1. Where are we?

2. Survival analysis

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

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

5. First example of the day

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

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

8. 1-year survival rate 6 patients dies the first year
25 patients started
24%

9. 1-year survival rate 3 patients less than 1 year 6/(25-3) = 27%
24% -27%

10. Actuarial / life table anelysis
Treatment for lung cancer

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

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

13. 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

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

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

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

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

18. Survival curves
How long will a lung cancer patient keep having cancer on this particular treatment?

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

20. Confidence interval of the Kaplan-Meier method
Fx at first terminal event

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

22. 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

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

24. Comparing Two Survival Curves
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

25. Comparing Two Survival Curves
Smells like Chi-Square statistics

26. Comparing Two Survival Curves
Hazard ratio

27. 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)
row total * column total
grand total

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

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