1. Lecture 11: cohort analysis (part 2): Survival analysis and Poisson regression
Jeffrey E. Korte, PhD
BMTRY 747: Foundations of Epidemiology II
Department of Public Health Sciences
Medical University of South Carolina
Spring 2015

2. Survival analysis Basic form: Kaplan-Meier method
Takes into account different follow-up for individuals, different risk sets at each event
Can produce Kaplan-Meier survival curves
Graphical method for comparing time-to-event between different exposure groups
Step function, with jumps at the failure times

3. Kaplan-Meier method 5 “failures” (events)
10 censored before disease onset or study end
5 still alive and healthy at study end
25% observed disease rate (this underestimates the true rate)
“Product-limit estimate” is a more accurate way to calculate the rate, taking into account person-time at risk
17 at risk when first event occurred; 14 at risk when 2nd event occurred, etc.

4. Kaplan-Meier method 1st event: 17 at risk 2nd event: 14 at risk
3rd event: 11 at risk
4th event: 9 at risk
5th event: 7 at risk
Probability of avoiding “failure” during entire study period: (16/17) * (13/14) * (10/11) * (8/9) * (6/7) = 0.605
Probability of “failure” = (1 – 0.605) = 0.395

5. Kaplan-Meier survival curve

6. Kaplan-Meier method
If there are no withdrawals (i.e. if there are no censored observations): product-limit estimate = [(# of cases)/(# in cohort)]
e.g. (4/5)*(3/4)*(2/3) = 2/5

7. Kaplan-Meier method
Can be used to compare 2 exposure groups (chi-square statistic): e.g. 5-year survival
Can be used to calculate stratified comparisons to account for possible confounding
Can be used to compare more than 2 exposure groups

8. Survival analysis with models
Regression models
More flexible and powerful than Kaplan-Meier methods
Can explicitly take into account time to event, rather than sequential risk sets with undefined length of time elapsed between events

9. Survival analysis Cox proportional hazards modeling
Basic assumption: hazard of outcome in exposed group is a constant multiple of hazard of outcome in unexposed group

10. Survival analysis (constant HR)

11. Survival analysis
Need to estimate B (constant multiplication factor determining risk in “exposed”)
Convenient to define it as an exponential function (therefore we model the log hazard)

12. Survival analysis
This gives Cox proportional hazards models similar characteristics to logistic regression models
Multiplicative model; assumption of exponential risk increases

13. Survival analysis
Model estimated in Cox proportional hazards regression:

14. Cox proportional hazards model
Notable differences with logistic regression:
Model does not estimate value or shape of baseline hazard function
Under the proportional hazards assumption, we can estimate betas even with baseline hazard function unspecified
We may specify a baseline hazard function if desired (e.g. Weibull proportional hazards model)

15. Cox proportional hazards model
Notable differences with logistic regression:
If hazard ratio is not constant over time, you must stratify on follow-up time
This could be solved by fitting an interaction term (main exposure * follow-up time)

16. Poisson regression Used with person-time data
Estimate adjusted relative rate

17. Poisson regression Similar to Cox model: Different from Cox model:
Uses time-to-event data
Different from Cox model:
Does not assemble individual comparison group for each event at the time it occurs
Rather, calculates rates (events per person-time) in different exposure-covariate strata to produce regression coefficients
Produces true estimate of rate ratio

18. Poisson regression
Assumes outcome (count variable) has a Poisson distribution
Count variable (integer)
Minimum 0, maximum is infinite
Mean and variance are equal
Distribution shape depends on mean
Skewed right for low mean (e.g. 1, 2)
Fairly normal for higher mean (e.g. 10 or higher)

19. Summary Poisson regression Cox proportional hazards regression
Can obtain adjusted rates, rate ratios
Requires person-years of follow-up to calculate denominator of rate
Cox proportional hazards regression
Can obtain adjusted hazard ratio based on time-to-event
Fair comparison of individuals at risk (i.e. better than logistic regression)