1. 01/20151 EPI 5344: Survival Analysis in Epidemiology Epi Methods: why does ID involve person-time? March 10, 2015 Dr. N. Birkett, School of Epidemiology, Public Health & Preventive Medicine, University of Ottawa

2. Background I just showed that we get the same answer from: –Exponential survival model –Person-time epidemiology estimate of ID Why? What is the basis for this link? 01/20152

3. The Issue (1) Epidemiological analysis focuses on: –Incidence Proportion or Cumulative Incidence (CI) –Incidence Density or Incidence Rate (ID). Standard formulae are: 01/20153

4. The Issue (2) How do these measures relate to survival analysis? Why does ID involve person-time? 01/20154

5. Incidence Density (rate) Rate of getting disease. –A number with units (time -1 ) –Ranges from 0  ∞ Often measured from time ‘0’ (recruitment) Can be measured for any time interval –Separate ID’s for each year of follow-up If the time units get smaller, we approach the ‘instantaneous ID’ 01/20155

6. Incidence Density (rate) Rate of getting disease (outcome) at time ‘t’ given (conditional on) on having survived to time ‘t’ Instantaneous ID is the same as the hazard Average ID is more common in epidemiology 01/20156

7. 7 Epidemiology formulae ignore ID variability over time and compute average ID (ID`) Actuarial method (density method) lets each interval have a different ID Linked to piecewise exponential model

8. Why does ID relate to person-time? 01/2015 Let’s look at a simple situation (assumption): No losses (i.e. no censoring) A constant ID over time (I) an exponential model Then, we have: 8

9. 01/20159 Area under S(t) from 0 to ‘t’ Graph of S(t)

10. Why does ID relate to person-time? 01/201510

11. 01/201511 So, how can we figure out the area under S(t)? Let’s look at the next slide

12. 01/201512 Area under S(t) from 0 to 1 Actually a curve but we assume it’s a straight line Graph of S(t)

13. 01/201513

14. 01/201514 In general, area under S(t) from ‘0’ to ‘t’ is given by: How does this help? In the formula we derived for ID, multiply top and bottom by ‘N’ (the initial # of people at risk) Now, CI(t) * N = # new cases by time ‘t’.

15. 01/201515 This is the standard Epidemiology definition of ID

16. Person-time approach to ID assumes that ID (hazard) is constant –Can be seen as estimating an average ID BUT, constant hazard gives the exponential survival model which does not reflect real-world S(t)’s. 01/201516

17. Why do we use constant ID Why does epidemiology ignore this and use a constant ID? –Lack of data –Lack of measurement precision –Tradition –”teaching” –Old fashioned methods or learning by rote 01/201517

18. What can we do different Piece-wise constant hazard approach is better Density methods Survival methods 01/201518

19. Density method (1) GOAL: to estimate CI for outcome by year ‘t*’ 1.Select a time interval (usually 1 year) 2.Divide follow-up time into intervals of this size 3.Within each interval, compute the ID of surviving the interval given you are disease-free at start: 01/201519

20. Density method (2) 4.Compute: 01/201520 5.Then, we have:

21. Density method (3) Very similar to the methods based on H(t). When h(t) is piecewise constant, we have: 01/201521

22. 01/201522