1. 01/20151 EPI 5344: Survival Analysis in Epidemiology Quick Review from Session #1 March 3, 2015 Dr. N. Birkett, School of Epidemiology, Public Health & Preventive Medicine, University of Ottawa

2. 01/20152 Objectives (for entire session) Primary goal is to address two key concepts: –Hazard estimation role in survival methods –Methods to compare two survival curves using non- parametric methods

3. 01/20153 Objectives (for entire session) Review –Survival concepts –Hazard Methods for estimation of hazard Proportional hazards Non-regression comparison of survival curves –Log-rank test –Variations of log-rank test

4. Review Material 01/20154

5. 5 Time Scale (1) Time of events is usually measured using ‘calendar dates’ Can be represented in graphic display by ‘time lines’ –The conceptual idea used in analyses Patient #1enters on Feb 15, 2000 & dies on Nov 8, 2000 Patient #2enters on July 2, 2000 & is lost (censored) on April 23, 2001 Patient #3Enters on June 5, 2001 & is still alive (censored) at the end of the follow-up period Patient #4Enters on July 13, 2001 and dies on December 12, 2002

6. 01/20156 Study course for patients in cohort 2001 2003 2013

7. 01/20157

8. 8 Histogram of death time -Skewed to right -pdf or f(t) -CDF or F(t) -Area under ‘pdf’ from ‘0’ to ‘t’ t F(t)

9. 01/20159 Survival curves (3) Plot % of group still alive (or % dead) S(t) = survival curve = % still surviving at time ‘t’ = P(survive to time ‘t’) Mortality rate = 1 – S(t) = F(t) = Cumulative incidence

10. 01/201510 Deaths CI(t) Survival S(t) t S(t) 1-S(t)

11. 01/201511 Essentially, you are re-scaling S(t) so that S * (t 0 ) = 1.0 Conditional Survival Curves

12. 01/201512 S * (t) = survival curve conditional on surviving to ‘t 0 ‘ CI * (t) = failure/death/cumulative incidence at ‘t’ conditional on surviving to ‘t 0 ‘ Hazard at t 0 is defined as: ‘the slope of CI * (t) at t 0 ’ Hazard (instantaneous) Force of Mortality Incidence rate Incidence density Range: 0  ∞

13. 01/201513 Some relationships If the rate of disease is small: CI(t) ≈ H(t) If we assume h(t) is constant (= ID): CI(t)≈ID*t

14. 01/201514 DEAD p1p1 1- p 1 p2p2 1- p 2 p3p3 1- p 3 Year 0 Year 1 Year 2 Year 3

15. 01/201515 Actuarial Method ABCDEFGH Year# people under follow-up # lost# people dying in this year Effective # at risk Prob die in year Prob survive this year S(t) 0-11000 011 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 ABCDEFGH Year# people under follow-up # lost# people dying in this year Effective # at risk Prob die in year Prob survive this year S(t) 0-11000 011 1-210119.50.1050.895 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 ABCDEFGH Year# people under follow-up # lost# people dying in this year Effective # at risk Prob die in year Prob survive this year S(t) 0-11000 011 1-210119.50.1050.895 2-380180.1250.8750.783 3-4 4-5 5-6 6-7 7-8 8-9 9-10 ABCDEFGH Year# people under follow-up # lost# people dying in this year Effective # at risk Prob die in year Prob survive this year S(t) 0-11000 011 1-210119.50.1050.895 2-380180.1250.8750.783 3-472160.1670.8330.652 4-54004010.652 5-640140.250.750.489 6-73103.5010.489 7-82102.5010.489 8-91101.5010.489 9-100000010.489

16. 01/201516 Kaplan-Meier method ‘i'time# deaths # in risk set Prob die in interval Prob survive interval S(t 1 ) 00--- 1.0 122190.1110.889 2 3 4 ‘i'time# deaths # in risk set Prob die in interval Prob survive interval S(t 1 ) 00--- 1.0 122190.1110.889 229180.1250.8750.778 3 4 ‘i'time# deaths # in risk set Prob die in interval Prob survive interval S(t 1 ) 00--- 1.0 122190.1110.889 229180.1250.8750.778 346150.2000.8000.622 4 ‘i'time# deaths # in risk set Prob die in interval Prob survive interval S(t 1 ) 00--- 1.0 122190.1110.889 229180.1250.8750.778 346150.2000.8000.622 461140.2500.7500.467

17. END OF REVIEW MATERIAL 01/201517

18. 01/201518