This document provides an overview of survival analysis concepts and methods. It defines time-to-event data and censoring, and describes how to calculate a Kaplan-Meier survival curve from censored data. It also discusses log-rank tests to compare survival curves between groups and the Cox proportional hazards regression model for assessing the effects of multiple covariates on survival.

1. Essentials of survival analysis
How to practice evidence based
oncology
European School of Oncology
July 2004
Antwerp, Belgium
Dr. Iztok Hozo
Professor of Mathematics
Indiana University NW
www.iun.edu/~mathiho

2. Time-to-Event
 Time-to-event data are generated when the measure of interest is
the amount of time to occurrence of an event of interest.
 For Example:
 – Time from randomization to death in clinical trial
 – Time from randomization to recurrence in a cancer clinical trial
 – Time from diagnosis of cancer to death due to the cancer
 – Time from diagnosis of cancer to death due to any causes
 – Time from remission to relapse of leukemia
 – Time from HIV infection to AIDS
 – Time from exposure to cancer incidence in an epidemiological
cohort study

3. Censoring
 Censoring occurs when we have some information, but we don’t know
the exact time-to-event measure.
 For example, patients typically enter a clinical study at the time
randomization (or the time of diagnosis, or treatment) and are followed
up until the event of interest is observed.
However, censoring may occur for the following reasons:
 a person does not experience the event before the study ends;
 death due to a cause not considered to be the event of interest (traffic
accident, adverse drug reaction,…); and
 loss to follow-up, for example, if the person moves.
We say that the survival time is censored. These are examples of right
censoring, which is the most common form of censoring in medical
studies. For these patients, the complete time-to-event measure is
unknown; we only know that the true time-to-event measure is greater
than the observed measurement.

4. Example:
X means an event occurred; O means that the subject was censored.

5. Example 2 (from Kleinbaum: “Survival Analysis”)
Patient Time (t) Censor (d)
1 23 1
2 47 1
3 69 1
4 70 0
5 71 0
6 100 0
7 101 0
8 148 1
9 181 1
10 198 0
11 208 0
12 212 0
13 224 0
Consider data from a retrospective
study of 13 women who had surgery
for breast cancer. The survival times
are:
23, 47, 69, 70+, 71+, 100+, 101+,
148, 181, 198+, 208+, 212+, 224+
(the “+” means that that particular patient was
censored)

6. Survival Curve - Calculus
 S(t) = cumulative survival function = proportion that survive until time t
f(t) = frequency distribution of age at death
h(t) = hazard function (i.e. death rate at age t) = event rate
 Relationships:
     
 

 
t
duuh
t
eduuftTPtS 0
   
dt
tdS
tf 
   
 
 
 
 
  tS
dt
d
tS
tf
TtPt
ttTtP
t
TtttTtP
th
t
t
lnlim
|
lim
0
0










7. Distribution Function, Survival Function
and Density Function
)(1)Pr()( tFtTtS 
 Probability Distribution function
 Probability Density function
 Survival function t
tF
tf



)(
)(
)Pr()( tTtF 

8. Creating a Kaplan-Meier curve
j
jj
n
dn 
 211 |   jjj tTtTPt
     
   
1
11
1
11
0132
21
|...|
|
n
dn
n
dn
n
dn
tTtTPtTtTP
tTtTPtTPtS
j
jj
j
jj
jj
jj













For each non-censored failure time tj (time-to-event time) evaluate:
•nj = number at risk before time tj
•dj = number of deaths from tj-1 to tj
•Fraction
= estimated probability of surviving past tj-1
given that you are at risk at time
The Product Limit Formula:

9. Kaplan-Meier Product Limit Estimate
Consider data from a retrospective study of 45 women who had surgery for breast cancer. The
survival times are: 23, 47, 69, 70+, 71+, 100+, 101+, 148, 181, 198+, 208+, 212+, 224+
j Interval n j d j S (t)
1 13 0 1.00 1.00
2 13 1 0.92 0.92
3 12 1 0.92 0.85
4 11 1 0.91 0.77
5 6 1 0.83 0.64
6 5 1 0.80 0.51
230  t
4723  t
6947  t
14869  t
181148  t
t181
j
jj
n
dn 
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 20 40 60 80 100 120 140 160 180 200

