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The Wikipedia page for the Pearson product-moment correlation coefficient has a section on variants of the idea. This includes the reflective correlation coefficient, which has had a citation needed tag since 2011.

$$R_{\text{reflective}} \left[X, Y \right] = \frac{\mathbb{E}\left[ XY \right]}{\sqrt{\mathbb{E}\left[X^2\right]\mathbb{E}\left[Y^2 \right]}}$$

Trying a few search engines, I have not found a plausible origin of this statistic. While its construction is quite straightforward from the Cauchy-Schwarz inequality, I would rather cite it if possible.

I followed the link provided by Sycorax, and quickly searched each result for any sign of citations. Here are the results of that search.

  • Higgens 2008 has a document preview available, but I do not currently have access to the whole document. The term "reflective" appears in the summary of tables including "Table 5. Descriptive Statistics: Reflective" and "Table 6. Merged Responses for the HQPD Characteristic: Reflective".
  • Li 2012 is not accessible to me. The document preview mentions correlation, but nothing about a "reflective" correlation.
  • Zhu 2014 mentions a "reflective correlation coefficient" but does not provide a citation.
  • Chen 2015 mentions and mathematically defines a sample "reflective correlation coefficient" as found in the Wikipedia article. They do not cite a source for it.
  • Nilsson et al 2015 give a definition of the reflective correlation coefficient that matches the Wikipedia article. They cite an IEEE standard that I have not been able to access yet.
  • Buckley and Doyle 2017 mention the term "reflective" multiple times in their paper, as well as the term "correlation". But here they appear to be independent usages.
  • Miyazawa 2017 mentions and uses a "reflective correlation coefficient" in a way consistent with the Wikipedia page, but does not provide a definition or citation.
  • Hodžić and Namas 2017 is beyond my current access. The available abstract and list of references is not clarifying here. But a related paper with the same authors and similar title mentions the "reflective correlation coefficient" as related to the Pearson correlation. No citation is given.
  • Rui et al. 2017 mention "reflective correlation coefficient", however it is not clear prima facie whether this is physical reflection or not. They do not cite a source for this term, nor give the equation of interest.
  • Malevich 2017 is not accessible to me. The document preview suggest that various correlations are calculated, but gives not hint of any of them being "reflective".
  • Miyazawa 2017 mentions a "reflective correlation coefficient" in a usage that appears consistent with the wikipedia article, but skimming the article did not show any definition or citation.
  • Kowalczyk et al 2018 mention "reflective correlation" in parentheses to "Pearson correlation", but do not cite, define, or clarify this term.
  • Nilsson 2018 which appears to define the reflective correlation coefficient, but does not cite a source.
  • Noda 2018 define reflective correlation in the same way as the Wikipedia article. They do not cite a source, but rather link to a Science Direct summary that does not contain the term.
  • Cordova 2018/2019 has an abstract in English that suggests the term "reflective" was used in a psychological rather than statistical sense. An equation appearing equivalent to the reflective correlation coefficient does not appear in the thesis.
  • Tølløse et al. 2021 uses the term "reflective correlation coefficient in the same sense as the Wikipedia page, however they give no citation for it.
  • Miyazawa 2022 mentions their use of a "reflective correlation coefficient", but an explicit definition or citation is not given.
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    $\begingroup$ Re "plausible origin:" this is the Pearson correlation for any pair of zero-mean variables. $\endgroup$
    – whuber ♦
    Commented Mar 21, 2022 at 18:37
  • $\begingroup$ @whuber Mathematically, that is right. Did Bravais/Galton/Pearson say this? Or someone else? $\endgroup$
    – Galen
    Commented Mar 21, 2022 at 18:40
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    $\begingroup$ Not that I'm aware of--I haven't read all their works. But this is such an obscure term, nobody should use it without first defining it; and its simple relationship with the Pearson coefficient means little analysis needs to be offered: anyone familiar with the latter will already know all the important properties of this variant. $\endgroup$
    – whuber ♦
    Commented Mar 21, 2022 at 18:41
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    $\begingroup$ @whuber All that is a agreeable to me. Extrapolating beyond what you said, it is possible that no citable source ever bothered to name this statistic. The name "reflective correlation" may have been an idiosyncratic choice by a contributor to Wikipedia. I will leave the question up in case there is something with historical precedence that can be cited. $\endgroup$
    – Galen
    Commented Mar 21, 2022 at 18:55
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    $\begingroup$ On my computer, a Google Scholar search turns up 28 papers using the exact phrase "reflective correlation coefficient." Few of the articles are freely accessible, so I can't proceed much further to determine which ones are relevant, if any. But if you can get access to them, it should be a simple matter of reading to determine which one(s) are first, or if one article cites an earlier article not searched by Google Scholar, etc. $\endgroup$
    – Sycorax ♦
    Commented Mar 21, 2022 at 20:46

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User Skbkekas in Wikipedia created the description of reflective correlation on Dec 4 2009. You can “talk” to him on Wikipedia, and ask why he called it such.

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  • $\begingroup$ While a useful piece of guidance, it is not an answer to the question. perhaps demote it to a comment? $\endgroup$
    – Galen
    Commented Mar 21, 2022 at 19:12
  • $\begingroup$ You asked who proposed the term, and I answered. $\endgroup$
    – Aksakal
    Commented Mar 21, 2022 at 19:13
  • $\begingroup$ I guess this is an indication that I did not make the importance of citability clear enough. Thanks for the feedback. $\endgroup$
    – Galen
    Commented Mar 21, 2022 at 19:15
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    $\begingroup$ I wouldn’t assume it is citable. Skbkekas made up a term, only he knows why $\endgroup$
    – Aksakal
    Commented Mar 21, 2022 at 19:18
  • $\begingroup$ Perhaps my fault, but we are not communicating well. I am not assuming the origin of the term is citable. I want to cite the origin if it is citable. $\endgroup$
    – Galen
    Commented Mar 21, 2022 at 19:28

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