Another approach to generalizing logistic regression to multinomial data has been developed by McFadden (1974). This method compares each category to all others, combined. McFadden, an economist, was concerned with modeling consumer choices, made to the exclusion of all other options. This explanation motivates models of the form log Pr[y=i | x] =ix Pr[ y = i | x ] for each choice i. (a) Solve for Pr[y = i] and show that this model is over-para- meterized. (What happens when you add the same vector to all s?) One solution to this problem is to restrict 1 = 0. (b) Compare this restricted model to the approach taken in Sect. 10.2 where the reference category corresponds to y = 1. Are these two approaches the same? How are they different? (c) Does this approach support independence of irrelevant alter- natives?

.9 Another approach to generalizing logistic regression to multinomial data has been developed by McFadden (1974). This method compares each category to all others, combined. McFadden, an economist, was concerned with modeling consumer choices, made to the exclusion of all other options. This explanation motivates models of the form log(Pr[y=ix]Pr[y=ix])=ix for each choice i. (a) Solve for Pr[y=i] and show that this model is over-parameterized. (What happens when you add the same vector to all s?) One solution to this problem is to restrict 1=0. (b) Compare this restricted model to the approach taken in Sect. 10.2 where the reference category corresponds to y=1. Are these two approaches the same? How are they different? (c) Does this approach support independence of irrelevant alternatives