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Unformatted text preview: STA 7249: Generalized Linear Models Assignment 2 1. ( Logistic discrimination . Exercise 4.12, McCullagh and Nelder, 1989. See exercise 5.15 for a generalization.) Suppose that a population of individuals is partitioned into two sub-populations or groups, G 1 and G 2 , say. It may be helpful to think of G 1 in an epidemiological context as the carriers of a particular virus, comprising 100 % of the population, and G 2 as the non-carriers, comprising the remaining 100(1- )%. Assume that the p-dimensional covariate vector X has the following distributions in the two groups: G 1 : X N p ( 1 , ) G 2 : X N p ( 2 , ) . Let X be an observation made on an individual drawn at random from the com- bined population and let Y represent the individuals group membership (1 or 2). The marginal odds (ignoring X ) that the individual belongs to G 1 are / (1- ). Show that the conditional odds of belonging to G 1 given X = x can be written in the form odds( Y = 1 | x ) = 1- exp( + x T ) , and express and as functions of 1 , 2 , and . Note that this result implies that the conditional distribution of Y given X = x has the logistic regression form: log odds( Y = 1 | x ) = * + x T , where * = logit( ) + . Comment: If and are known or can be estimated, then we may predict the group membership of the individual by Y = 1 if the posterior odds are greater than 1 and as Y = 2 otherwise (assuming that the costs of both types of possible misclassification error are equal). Given training data consisting of measurements of X for n 1 individuals drawn at random from G 1 and n 2 individuals drawn at random from G 2 , and can be estimated either via normal-theory maximum likelihood or via logistic regression. The advantage of the normal-theory approachlikelihood or via logistic regression....
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This note was uploaded on 01/15/2012 for the course STA 7249 taught by Professor Daniels during the Spring '08 term at University of Florida.
- Spring '08