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%
Patient 7
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
• p i = 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 KaplanMeier method
• Fx at first terminal event
SE ( Si )  Si
di
 n n  d 
i
i
i
SE S1   0.9231
1
 0.0739
1313  1
Confidence interval of the KaplanMeier 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
1.
2.
3.
4.
5.
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
O  E

2
  
E
all_treatments
2
 23  17.07   12  17.93
2 
2
17.07
df  1
p  0.05
17.93
2
 4.02
Comparing Two Survival Curves
•
Hazard ratio
Hazard ratio 
O1 E1 23 17.07

 2.01
O2 E2 12 17.93
Comparing Two Survival Curves
•
Mantel Haenszel test
 a  b n

OR 
c  d n
•
•
•
Is the OR significant
different from 1?
Look at cell (1,1)
Estimated value, E(ai)
row total * column total
grand total
•
Variance, V(ai)
 (a  c)(b  d )(a  b)(c  d ) 
V (ai )  

2
n
n

1
 


Comparing Two Survival Curves
•
•
Mantel Haenszel test
  a  E (a ) 2 
 i  i   1.12
M H 
V (ai )





df = 1; p>0.05