Hi all,
I'm having some theroretical ponderings with polytomous items and item-total correlation. In the binary case, we have a modification of Pearson correlation of the item X and scale Y
(M1-M0)*var(X)^(1/2)/var(Y)^(1/2)
Does anyone know a polytomous modification/generalization of this form of coefficient? There could be something like [(M1-M0) + (M2-M1)...] x var(X)^(1/2) in the denominator.
See better the formulae in the appendix.
Hello Jari,
The point-biserial formula was derived as a short cut to the usual definitional or computational version of the Pearson product-moment correlation formula in the days before statistical computing software or cheap/free calculators. Both the short-cut and PPMC formulae yield the same numeric result. In the case of polytomous items, so long as one is willing to assume interval scale strength to the polytomous scores, the usual PPMC formula will also apply. If interval scale isn't a realistic claim for the data set, then a rank-order correlation would need to be used.
That doesn't address your question directly, but I strongly suspect that the corresponding "short-cut" polytomous generalization version would be more trouble to code than the basic PPMC formula. However, as an intellectual exercise, it may be of interest to some folks.
Good luck with your work!
Thank David,
It really may be more or less an intellectual problem rather than a practical/computational one. I have two reasons to ask the other form. 1) By using the (M1-M0) version of PPMC formula, it is easy to show why PPMC always underestimates the items discrimination power and why it can reach the perfect 1 with only one specific data structure. 2) this version makes it easy to show why the item-rest correlation always gives a lower estimate than PPMC. These can be proven in the dichotomous case. If finding the version for the polytomous case, it may make it possible to show it in general case too. Hence...
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