Kaplan-Meier survival curves and the log rank test Dr Douwe Postmus (d.postmus@umcg.nl)

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

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

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

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

Actual versus estimated survival

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

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

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

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

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

Graphically

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*(1 - 0.282) = 0.718 1 - 2 0.718 628 137 64 137 / 628 = 0.218 0.718*(1 - 0.219) = 0.561 2 - 3 0.561 427 76 50 76 / 427 = 0.178 0.561*(1 - 0.178) = 0.461 3 - 4 0.461 301 56 46 56 / 301 = 0.186 0.461*(1 - 0.186) = 0.375 4 - 5 0.375 199 27 172 27 / 199 = 0.136 0.375*(1 - 0.136) = 0.324 Hazard: probability of experiencing the event within the interval conditional on being alive at the start of the interval

Actual versus estimated survival

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

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

Actual versus estimated survival

Creating KM survival curves in SPSS

Creating KM survival curves in SPSS

KM survival curves for the three groups

Log rank test

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

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

contact: d.postmus@umcg.nl