1. Midterm Exam • Yes, I know the first poll had a bug • Second poll is up and running: • www.tinyurl.com/epiexam2 • I will accept responses until Monday Morning and will announce the results in Monday’s class • There will have to be an overwhelming majority for me to change the exam date

2. HSS4303B – Intro to Epidemiology Jan 28, 2010 - Kaplan-Meier Survival Curves

3. Abstracts • Don’t forget…. Due by midnight tonight • Email assignments to hss4303@gmail.com (already received 6 as of 2:AM) • Include name and student number in subject heading • Do not CC me or Erin!!!!!(!!!!)

4. Have you ever wished to have some of your work published but never had an opportunity? Well now’s your chance! Submit your work to the IJHS today! Details: www.IJHS.ca Inquiries: IJHS@hssuottawa.ca Deadline for submissions: February 20th, 2010

5. Early example of survival analysis, 1669 Christiaan Huygens' 1669 curve showing how many out of 100 people survive until 86 years. From: Howard Wainer­ STATISTICAL GRAPHICS: Mapping the Pathways of Science. Annual Review of Psychology. Vol. 52: 305-335.

6. Early example of survival analysis What was a person’s chance of surviving past 20? Past 36? This is survival analysis! We are trying to estimate this curve—only the outcome can be any binary event, not just death.

7. Probabilities P(T>76)=.01 P(T>36) = .16 P(T>20) ~ 0.32, etc.

8. Retrospective cohort study:From December 2003 BMJ: Aspirin, ibuprofen, and mortality after myocardial infarction: retrospective cohort study Curits et al. BMJ  2003;327:1322-1323.

9. Survival Analysis: Terms • Time-to-event: The time from entry into a study until a subject has a particular outcome • Censoring: Subjects are said to be censored if they are lost to follow up or drop out of the study, or if the study ends before they die or have an outcome of interest. They are counted as alive or disease-free for the time they were enrolled in the study. • If dropout is related to both outcome and treatment, dropouts may bias the results

10. What is Survival Time? • Survival time refers to a variable which measures the time from a particular starting time (e.g., time initiated the treatment) to a particular endpoint of interest (e.g., attaining certain functional abilities) • It is important to note that for some subjects in the study a complete survival time may not be available due to censoring

11. Censored Data Some patients may still be alive or in remission at the end of the study period The exact survival times of these subjects are unknown These are called censored observation or censored times and can also occur when individuals are lost to follow-up after a period of study

12. Right Censoring • Pretending that all subjects began at the same time

13. Choice of time of origin. Note varying start times.

14. Count every subject’s time since their baseline data collection. Right-censoring!

17. Kaplan-Meier Survival Curve (K-M) • K-M curves represent the proportion of the study population still surviving (or free of disease or some other outcome) at successive times • as the number of subjects in each intervention group decreases over time, the curves are more precise in the earlier periods (left hand side of the survival curves) than later periods (right hand side of the survival curves)

18. Introduction to Kaplan-Meier Non-parametric estimate of the survival function: Simply, the empirical probability of surviving past certain times in the sample (taking into account censoring).

19. Introduction to Kaplan-Meier • Non-parametric estimate of the survival function. • Commonly used to describe survivorship of study population/s. • Commonly used to compare two study populations. • Intuitive graphical presentation.

20. Treatment #1 Treatment #2

21. Subject A Subject B Subject C Subject D Subject E 1. subject E dies at 4 months X Beginning of study End of study  Time in months  Survival Data (right-censored)

22. 100% Probability of surviving to 4 months is 100% = 5/5 Fraction surviving this death = 4/5 Subject E dies at 4 months  Time in months  Corresponding Kaplan-Meier Curve

23. Subject A 2. subject A drops out after 6 months Subject B Subject C 3. subject C dies at 7 months X Subject D Subject E 1. subject E dies at 4 months X Beginning of study End of study  Time in months  Survival Data

24. 100% Fraction surviving this death = 2/3 subject C dies at 7 months  Time in months  Corresponding Kaplan-Meier Curve

25. Subject A 2. subject A drops out after 6 months Subject B 4. Subjects B and D survive for the whole year-long study period Subject C 3. subject C dies at 7 months X Subject D Subject E 1. subject E dies at 4 months X Beginning of study End of study  Time in months  Survival Data

26. 100%  Time in months  Corresponding Kaplan-Meier Curve P(surviving intervals 1 and 2)=P(surviving interval 1)*P(surviving interval 2) Product limit estimate of survival = P(surviving interval 1/at-risk up to failure 1) * P(surviving interval 2/at-risk up to failure 2) = 4/5 * 2/3= .5333

27. The product limit estimate • The probability of surviving in the entire year, taking into account censoring • = (4/5) (2/3) = 53% • NOTE:  40% (2/5) because the one drop-out survived at least a portion of the year. • AND <60% (3/5) because we don’t know if the one drop-out would have survived until the end of the year.

