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 NOTATION Case set indicator Let Dd I D d indicator for the event D d The events Dd d R indicate possible case sets Note D is a set of indices Dd is an indicator variable 12 H CONDITIONAL LIKELIHOOD AND LOG LIKELIHOOD l D H D H D P s R s H s L D H log D log X s H s s R 13 SCORE EXPRESSIONS U D H U Z D EH X Z d Dd pd h d R X Z i Di pi h i R Suppressing 0D P 0 s R s H s U P D u R u H u 0 X s s H s ZD P s u R u H u s R X ZD Z s ps H s R Z D EH the first expression 14 SCORE EXPRESSIONS Continuing U Z D X Z s ps H s R X d R X Z dDd X Z s ps H s R Z d Dd pd H d R the second expression Useful Identities Z D EH X i D X i R X Zi Further using the X Z i pi H i R Z i Di X Z i pi H i R Z i Di pi H i R giving the third expression 15 OBSERVED AND EXPECTED INFORMATION EXPRESSIONS vobs D H vobs X ps H Z s EH 2 Z 2 i pi H qi H X i R Z 0s Ds ps H s R s R X X Z jZ T k pj k H pj H pk H j6 k R X Z 0i Di pi H i R vexp D H vexp X ps H Z s EH 2 s R X i R Z 2 i pi H qi H X Z jZ T k pj k H pj H pk H j6 k R is suppressed in the information expressions 16 OBSERVED INFORMATION FIRST EXPRESSION vobs X pd H Z d EH 2 d R X Z 0 Dd pd H d R Proof From the second score expression d U d X Z 0 Dd pd H Z d p0d H T vobs d R Now recall that pd H P d s R H d s H s So sup pressing p0d H P 0 0d H d s R s H s P d H d P H s s R s H s 2 s R s X Z d pd H pd H Z s ps H s R 17 OBSERVED INFORMATION Thus X Z d p0d H T d R X 2 Z s ps H 2 Z d pd H d R d R X 2 p E Z 2 d H H d d R X pd H Z d EH 2 d R X Putting the pieces together we get the result 18 EXPECTED INFORMATION EXPRESSIONS Note that these differ from the observed information vobs in that they do not have Z 0s Ds ps H or X s R X Z 0i Di pi H i R parts Think of case set probability pd H or case probability pi H under the model Expectation over case set Dd or case Di indicators using the model probabilities of the terms above is zero I e very heuristically E I Ds ps H 0 This is heuristic Theory shows that these will be asymptotically zero if model fits the data 19 BASIC LIKELIHOOD PROPERTIES Suppose that There exists a point in the model space 0 that corresponds to the Z intensity for failure D i e p t y z 0 yp t z 0 p t Y Z Di are Z conditional independent Then The H conditional expectation of the score is zero The H conditional variance of the score is equal to the H conditional expectation of the observed or expected information Asymptotic properties will require more specific assumptions 20 H conditional likelihood quantities in terms of sets set indicators and subjects Set notation l P D s R H D s H s P L log D log U Z D EH P P 2 Z 0 0 p Z E s H D s R s H s R Z s ps H P 2 s R ps H Z s …