Introduction to Survival Analysis

1. Introduction to Survival Analysis

2. Consider the following clinical out-come evaluation situation: • Goal: To determine the effectiveness of a new therapeutic intervention with sexually abused children.

3. Consider the following clinical outcome evaluation: Procedure: • Identify N=9 children who are referred to your agency after being sexually abused. 2. Obtain a baseline measure on each (e.g., number of behavioral or emotional symptoms). 3. Provide therapeutic intervention. 4. Obtain post-treatment measurement.

4. Results

5. How would we summarize the data?

6. Outcome Analysis • Are the children showing significant improvement in number of symptoms? • What type of statistical test should I do? • Answer: A paired t-test (pre/post data)

7. Pre/Post Findings

8. Pre/Post Findings • Interpretation: There was a significant improvement in number of symptoms between the beginning of treatment and the end of treatment.

9. Pre/Post Findings Concern: But would the children have improved the same amount without our treatment?

10. Pre/Post Findings Concern: But would the children have improved the same amount without our treatment? • We need a control group!

11. How Do We Compare Pre/Post Data Between Groups?

12. How Do We Compare Pre/Post Data Between Groups? • Answer: Independent t-test on change in number of symptoms.

13. Analysis

14. Analysis

15. Findings • Interpretation: No significant difference in symptom reduction between control and treatment groups

16. New Type of Problem • How would you analyze the following problem?

17. TIME TO FAILUREDid our client fail over time? Yes/No • Recidivism • Rehospitalization • Acting Out • Relapse • Dropping Out • Death

18. Time is a Variable • The question is not simply, “Did our client experience a failure?” • Rather, “Did our client last longer before failure?” • e.g. “Did our clients go longer before being rehospitalized?”

19. Chi-Square?? • Problem: What time period are you talking about?

20. Particularly Difficult In Agencies • Unlike controlled experiments, it is difficult to study cohort groups. • Clients come and go in unpredictable manner. • How do you randomize over time?

21. Path Of Failure Over Time Is Important Rx as Usual 100% % of clients NOT rehospitalized 6 months 1 year Time to Rehospitalization

22. Additionally, Path Of Failure Is Important Rx as Usual 100% % of clients NOT rehospitalized New Rx 6 months 1 year Time to Rehospitalization

23. Survival Analysis • Compares number of failures over TIME • Recorded as binary yes/no • Path of that Failure

24. Instead Of Looking at Symptom Reduction, Let’s Look at the Symptom Avoidance? EXAMPLE: Sexually Reactive Children Sexually Reactive" children are pre-pubescent boys and girls who have been exposed to, or had contact with, inappropriate sexual activities. The sexually reactive child may engage in a variety of age-inappropriate sexual behaviors as a result of his or her own exposure to sexual experiences, and may begin to act out, or engage in, sexual behaviors or relationships that include excessive sexual play, inappropriate sexual comments or gestures, mutual sexual activity with other children, or sexual molestation and abuse of other children.

25. Suppose we are interested in decreasing or avoiding sexual reactivity. • What kinds of things are going to change with respect to our methods for evaluating the data?

26. What would the data look like? Summary: Median survival = 10 months or 6-month survival = 58% +/- 19% S = start; 0 = last measurement of observed non-reactive behavior; x = Reactive behavior occurred

27. Summary: What is Survival Analysis? • Outcome variable: Time until an event occurs • Time = Survival time • Event (Assume single event) = failure • Death • Relapse • Sexual reactivity • Assault • Rehospitalization

28. Why censor? Study ends (no event) Lost to follow-up Withdraws Censored Data • Censoring: Don’t know survival time exactly

29. Terminology and Notation • t = observed survival time • = (0,1) random variable • 1 if failure • 0 if censored • S(t) = survivor function • h(t) = hazard function • t(j) = time period Probability of survival beyond a certain point in time Failure rate

30. How do we use this data in a survival analysis?

32. t = observed survival time • = (0,1) random variable • 1 if failure • 0 if censored)

38. Graphical Visual of Table Out of the 9 children survived > 0 months: 0 failed 1 was censored Out of the 8 children survived > 3 months: 1 failed 1 was censored Out of the 6 children survived > 5 months): 2 failed 2 were censored Out of the 2 children survived > 10 months: 1 failed 1 was censored t(0) t(1) t(2) t(3) t(j) = failure time period; t = month when failure occurred; m = # of failures; q = # censored; R = risk set

