does somebody know of a name (even if it is not widely established) and can point to sources which define it and deal with it?

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One option is create new features. One feature that is completely nominal, and another feature that is completely ordinal.

Otherwise, the scale types are "rounded down". There is no danger in treating ordinal data as nominal. However, treating truly nominal data as ordinal can cause analysis problems.

There is not much literature on mixing scale types because that could easily lead to misinterpretation of results.

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