I've checked related links to understand what a Polychoric Correlation measures, got some 404's and other unanswered questions. I've seen it used to determine independence between Nominal/Interval ( "Contunuized"*, aka, Jointly Normal ) on YouTube (https://www.youtube.com/watch?v=besaBez9giw&t=597s), as ways of measure correlation between (underlying continuous ) variables, and I've seen it used to describe the distribution of some (underlying continuous ) trait of fertility of cattle given data on cattle births (From John Uebersax' page ).

I (think I) understand the general underlying setup for the polychoric. We start with a paired discrete/Interval/Ordinal data set . We assume each data set is discrete output comes from an underlying Normal Distribution , and the pair of variables is jointly-normal. The poly/tetra choric is then the Pearson correlation between these underlying continuous variables . Is this correct? If so, does it have / can it be used for the purposes described in the above paragraph?

Edit: Also, are normality of each and joint normality required? How robust is it (i.e., how well does the test fare under departures from normality assumptions). Because doesn't joint normality equivalent to individual Normality?