28. Comparing 2 groups

29. KM Curves comparing Thiotepa to Placebo Thiotepa Control

30. What happens to our precision as the experiment lingers on in time? • Survival estimates can be unreliable toward the end of a study when there are small numbers of subjects at risk of having an event.

31. Limitations of Kaplan-Meier • Mainly descriptive • Doesn’t control for covariates • Requires categorical predictors • Can’t accommodate time-dependent variables

32. Example of K-M Curve • In 1982, 38 infertile women underwent a special laparascopic technique, and attempted to get pregnant • They were each followed for 24 weeks to see who would conceive and when What is the outcome variable? Is this population likely to be right-censored?

33. (This is how it’s going to look)

34. Months to conception or censoring in 38 sub-fertile women after laparoscopy Conceived (event) Did not conceive (censored) 1 2 1 3 1 4 1 7 1 7 1 8 2 8 2 9 2 9 2 9 2 11 3 24 3 24 3 4 4 4 6 6 9 9 9 10 13 16 Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

35. Months to conception or censoring in 38 sub-fertile women after laparoscopy Conceived (event) Did not conceive (censored) 1 2 1 3 1 4 1 7 1 7 1 8 2 8 2 9 2 9 2 9 2 11 3 24 3 24 3 4 4 4 6 6 9 9 9 10 13 16 Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

36. Corresponding Kaplan-Meier Curve 6 women conceived in 1st month (1st menstrual cycle). Therefore, 32/38 “survived” pregnancy-free past 1 month.

37. S(t=1) = 32/38 = 84.2% S(t) represents estimated survival probability: P(T>t) Here P(T>1). Corresponding Kaplan-Meier Curve

38. Months to conception or censoring in 38 sub-fertile women after laparoscopy Conceived (event) Did not conceive (censored) 1 2.1 1 3 1 4 1 7 1 7 1 8 2 8 2 9 2 9 2 9 2 11 3 24 3 24 3 4 4 4 6 6 9 9 9 10 13 16 Important detail of how the data were coded:Censoring at t=2 indicates survival PAST the 2nd cycle (i.e., we know the woman “survived” her 2nd cycle pregnancy-free). Thus, for calculating KM estimator at 2 months, this person should still be included in the risk set. Think of it as 2+ months, e.g., 2.1 months. Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

39. Corresponding Kaplan-Meier Curve

40. S(t=2) = ( 84.2%)*(84.4%)=71.1% Corresponding Kaplan-Meier Curve 5 women conceive in 2nd month. The risk set at event time 2 included 32 women. Therefore, 27/32=84.4% “survived” event time 2 pregnancy-free.

41. Months to conception or censoring in 38 sub-fertile women after laparoscopy Conceived (event) Did not conceive (censored) 1 2.1 1 3.1 1 4 1 7 1 7 Risk set at 3 months includes 26 women 1 8 2 8 2 9 2 9 2 9 2 11 3 24 3 24 3 4 4 4 6 6 9 9 9 10 13 16 Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

42. Corresponding Kaplan-Meier Curve

43. S(t=3) = ( 84.2%)*(84.4%)*(88.5%)=62.8% Corresponding Kaplan-Meier Curve 3 women conceive in the 3rd month. The risk set at event time 3 included 26 women. 23/26=88.5% “survived” event time 3 pregnancy-free.

44. Months to conception or censoring in 38 sub-fertile women after laparoscopy Conceived (event) Did not conceive (censored) 1 2 1 3.1 1 4 1 7 1 7 1 8 2 8 2 9 2 9 2 9 2 11 3 24 3 24 3 4 4 4 6 6 9 9 9 10 13 16 Risk set at 4 months includes 22 women Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

45. Corresponding Kaplan-Meier Curve

46. S(t=4) = ( 84.2%)*(84.4%)*(88.5%)*(86.4%)=54.2% Corresponding Kaplan-Meier Curve 3 women conceive in the 4th month, and 1 was censored between months 3 and 4. The risk set at event time 4 included 22 women. 19/22=86.4% “survived” event time 4 pregnancy-free.

47. Months to conception or censoring in 38 sub-fertile women after laparoscopy Conceived (event) Did not conceive (censored) 1 2 1 3 1 4.1 1 7 1 7 1 8 2 8 2 9 2 9 2 9 2 11 3 24 3 24 3 4 4 4 6 6 9 9 9 10 13 16 Risk set at 6 months includes 18 women Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

48. Corresponding Kaplan-Meier Curve

49. S(t=6) = (54.2%)*(88.8%)=42.9% Corresponding Kaplan-Meier Curve 2 women conceive in the 6th month of the study, and one was censored between months 4 and 6. The risk set at event time 5 included 18 women. 16/18=88.8% “survived” event time 5 pregnancy-free.

50. S(t=13)  22% (“eyeball” approximation) Skipping ahead to the 9th and final event time (months=16)…