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Is there a good expository account that explains and justifies Stevens's classification of "levels of measurement" into nominal, ordinal, interval, and ratio, that is comprehensible to all mathematicians?

Stephan Kolassa
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Michael Hardy
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  • Being comprehensible to all mathematicians seems like a very strong criterion that will be fragile to the problem of empirical induction. Fortunately many mathematicians will be sufficiently familiar with the requisite group theory and representation theory to understand its meaning and implications. – Galen Jun 17 '23 at 21:59
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    Why not Steven's 1946 paper itself? Is that not sufficiently expository for a mathematician? – Galen Jun 17 '23 at 21:59
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    @Galen : Partly I wondered whether it might include circumstances in which some level other than those four might make sense, but perhaps based on the same sort of reasons that justify those four. – Michael Hardy Jun 17 '23 at 23:27
  • Your question as written doesn't mention going outside of Steven's typology, but you can certainly explore that further. What is required to "do the thing that Steven did in 1946" is identify an algebraic structure that we think models data. A hot topic in deep learning are graph neural networks which are often designed to have permutation symmetry on the vertex and edge sets while being non-invariant to the connectivity properties of the input graph. But I wouldn't say that fits nicely with the word "level" in "level of measurement". I'm not aware of an interesting total order on structures. – Galen Jun 18 '23 at 05:31
  • Definitely Stevens, not Steven, so possible possessives are Stevens' or Stevens's but not Steven's – Nick Cox Jun 18 '23 at 07:32
  • @nickcox thank you for the correction on the possessive forms of "Stevens". It was a hasty oversight on my part. – Galen Jun 18 '23 at 13:23
  • Fine. It's a pity that there isn't more scope to edit one's own comments. – Nick Cox Jun 18 '23 at 13:46

1 Answers1

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The original exposition of these concepts in Stevens (1946) ought to be simple to understand for any mathematician, as would virtually all of the later literature that elaborates on these concepts. In this original paper, Stevens explains these measurement concepts by reference to their invariance under various types of mathematical transforms, leading to different types of "mathematical group structure". From Table 1 (p. 678) he notes the following types of invariance for the measurement levels (paraphrased from the table):

  • Nominal: Invariant to any permutation group $x' = f(x)$ where $f$ is a one-to-one substitution.

  • Ordinal: Invariant to any isotonic group $x' = f(x)$ where $f$ is any monotonic increasing function.

  • Interval: Invariant to any general linear group $x' = ax+b$.

  • Ratio: Invariant to any similarity group $x' = ax$.

This exposition, framed in terms of invariance properties to various types of functions, is precisely the type of exposition that would be most suitable for a mathematician (or other mathematically trained person). Indeed, I don't think it's a stretch to say that the original paper on this topic, and much of the follow-up literature, is written primarily for mathematicians/statisticians who are conversant with different classes of functions and invariance properties with respect to these functions.

Ben
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    This is bang on the question, but it might be helpful to add that (1) Stevens kept revisiting the main idea, with his last fling posthumously in his book Psychophysics (1975); (2) while there is some pojnt to these distinctions their use in some circles (chiefly parts of psychology and sociology) far exceeds (in the view of many statistical people) their total merit. For example, the distinctions can be used to explain why coefficient of variation (SD/mean) is sometimes useful for precipitation (a ratio variable), but meaningless for Fahrenheit or Celsius temperatures (interval scale). – Nick Cox Jun 18 '23 at 07:21
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    However, this terminology often goes with inhibitions and prohibitions that aren't always coherent in principle or helpful in practice, such as that you shouldn't take means of ordinal variables (but you should use methods geared to such data, some of which hinge on that taking the mean of ranks). https://www.jstor.org/stable/2684788 (find discussions too) and https://www.jstor.org/stable/2983326 raise many of the issues. – Nick Cox Jun 18 '23 at 07:27
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    Some of the literature misses simple points, so cats and dogs may be categories on a nominal scale, but their frequencies are integers and yield to generalized linear models. Sometimes absence is as revealing as presence in that many textbooks ignore this terninology altogether. – Nick Cox Jun 18 '23 at 07:30
  • Building on Nick's example of cats and dogs, we can also argue that biological evolution suggests a lot more structure between cats and dogs than just being distinct nominal variables. – Galen Jun 18 '23 at 16:16
  • And on the language side of things we know from the topic of word embedding that words often have more structure to them than just being distinct nominal categories. A classic example: $\operatorname{encode}(\text{king}) - \operatorname{encode}(\text{man}) + \operatorname{encode}(\text{woman}) \approx \operatorname{encode}(\text{queen})$. – Galen Jun 18 '23 at 16:19
  • You refer to "virtually all of the later literature". The "literature" that widely circulates, even in statistics graduate programs, is dogmatic: it says there are these four levels, and explains what they are, and doesn't give any arguments, and it shows up that way in courses on psychology, economics, geography, sociology, and probably all other fields that use statistics. – Michael Hardy Jun 18 '23 at 19:34
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    I am a geographer and I agree that NOIR is all over (too) many texts. I don't see it appearing much in econometrics books. Perhaps it does appear more in lower level books on statistics for economics or business. Most expositions seem to miss (to riff on a major example) that binary outcomes are not categorical in any inhibiting sense, but through 9, 1 coding allow means to be calculated and full analyses in terms of probability of each outcome. And logit and probit ideas were in the literature before 1946. – Nick Cox Jun 18 '23 at 20:40
  • @NickCox Isn't this precisely the point of these definitions? The mean of the ordinals themselves is not invariant under transformation with any monotonically increasing function and therefore should not be used. The mean of the rank however is invariant under such transformations and therefore can be used without problems. – quarague Jun 19 '23 at 08:53
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    Some writers agree with you. but it's backwards in my view to let the definitions determine your analysis. If I have grades say 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5 and 1,1,1,1,1,2,2,2,2,3,3,3,4,4,5 then the mean grade carries more information than any other summary, and it's perverse to reduce such data to ranks. Means are just what many do with grade-point averages or means of evaluations. The crux is (1) what is often said in texts (2) is it correct in any sense (3) is it helpful as a guide to analysis. I've seen people say: If the data are ranks, taking a mean is wrong, which is not helpful. – Nick Cox Jun 19 '23 at 09:04
  • @Galen This is a structure that an embedding may or may not have -- depending on the individual embedding. The example you're citing as been repeated a thousand times in the literature, regardless of the fact that this works for some works and some embeddings, and much less so for others. – MightyCurious Jun 19 '23 at 19:25
  • @MightyCurious Of course, hence I referred to it as a "classic example" rather than a theorem/proposition with a universal quantifier over embeddings. – Galen Jun 19 '23 at 19:41