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There is clear meaning of Pearson product-moment correlation coefficient:

it is cosine of angle between two vectors based on variables.

Also there are 12 other ways to clearify the meaning of Pearson correlation.

Is there any similar for Polychoric correlation coefficient? Besides just "correlation between two ordinal variables" and without complicated formulas.

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    $\begingroup$ Check Chi's answer to this question for a start: Differences between tetrachoric and Pearson correlation and john-uebersax.com/stat/tetra.htm $\endgroup$
    – user20650
    Commented Dec 6, 2013 at 0:44
  • $\begingroup$ @user20650 I already have studied this link before. It is a good paper, but not complete and mainly aimed at Tetrachoric. I think there should be some textbooks with better description. $\endgroup$
    – drobnbobn
    Commented Dec 6, 2013 at 0:53
  • $\begingroup$ The referenced answer is presumably @chl's here: Differences between tetrachoric and Pearson correlation. $\endgroup$
    – gung - Reinstate Monica
    Commented Aug 16, 2017 at 20:27
  • $\begingroup$ Polychoric correlation is an inferred Pearson correlation. It is not correlation between ordinal variables, it is correlation between "underlying" continuous variables whereof the ordinal variables are seen as the result of binning. When there is only 2 levels in the ordinal variables, it is called tetrachoric correlation. $\endgroup$
    – ttnphns
    Commented Aug 16, 2017 at 20:31

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I found Kolenikov and Angeles "The Use of Discrete Data in Principal Component Analysis" working paper to be helpful (published version here if you have access). Slides here as well.

To quote the authors (from the help-file for their polychoric Stata command):

The polychoric correlation of two ordinal variables is derived as follows. Suppose each of the ordinal variables was obtained by categorizing a normally distributed underlying variable, and those two unobserved variables follow a bivariate normal distribution. Then the (maximum likelihood) estimate of that correlation is the polychoric correlation.

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  • $\begingroup$ It looks like it is good source. (Nevertheless, Principal Component Analysis and Factor Analysis are different things; and this is not a textbook.) $\endgroup$
    – drobnbobn
    Commented Dec 6, 2013 at 1:35
  • $\begingroup$ I managed to overlook that part somehow. Sorry! $\endgroup$
    – dimitriy
    Commented Dec 6, 2013 at 1:40
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    $\begingroup$ @gary Found the file using Wayback Machine and edited the link. $\endgroup$
    – dimitriy
    Commented Aug 16, 2017 at 20:22
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    $\begingroup$ Could you possibly add a brief summary for the answer to be self-contained? $\endgroup$
    – Tim
    Commented Aug 16, 2017 at 20:24
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    $\begingroup$ @Tim Let me know if that's better. $\endgroup$
    – dimitriy
    Commented Aug 16, 2017 at 22:57

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