Yules Q (cf. Exercise 2.7.6.) is one of many measures of association that have been proposed for

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Yule’s Q (cf. Exercise 2.7.6.) is one of many measures of association that have been proposed for 2 × 2 tables. Agresti (1984, Chapter 9) has a substantial discussion of measures of association. It has been suggested that measures of association for 2 × 2 multinomial tables should depend solely on the conditional probabilities of being in the first column given the row, i.e., p11/p1· and p21/p2·, or, alternatively, on the conditional probabilities of being in the first row given the column, i.e., p11/p·1 and p12/p·2. Moreover, it has been suggested that the measure of association should not depend on which set of conditional probabilities are used. Show that any measure of association f

p11 p1·

, p21 p2·



can be written as some function of the odds g

p11 p12

, p21 p22 

.

Show that if f

p11 p1·

, p21 p2·



= f

p11 p·1

, p12 p·2



for any sets of probabilities, then g(x, y) = g(ax, ay) for any x, y, and a.

Use this to conclude that any such measure of association is a function of the odds ratio.

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