Chapter14 - Logistic Regression STAT 563 Spring 2007...

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Logistic Regression STAT 563 Spring 2007
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General Linear Models Family of Regression Models Outcomes variable determines the choice of the model
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Binomial Distribution Example Assume 5% of the population has Coronary Heart Disease (CHD). If we pick 500 people randomly, how likely is it that we get 30 or more people with CHD? The number of people with CHD we pick is a random variable Y which follows a binomial distribution with N=500 and π =0.05 (when picking people with replacement). We are interested in the probability P[Y>=30]
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Binomial Distribution Probability Mass Function Let π denote the probability of ‘response’ (in this case, having a CHD) in a given and let Y denote the number of “responses” out of the N trials. Then the probability of outcome y for Y equals (called Binomial distribution) N y y N y N y Y P y N y ..., 2 , 1 , 0 , ) 1 ( )! ( ! ! ) ( = - - = = - π π
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Generalized Linear Model A Quick Introduction In a Generalized Linear Model (GLM), the outcome Y is assumed to be generated from a particular distribution function in the exponential family (a large collection of distributions)
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Generalized Linear Model (GLM) Definition A vector of observations y of length N is assumed to be a realization of a vector of iid random variables Y with mean μ . A set of covariates x 1 , x 2 ,……,x p defines a linear predictor: η = Σ β j x j The classical linear model can be written in a tripartite form:
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Classical Linear Model Random Component Y has an independent normal distribution with constant variance σ 2 and mean μ Systematic Component – Covariates x 1 , x 2 ,……,x p produce a linear predictor η = Σ β j x j The Link η = μ
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Generalized Linear Model Consider exponential family Normal distribution can be expressed in terms of an exponential family Replace the identity link by a monotonic differentiable function η= g( μ ) )] ( exp[ ) ( ) ( ) : ( i i i i i i g y y b a y f μ μ μ =
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Exponential Family Write Bernoulli as an exponential family Natural parameter = ln[ π/1-π] - - = - - = - = - π π π π π π π π π 1 ln exp ) 1 ( 1 ) 1 ( ) 1 ( ) : ( 1 y y f y y y a( π29 g( π ) b(y)=1 η= ln [π/1-π] Logit of π
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Exponential family Write Poisson as an exponential family Natural parameter=log( μ ) η =log( μ ) ) log exp( ! 1 ) exp( ! ) : ( μ μ μ μ μ y y y e y f y - = = - a( μ29 b(y) g( μ29
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Link functions Other suitable link functions for binomial data are the Probit g( μ29 = Φ -1 (μ29 Complementary log-log g( μ29= ln(-ln(1- μ )) The most typical is the logit function. GLMs with logit as the link function are called logistic regression models as we discuss now
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Example Data AGE in years Presence or absence of evidence of significant coronary heart disease (CHD) for 100 subjects, coded as 0 (absent) and 1(present) AGRP, age group variable ID, identifier variable Goal Explore the relationship between age and the presence or absence of CHD
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