Survival Analysis in Cancer Treatment: ANOVA, Kaplan-Meier, and Logrank Methods

1. Lecture 3 Survival analysis

2. Problem • Do patients survive longer after treatment A than after treatment B? • Possible solutions: • ANOVA on mean survival time? • ANOVA on median survival time?

3. Progressively censored observations • Current life table • Completed dataset • Cohort life table • Analysis “on the fly”

4. First example of the day

5. Person-year of observation • In total: 15.122 days ~ 41.4y • 11 patients died: 11/41.4y = 0.266 y-1 26.6 death/100y • 1000 patients in 1 y or • 100 patients in 10y

6. Mortality rates • 11 of 25 patients died • 11/25 = 44% • When is the analysis done?

7. 1-year survival rate • 6 patients dies the first year • 25 patients started • 24%

8. 1-year survival rate • 3 patients less than 1 year • 6/(25-3) = 27% • Patient 7 • 24% -27%

9. Actuarial / life table anelysis • Treatment for lung cancer

10. Actuarial / life table anelysis • A sub-set of 13 patients undergoing the same treatment

11. Actuarial / life table anelysis • Time interval chosen to be 3 months • ni number of patients starting a given period

12. Actuarial / life table anelysis • di number of terminal events, in this example; progression/response • wi number of patients that have not yet been in the study long enough to finish this period

13. Actuarial / life table anelysis • Number exposed to risk: ni – wi/2 Assuming that patients withdraw in the middle of the period on average.

14. Actuarial / life table anelysis • qi = di/(ni – wi/2) Proportion of patients terminating in the period

15. Actuarial / life table anelysis • pi = 1 - qi Proportion of patients surviving

16. Actuarial / life table anelysis • Si = pi pi-1 ...pi-N Cumulative proportion of surviving Conditional probability

17. Survival curves • How long will a lung canser patient keep having canser on this particular treatment?

18. Kaplan-Meier • Simple example with only 2 ”terminal-events”.

19. Confidence interval of the Kaplan-Meier method • Fx after 32 months

20. Confidence interval of the Kaplan-Meier method • Survival plot for all data on treatment 1 • Are there differences between the treatments?

21. Comparing Two Survival Curves • One could use the confidence intervals… • But what if the confidence intervals are not overlapping only at some points? • Logrank-stats • Hazard ratio • Mantel-Haenszel methods

22. Comparing Two Survival Curves • The logrank statistics • Aka Mantel-logrank statistics • Aka Cox-Mantel-logrank statistics

23. Comparing Two Survival Curves • Five steps to the logrank statistics table • Divide the data into intervals (eg. 10 months) • Count the number of patients at risk in the groups and in total • Count the number of terminal events in the groups and in total • Calculate the expected numbers of terminal events e.g. (31-40) 44 in grp1 and 46 in grp2, 4 terminal events. expected terminal events 4x(44/90) and 4x(46/90) • Calculate the total

24. Comparing Two Survival Curves • Smells like Chi-Square statistics

25. Comparing Two Survival Curves • Hazard ratio

26. Comparing Two Survival Curves • Mantel Haenszel test • Is the OR significant different from 1? • Look at cell (1,1) • Estimated value, E(ai) • Variance, V(ai)

27. Comparing Two Survival Curves • Mantel Haenszel test • df = 1; p>0.05

28. Hazard function d is the number of terminal events f is the sum of failure times c is the sum of censured times