58. * Goodman and Kruskal's tan, also called the concentration measure, is a PRE measure of association for nominal vanables. We illustrate it for Table 8.35. Rule 1 for predicting attitude without using religious preference is based on the marginal distribution for that response variable. To illustrate. since the proportion 381/1166 = .327 of the sample is classified Favor, we guess a classification of Favor for that same proportion of the sam- ple. We predict Oppose for the remaining 785 people. For rule 2. in predicting attitudes. we now do it for the conditional distribution of the response variable within each level of religious affiliation. For example, we predict Favor lor a random sample of 103 of 290 Liberal Protestants and predict Oppose for the other 187 Liberal Protestants.

a) Show that the number of errors expected in predicting attitude using rule 1 is E =381(785/1166) +-785(381/1166) = 513.0

b) Show that the expected number of errors in predicting attitude using rule 2 is E = 490 4

c) Show that tau, the proportional reduction in error, equals .044. (Properties of this mea- sure include (i) 01, (ii) = 0 corresponds to independence. (ii) = 1 when for each category of the predictor variable, all the observations occur in one category of the response variable.)