EPI 5344: Survival Analysis in Epidemiology Hazard March 4 , 2014

1. EPI 5344:Survival Analysis in EpidemiologyHazardMarch 4, 2014 Dr. N. Birkett, Department of Epidemiology & Community Medicine, University of Ottawa

2. Hazard • Simplest model assumes a constant hazard • Yields an exponential survival curve • Leads to basic epidemiology formulae for incidence, etc. • More next week • Can extend it using the piecewise model • Fits a different constant hazard for given follow-up time intervals.

3. Hazard estimation (1) • If hazard is not constant, how does it vary over time?

4. Hazard estimation (2) • How can we estimate the hazard? • Parametric methods (not discussed today) • Non-parametric methods • We can estimate: • h(t) • H(t) • Prefer to estimate H(t) • h(t) shows much random variation • Nelson-Aaalen method is main approach. • Start by dividing follow-up time into intervals (similar to piece-wise method)

5. h(t) direct estimation

6. Hazard estimation (3) • Direct hazard estimation has issues • Unstable estimates due to small event numbers in time intervals • Works ‘best’ for actuarial method since intervals are pre-defined • Length is generally the same for each interval (ui).

7. Hazard estimation (4) • hazard estimation with actuarial method • Length is generally the same for each interval (ui). Standard ID formula from Epi

8. Hazard estimation (5) • Person-time variant • Divide follow-up time into fixed intervals • Compute person-time in each interval (rather than using approximation). • Gives a slightly smoother curve

9. Hazard estimation (6) • Kaplan-Meier • ‘interval’ is time between death events • Varies irregularly • Formula has same structure as above di = # with event ui = ti+1 – ti ni = size of risk set at ‘t’ Estimates are ‘unstable’ so usually estimate cumulative hazard or use Kernel Smoothing

10. Hazard estimation (7) • Cumulative hazard: H(t) • Measures the area under the h(t) curve. • If we use a piece-wise constant h(t), then H(t) is the sum of the pieces. • For each ‘piece’ before time ‘t’, compute • product of the estimated ‘hi’ for the interval multiplied by the length of the interval. • Add these up across all ‘pieces’ before time ‘t’. • width of last ‘piece’ is up to ‘t’ only • Relates to the density method from epi (discussed elsewhere)

11. h(t) direct estimation

12. Hazard estimation (8) • Several ways to estimate H(t) • Nelson-Aalen estimator • Kaplan-Meier • -log(S(t)) • Actuarial method • Person-time method

13. Hazard estimation (8a) • Kaplan-Meier method • estimate h(t) and sum up across intervals • Actuarial method & Person-time method • estimate h(t) and sum up across intervals • -log(S(t)) • from our basic formulae, we have:

14. Hazard estimation (9) • All methods should give similar estimates Nelson-Aalen estimator for H(t) is most commonly used. • Similar approach to Kaplan-Meier method • Compute at each time when event happens: di = # with event at ‘ti’ ni= size of risk set at ‘ti’

15. Numerical example

16. Nelson-Aalen estimate of H(t)

18. Hazard estimation (10) H(t) has many uses, largely based on:

19. Nelson-Aalen (2) Estimating H(t) gives another way to estimate S(t). Uses formula:

21. Key for testing proportional hazards assumption (later)

22. Suppose the hazard is a constant (λ), then we have: Plot ‘ln(S(t))’ against ‘t’. • A straight line indicates a constant hazard. • Approach can be used to test other models (e.g. Weibull).

23. Smoothing & hazard estimation • With the Kaplan-Meier method, there is one ‘step’ in the survival curve for each event time. • Direct estimation of h(t) is very unstable • Instead, start with the Nelson-Aalan method to estimate H(t) • At each event time, compute • Apply a smoothing method to generate an estimate of h(t)

24. Example (from Allison) • Recidivism data set • 432 male inmates released from prison • Followed for 52 weeks • Dates of re-arrests were recorded • Study designed to examine the impact of a financial support programme on reducing re-arrest

26. Adjusted hazard estimates using actuarial method: last interval ends at 53 weeks, not 60 weeks Simple hazard estimates using actuarial method

29. Proportional Hazards

30. Proportional Hazards (1) • Suppose we have two groups followed over time (say treatment groups in an RCT). • How will the hazards in the two groups relate? • There need be no specific relationship • They could even go in opposite directions

31. Hazard functions for 2 hypothetical groups in a RCT

32. Proportional Hazards (2) • Often, it is reasonable to place restrictions on how the hazards relate • Consider a situation where the hazard is constant over time: • Experimental: λe • Control: λc

33. λc λe The ratio of the hazard in one group to the other is constant for all follow-up time. • A simple example of Proportional Hazards

34. Proportional Hazards (3) • What if the hazard is not constant over time? • Relationship between curves can be complex • It is common to make the assumption that the hazard curves are proportional over all follow-up time

35. λc λe The hazard in the experimental group is a constant multiple of that in the control group for all follow-up time. • Proportional Hazards (PH)

36. Proportional Hazards (4) • PH is easier to see if we look at the logarithm of the hazards. • The difference in the log-hazards is constant over time. • Means that the curves are a fixed distance apart

37. λc λe

38. Proportional Hazards (5) • If PH is true, then we frequently designate one group as the reference group (0). Re-write this to get:

39. Proportional Hazards (6) • In above equation, HR can be affected by patient characteristics • Age • Sex • Residence • Baseline disease severity • Can model this as:

40. Proportional Hazards (7) • Most common form for this model is: Model underlies the Cox regression approach.

41. Reminder & Warning • Proportional Hazards is an ASSUMPTION • It need not be true • Not all probability models for survival curves leads to PH • PH is less likely to be true when the follow-up time gets very long