Where are we?

Survival analysis

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

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

First example of the day

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

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

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

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

Actuarial / life table anelysis Treatment for lung cancer

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Comparing Two Survival Curves Smells like Chi-Square statistics

Comparing Two Survival Curves Hazard ratio

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

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

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