1. Binomial, Poisson, Survival Analysis (in conclusion) stats methodologist meeting 13 June 2013

2. 1. Previously… established that a simple survival process (hazard λ) is actually equivalent to a ‘filtered’ Poisson process (of the same rate λ) which dichotomises outcomes into 0/>0 for grouped survival data, this led to a glm with binomial distribution and complementary log-log link (optimal model) what about (usual case) non-grouped data, where hazard varies continuously with time? [‘Competing’ models are Cox PH and Poisson regression]

3. how does a Poisson process relate to a survival process with continuously varying hazard? survival process - hazard λ(t) over some time interval – is ‘like’ a sequence of joined-up Poisson processes [Expectations λ(t).δt] so, over the whole interval, is equivalent to a ‘filtered’ Poisson process with Expectation =∫λ(t).dt and, within time intervals (length T) in which the hazard is (approximately) constant (=λ), effectively have a Poisson process of Expectation =λ.T thus, if an individual dies at a fraction f into such an interval, can define a ‘fair exposure’ to that hazard =f.T

4. Cox PH & Poisson Regression: what’s the difference? 1. Cox PH –focuses only on that part of the likelihood where events (deaths) occur –see: ‘Partial Likelihood’; D. R. Cox; Biometrika (1975), 62, pp. 269-276 [follows his 1972 JRSS paper] 2. Poisson Regression –must partition time into intervals within which hazard is approx. constant (so a ‘fair exposure’ can be defined within each interval)

5. Cox PH & Poisson Regression: when are they equivalent? Poisson Regression partitions time into intervals within which hazard is (hopefully) approximately constant to the extent this is successful, PR and PH will be very similar in the limit, with a partition as ‘fine’ as possible, PR is exactly equivalent to Cox PH – see: ‘Covariance Analysis of Censored Survival Data using Log-Linear Analysis Techniques’ by Laird & Olivier: JASA, Vol 76, No. 374 (1981), pp. 231-240

6. Cox PH & Poisson Regression: which is best? 1. Cox PH –no assumptions made about the underlying hazard (the part of the likelihood where events don’t occur is discarded) –can take a while to fit for big datasets 2. Poisson Regression –quick to fit –depends on an adequate definition of ‘fair exposure’ 3. Suggestion (where both are plausible candidates): –Use PR to explore possible models quickly and efficiently –Check chosen partition is adequate by seeing how much estimates change with a finer partition –fit Cox model at the end