An asymptotic partial correlation test for the Goodman-Kruskal lambda

Goodman and Kruskal introduced a measure of predictive association when predicting the category of a variable A from a category of a variable B. This measure, denoted lambda, is the asymmetric proportional reduction in error measure in predicting an individual's A category that can be eliminated by using knowledge of the B classification. It takes values on the unit interval, with a zero value meaning no predictive gain, while a value of unity indicates a perfect predictive association between A and B. A test of H(0): lambda = 0 versus H(1): lambda > 0 is analogous to a test for the significance of the correlation coefficient. A test of the partial lambda coefficient, which is analogous to a test of the partial correlation coefficient, answers the question of whether knowledge of an additional third (or higher) classification or categorical variable results in a significant increase in predicting the variable A. Suich and Turek developed an exact test for the partial lambda coefficient, but only for the situation where the predicted categorical variable A is dichotomous. The present paper completes the previous work by developing an asymptotic test where the predicted category A is any polytomous variable.