1. Kaplan-Meier survival curves and the log rank test
Dr Douwe Postmus

2. Content What makes the analysis of time-to-event data special?
Kaplan-Meier estimator of the survival curve
Log rank test to compare the survival curves between two or more groups

3. Example
Population: patients admitted to the hospital with symptoms of heart failure (HF)
Outcome: time from hospital discharge to HF hospitalization or death from any cause
Parameter of interest: survival curve S(t)
Proportion of patients with an event time larger than t

4. Survival curve for the population
Survival curve S(t): proportion of patients in the population with an event time larger than t
1-year survival: S(1)=0.72
2-year survival: S(2)=0.56
3-year survival: S(3)=0.45
4-year survival: S(4)=0.37
5-year survival: S(5)=0.30
S(t) is generally unknown and needs to be estimated from the data

5. Random sample of n=1000
Estimated survival based on the event times in the sample
1-year survival: 708 / 100 = 0.71
2-year survival: 542 / 100 = 0.54
3-year survival: 445 / 100 = 0.45
4-year survival: 370 / 100 = 0.37
5-year survival: 300 / 100 = 0.30

6. Actual versus estimated survival

7. Right censoring
Administrative censoring: the event is observed only if it occurs prior to some pre-specified time
Studies with a fixed follow-up time (e.g., maximum of 2 years per patient)
Studies with a fixed duration (e.g., 5 years between start and end of study)
Loss to follow-up: subjects who drop out from the study before it is terminated

8. Graphically
Start of study
End of study x o
o censored o x
x event x o

9. Random sample (n=1000) with right-censored observations

10. How to estimate the 1-year survival?
For the 578 patients whose event times were observed we know that
296 survived for more than 1 year
282 experienced the event within the first year
For the 442 patients whose event times were censored we know that
332 survived for more than 1 year
90 either experienced the event within the first year or survived for more than one year

11. 1-year survival: lower and upper bounds
Lower bound: count the 90 patients who either experienced the event within the first year or survived for more than one year as if they experienced the event
Upper bound: count the 90 patients who either experienced the event within the first year or survived for more than one year as if they survived

12. Graphically

13. Estimation based on conditional probabilities
Interval
Survival
at start
interval
n.risk
n.event
n.censored
Hazard
Survival at end interval
0 - 1 1
1000
282 90
282 / 1000 = 0.282
1*( ) = 0.718
1 - 2
0.718
628
137 64
137 / 628 = 0.218
0.718*( ) = 0.561
2 - 3
0.561
427 76 50
76 / 427 = 0.178
0.561*( ) = 0.461
3 - 4
0.461
301 56 46
56 / 301 = 0.186
0.461*( ) = 0.375
4 - 5
0.375
199 27
172
27 / 199 = 0.136
0.375*( ) = 0.324
Hazard: probability of experiencing the event within the interval conditional on being alive at the start of the interval

14. Actual versus estimated survival

15. Kaplan-Meier estimator
Survival curve estimated based on conditional probabilities
Takes all the unique event and censoring times and sorts them in ascending order (from low to high)
Uses the periods between the sorted event and censoring times as the intervals

16. KM survival curve for the example (time in days instead of years)

17. Actual versus estimated survival

19. Creating KM survival curves in SPSS

20. Creating KM survival curves in SPSS

21. KM survival curves for the three groups

22. Log rank test

23. Limitations of the log rank test
The log rank test can be used to compare the survival curves of two or more groups
Stratification can be used to adjust for the effect of a second categorical covariate
Treatment effect adjusted for gender (i.e., separate survival curves for male and female patients)
Examples of research questions for which the log rank test cannot be used
Is age associated with the time to HF hospitalization or death from any cause?
Treatment effect adjusted for several covariates

24. Next lecture Date Location Speaker Topic 11 June TBD H. Burgerhof
Meta-analyses on continuous outcomes, odds ratios, and diagnostic tests

25. contact: d.postmus@umcg.nl