This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 subpopulations 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 noncarriers, comprising the remaining 100(1 )%. Assume that the pdimensional 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 normaltheory maximum likelihood or via logistic regression. The advantage of the normaltheory approachlikelihood or via logistic regression....
View
Full
Document
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
 Daniels

Click to edit the document details