39. Graphical Visual of Table Out of the 9 children survived > 0 months: 0 failed 1 was censored Out of the 8 children survived > 3 months: 1 failed 1 was censored Out of the 6 children survived > 5 months): 2 failed 2 were censored Out of the 2 children survived > 10 months: 1 failed 1 was censored t(0) t(1) t(2) t(3) Based upon this data set, what is the probability that a child entering our program will remain non-sexually reactive for each failure time period? S(3)= 7/8 = .875 Interpretation: Having started the program, there is a 87.5% chance that a given child will survive to 3 months without becoming reactive S(5)= 4/6 = .66.7 Interpretation: Having survived 3 months, there is a 66.7% chance that a given child will survive to 5 months without becoming reactive S(10)= 1/2 = .500 Interpretation: Having survived 5 months, there is a 50.0% chance that a given child will survive to 10 months without becoming reactive S(0)= 9/9 = 1 Interpretation: Having been admitted to the program, there is 100% chance all children will start the Rx

40. p(Heads) = .50 p(Tails) = .50 Probability Paths What are your chances of flipping heads the 1st time? Answer: 50%

41. p(Heads) = .50 .50 p(Tails) = .50 .50 Probability Paths Having already flipped heads once, what is the chance your next flip is heads? Answer: 50%

42. p(Heads) = .50 .50 .50 p(Tails) = .50 .50 .50 Probability Paths Having already flipped heads twice, what is the chance your next flip is heads? Answer: 50%

43. p(Heads) = .50 .50 .50 p(Tails) = .50 .50 .50 Probability Paths What are your chances of flipping heads three times in a row? Answer: .50 x .50 x .50 = .125 or there is a 12.5% chance of flipping heads three times in a row.

44. p(Heads) = .50 .50 .50 p(Tails) = .50 .50 .50 Probability Paths What are your chances of flipping two heads and one tail in that order? Answer: .50 x .50 x .50 = .125 or there is a 12.5% chance of flipping heads three times in a row.

45. 1.00 .875 0 .125 Kaplin-Mier Method Using Probability Paths S(3)= 7/8 = .875 Interpretation: Having started the program, there is a 85.5% chance that a given child will survive to 3 months without becoming reactive S(5)= 4/6 = .66.7 Interpretation: Having survived 3 months, there is a 66.7% chance that a given child will survive to 5 months without becoming reactive S(10)= 1/2 = .500 Interpretation: Having survived 5 months, there is a 50.0% chance that a given child will survive to 10 months without becoming reactive S(0)= 9/9 = 1 Interpretation: Having been admitted to the program, there is 100% chance all children will start the Rx What is the chance of a child surviving (being non-reactive) until 3 months after being assigned to the program? Answer: S(t(3)) = 1x .875 = .875 or there is a 87.5% chance of a child being non-reactive for 3 months after being assigned to the program.

46. 1.00 .875 .667 0 .125 .333 Kaplin-Mier Method Using Probability Paths S(3)= 7/8 = .875 Interpretation: Having started the program, there is a 85.5% chance that a given child will survive to 3 months without becoming reactive S(5)= 4/6 = .66.7 Interpretation: Having survived 3 months, there is a 66.7% chance that a given child will survive to 5 months without becoming reactive S(10)= 1/2 = .500 Interpretation: Having survived 5 months, there is a 50.0% chance that a given child will survive to 10 months without becoming reactive S(0)= 9/9 = 1 Interpretation: Having been admitted to the program, there is 100% chance all children will start the Rx What is the chance of a child surviving (being non-reactive) until 5 months after being assigned to the program? Answer: S(t(5)) = 1 x .875 x .667 = .584 or there is a 58.4% chance of a child being non-reactive for 5 months after being assigned to the program.

47. 1.00 .875 .667 .50 0 .125 .333 .50 Kaplin-Mier Method Using Probability Paths S(3)= 7/8 = .875 Interpretation: Having started the program, there is a 85.5% chance that a given child will survive to 3 months without becoming reactive S(5)= 4/6 = .66.7 Interpretation: Having survived 3 months, there is a 66.7% chance that a given child will survive to 5 months without becoming reactive S(10)= 1/2 = .500 Interpretation: Having survived 5 months, there is a 50.0% chance that a given child will survive to 10 months without becoming reactive S(0)= 9/9 = 1 Interpretation: Having been admitted to the program, there is 100% chance all children will start the Rx What is the chance of a child surviving (being non-reactive) until 3 months after being assigned to the program? Answer: S(t(10)) = 1 x .875 x .667 x .500 = .292 or there is a 29.2% chance of a child being non-reactive for 10 months after being assigned to the program.

48. Use survival data summary table to make survival curve 1 Theoretical S(t) S(t) 0 8 t

49. 1 .80 .60 Survival Probability S(t) in practice: The Kaplin-Mier Survival Curve .40 .20 0 2 4 6 8 10 12

50. We can compare the Kaplin-Mier survival curves of two Rx groups 1 Are these two curves significantly different? Simpliest method is known as the Log-rank Test .80 .60 Survival Probability .40 .20 0 2 4 6 8 10 12