Interactive · from Volume 1
Polychoric and polyserial
Two items measuring the same thing, answered on a rating scale. Pearson treats the category numbers as if they were quantities, which they are not, and pays for it. Polychoric estimates the correlation between the continuous things the items stand for. Change the number of categories and watch the two diverge.
At two categories this is the tetrachoric case. Push it up and Pearson climbs toward the truth, which is why treating a long rating scale as continuous is usually forgiven and a short one is not.
Two-step estimation, the same as R's polycor::polychor and polyserial: the thresholds come from the observed marginals, then the correlation is found by maximum likelihood with those thresholds fixed. Checked against polycor on this data at 2, 3, 5 and 7 categories; the polychoric agrees to four decimal places.
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