Goodman and Kruskal (1954) proposed an association measure (tau) for nominal variables based on variation measure a.
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Goodman and Kruskal (1954) proposed an association measure (tau) for nominal variables based on variation measure
a. Show V(Y) is the probability that two independent observations on Y fall in different categories (called the Gini concentration index).
Show that V(Y) = (J when ?+j?= 1 for some j and V(Y) takes maximum value of (J ? 1)/J when ?+j?= 1/J for all j.
b. For the proportional reduction in variation, show that E[V(Y|X)]= 1 ? ?i ?j ?2ij/?i+. [The resulting measure (2.12) is called the concentration coefficient. Like U, ? = 0 is equivalent to independence. Haberman (1982) presented generalized concentration and uncertainty coefficients.]
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a The formula for VY is VY 1 Jj1 j2 where j is the probability that a randomly selected observation ...View the full answer
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