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 Dec 6 '13 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 Dec 6 '13 at 0:53
  • $\begingroup$ The referenced answer is presumably @chl's here: Differences between tetrachoric and Pearson correlation. $\endgroup$ – gung Aug 16 '17 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 Aug 16 '17 at 20:31

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.

  • $\begingroup$ 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 Dec 6 '13 at 1:35
  • $\begingroup$ I managed to overlook that part somehow. Sorry! $\endgroup$ – Dimitriy V. Masterov Dec 6 '13 at 1:40
  • $\begingroup$ @DimitriyV.Masterov: Seems like link is dead. Any chance you have a newer one? $\endgroup$ – gary Aug 16 '17 at 20:14
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    $\begingroup$ @gary Found the file using Wayback Machine and edited the link. $\endgroup$ – Dimitriy V. Masterov Aug 16 '17 at 20:22
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    $\begingroup$ Could you possibly add a brief summary for the answer to be self-contained? $\endgroup$ – Tim Aug 16 '17 at 20:24

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