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Let D denote the event that a subject has a certain disease. Let S denote the event that a subject is sampled from the population under study. Let the probabilities related with this sampling be r0 and r1 where r0  = P(S|D) and r1= P(S|Dc). Note that Dc denotes the complement of D, which refers to the event that a subject does not have the disease under study. Suppose a  logistic regression holds for probability of the disease, that is P(D|x) = exp(α+Xβ) / (1+exp(α+Xβ)).  i.    Using Bayes Theorem, show that P(D|S,x) also follows a logistic regression model with the same effect coefficients β but with a different intercept, namely α* = α +log(r0/ r1).  ii.    Comment on what the result in (i) means in practice?

Let D denote the event that a subject has a certain disease. Let S denote the event that a subject is sampled from the population under study. Let the probabilities related with this sampling be r0 and r1 where r0  = P(S|D) and r1= P(S|Dc). Note that Dc denotes the complement of D, which refers to the event that a subject does not have the disease under study. Suppose a  logistic regression holds for probability of the disease, that is P(D|x) = exp(α+Xβ) / (1+exp(α+Xβ)).  i.    Using Bayes Theorem, show that P(D|S,x) also follows a logistic regression model with the same effect coefficients β but with a different intercept, namely α* = α +log(r0/ r1).  ii.    Comment on what the result in (i) means in practice?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 29E
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. Let D denote the event that a subject has a certain disease. Let S denote the event that a subject is sampled from the population under study. Let the probabilities related with this sampling be r0 and r1 where r0  = P(S|D) and r1= P(S|Dc). Note that Dc denotes the complement of D, which refers to the event that a subject does not have the disease under study. Suppose a  logistic regression holds for probability of the disease, that is P(D|x) = exp(α+Xβ) / (1+exp(α+Xβ)). 
i.    Using Bayes Theorem, show that P(D|S,x) also follows a logistic regression model with the same effect coefficients β but with a different intercept, namely α* = α +log(r0/ r1). 
ii.    Comment on what the result in (i) means in practice?

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