Timeline for Exposition of "levels of measurement" for mathematicians
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jun 19, 2023 at 19:41 | comment | added | Galen | @MightyCurious Of course, hence I referred to it as a "classic example" rather than a theorem/proposition with a universal quantifier over embeddings. | |
| Jun 19, 2023 at 19:25 | comment | added | MightyCurious | @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. | |
| Jun 19, 2023 at 9:04 | comment | added | Nick Cox | 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. | |
| Jun 19, 2023 at 8:53 | comment | added | quarague | @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. | |
| Jun 18, 2023 at 20:40 | comment | added | Nick Cox | 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. | |
| Jun 18, 2023 at 19:34 | comment | added | Michael Hardy | 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. | |
| Jun 18, 2023 at 16:19 | comment | added | Galen | 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: $\operatorname{encode}(\text{king}) - \operatorname{encode}(\text{man}) + \operatorname{encode}(\text{woman}) \approx \operatorname{encode}(\text{queen})$. | |
| Jun 18, 2023 at 16:16 | comment | added | Galen | 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. | |
| Jun 18, 2023 at 10:01 | history | edited | Ben | CC BY-SA 4.0 | added 8 characters in body |
| Jun 18, 2023 at 7:30 | comment | added | Nick Cox | 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. | |
| Jun 18, 2023 at 7:27 | comment | added | Nick Cox | 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). jstor.org/stable/2684788 (find discussions too) and jstor.org/stable/2983326 raise many of the issues. | |
| Jun 18, 2023 at 7:21 | comment | added | Nick Cox | 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). | |
| Jun 18, 2023 at 7:16 | history | edited | Nick Cox | CC BY-SA 4.0 | deleted 1 character in body |
| Jun 18, 2023 at 7:00 | history | answered | Ben | CC BY-SA 4.0 |