1. Topic 19: Survival Analysis T = Time until an event occurs. Events are, e.g., death, disease recurrence or relapse, infection, pregnancy.

2. Interpretation of hazard function Hazard function gives instantaneous risk of failure. For healthy person it is constant whereas for Lukaemia patients, it is decreasing.

3. Example: Suppose T has an Exponential Distribution.

4. Properties of exponential survival r.v.

5. In real applications, we don’t expect “used” to be as good as “new”. A more realistic survival variable is Weibull random variable with density

7. Aim of Survival Analysis is the estimation and comparison of S(t) and h(t). Time until relapse (in weeks) of acute myelogenous leukaemia (AML) patients receiving maintenance chemotherapy 9, 13, 13, 18, 23, 28, 31, 34, 45, 48, 161

8. Censoring At the end of study, some patients are yet to experience a relapse. Their remission times cannot be ascertained exactly; all we can say is that these times are more than the difference between termination and entry time. We say these remission times are right censored. Note that Censoring can also be the result of dropout and loss to follow up. We need to take into account Censoring while estimating the survival function. (Inherent feature of survival data)

9. Estimating survival function from censored data (Kaplan-Meier) 9, 13, 13+, 18, 23, 28+, 31, 34, 45+, 48, 161+ Shift the probability mass of the censored observations to the right equally

10. Kaplan-Meier estimate is also called the product limit estimate Let t(1) < t(2) < t(3), … denote the ordered uncensored time points. Let d(j) denote the number of uncensored events at t(j) and n(j) denote total number of individuals who are at risk at t(j). Then S[t(j)]=[1-d(j)/n(j)]S[t(j-1)]. INTUITION: P[T> u(j)]=[ 1 - P[u(j-1) u(j-1)] ] P[T>u(j-1)]. Next we compute K-Meier estimator of the AML-data. Order the UNCENSORED times as 9 < 13 < 18 < 23 < 31 < 34 < 48.

11. K-M Estimates of the survival function:

12. Re-distribute to the right algorithm (Kaplan -Meier Estimate) Time

13. In SPSS, click: Analyze, survival, Kalpan-Meier. In Status, take Var2, where 1 denotes an uncensored event and 0 a censored event. In the `Options’, plot the survival curve.

14. S[t(j)]=[1-d(j)/n(j)]S[t(j-1)].

15. Next we compare survival Functions of following two groups: Time to relapse (in weeks) Maintained group: 9, 13, 13+, 18, 23, 28+, 31, 34, 45+, 48, 161+ Non-maintained group: 5, 5, 8, 8, 12, 16+, 23, 27, 30, 33, 43, 45

17. Comparing two groups Maintained group: 9, 13, 13+, 18, 23, 28+, 31, 34, 45+, 48, 161+ Non-maintained group: 5, 5, 8, 8, 12, 16+, 23, 27, 30, 33, 43, 45

18. SPSS: Click `Status’ for censoring, `Factor’ for two groups and `Options’ for plotting survival curves. Then click `Compare Factor’ for log rank test.