# USC PM 518a - 2011-06-19 likelihood quantities (23 pages)

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## 2011-06-19 likelihood quantities

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## 2011-06-19 likelihood quantities

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Lecture Notes

Pages:
23
School:
University of Southern California
Course:
Pm 518a - Statistical Methods for Epidemiological Studies

Unformatted text preview:

LIKELIHOOD QUANTITIES FOR BINARY DATA Some review of Chapter 3 and Chapter 4 H conditional likelihood for binary data that encompasses the conditional logistic Poisson likelihoods Expressions for score and information 1 SETTING Grouped time shrink intervals to get continuous time Focus on a single risk set not necessarily event time General odds model not necessarily POM or PHM 2 PROBABILITY STRUCTURE i pi 1 pi p Zi Z 1 p Zi Z subject i Z odds d Q i d i 3 CONDITIONAL EVENT AND PROBABILITY Let H be a random event that we will condition on Define pd h pr D d H h Z pr D d H h Z H conditional D probability or intensity h d pr H h Dd 1 Z pr H h D e or H R e D d Z e g H R 4 CONDITIONAL PROBABILITY GIVEN AN EVENT H h Theorem Let D be generated according to the independent binary data probability structure I e pd R pr D d d qR Then the H conditional D probability is d h d s R s h s pd h pr D d H h Z P Proof Bayes Theorem 5 GENERAL ODDS MODEL Consider a general odds models not just POM t z model for odds as a function of t and z a i t Zi t model odds for i at s Q i s i 0i Z i working covariate for i i b at a Note that for POM is a vector that has both base line odds t and odds ratio b 0 i is the derivative of wrt 6 H CONDITIONAL LIKELIHOOD Likelihood H conditional odds model plugs in the odds model t Zi t for the true subject odds i D H D l D H pD H P s R s H s 7 DEFINITIONS Working covariate for set d at 0d Z d d Conditional expectation of the working covariate at X Eh Z d pd h d R Probability that i is a case at X pi h pd h d R d3i Probability that i and j are cases at X pi j h pd h d R d3i j 8 USEFUL IDENTITIES WORKING COVARIATE FOR A SET P 0d Lemma Z d i d Z i where d 0 d is the derivative of d with respect to Proof Homework 9 USEFUL IDENTITIES THE CONDITIONAL EXPECTATION OF THE WORKING COVARIATE Eh Lemma Eh P i R Z i pi h P d R Z d pd h Proof Suppressing the Eh X Z d pd h d R X X Z i pd h d R d3i X X Zi pd h i R d R d3i X Z i pi h i R 10 LIKELIHOOD QUANTITIES 11 NEW

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