10. 0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
80.00%
90.00%
100.00%
0 50 100 150 200

11. Survival Curves – more examples
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 100 200 300 400 500 600 700 800 900
Days Since Index Hospitalization
Warf
ASA
No Rx
Age 76 Years and Older (N = 394)

12. Log-Rank test for two groups
 Suppose we have two groups,
each with a different treatment.
Usually, we represent this kind
of situation in a 2x2 table.
Event No Event
Intervention 45 198
Control 52 203
or
# at Risk # Events
Intervention n1 = 243 m1 = 45
Control n2 = 255 m2 = 52
TOTAL: N = 498 M = 97
Expected number of events:
Intervention 47.33
Control 49.67
Observed- Expected: -2.33
Variance: 19.55
   
 risktotal
eventstotal
groupExpected
__#
__#
# 
 
 12
21



NN
MNMnn
Var
ExpectedObservedEO 

13.  If the data are given through time, we have a series of
2x2 tables.
 Expected number of events
If the two groups were the same – what would the
expected number of events be?
 Observed minus expected
This is a measure of deviation of one treatment from
their average (the expected)
 Log-rank statistic measures whether the data in the two
groups are statistically “different”.
Log-Rank test for two groups

14. Comparing Survival Functions
 Question: Did the treatment make a
difference in the survival experience of
the two groups?
 Hypothesis: H0: S1(t)=S2(t) for all t ≥ 0.
 Three often used tests:
1. Log-rank test (aka Mantel-Haenszel Test);
2. Wilcoxon Test;
3. Likelihood ratio test.

15. Log-rank example (from Kleinbaum: “Survival Analysis”)
Time n1 m1 n2 m2 Expected Obs-Exp Var
1 21 0 21 2 1.00 -1.00 0.488
2 21 0 19 2 1.05 -1.05 0.486
3 21 0 17 1 0.55 -0.55 0.247
4 21 0 16 2 1.14 -1.14 0.477 Log-rank Statistic
5 21 0 14 2 1.20 -1.20 0.466
6 21 3 12 0 1.91 1.09 0.651
7 17 1 12 0 0.59 0.41 0.243
8 16 0 12 4 2.29 -2.29 0.871
10 15 1 8 0 0.65 0.35 0.227
11 13 0 8 2 1.24 -1.24 0.448 Chi-square p-value
12 12 0 6 2 1.33 -1.33 0.418
13 12 1 4 0 0.75 0.25 0.188
15 11 0 4 1 0.73 -0.73 0.196
16 11 1 3 0 0.79 0.21 0.168
17 10 0 3 1 0.77 -0.77 0.178
22 7 1 2 1 1.56 -0.56 0.302
23 6 1 1 1 1.71 -0.71 0.204
Total: 9 21 -10.25 6.257
16.7929
0.00004

16. 0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0 10 20 30

17. Survival data vs. two-by-two table = different
Timen1 m1 q1 S1 n2 m2 q2 S2
0 21 0 100% 21 0 100%
1 21 0 100% 21 2 90%
2 21 0 100% 19 2 81%
3 21 0 100% 17 1 76%
4 21 0 100% 16 2 67%
5 21 0 100% 14 2 57%
6 21 3 1 86% 12 0 57%
7 17 1 1 81% 12 0 57%
8 16 0 81% 12 4 38%
10 15 1 2 75% 8 0 38%
11 13 0 75% 8 2 29%
12 12 0 75% 6 2 19%
13 12 1 69% 4 0 19%
15 11 0 69% 4 1 14%
16 11 1 3 63% 3 0 14%
17 10 0 63% 3 1 10%
22 7 1 54% 2 1 5%
23 6 1 5 45% 1 1 0%
Total: 9 21
Event no-event Total
Rx1 9 12 21
Rx2 21 0 21
Total 30 12 42
Surv. Rx1 =12/21= 57.1%
Surv. Rx2 =0/21= 0.0%

18. Log-Rank test for several groups
 The null hypothesis is that all the survival curves are
the same.
 Log-rank statistic is given by the sum:
 This statistic has Chi-square distribution with (# of
groups – 1) degrees of freedom.
 
 
 






groupsof
i i
ii
groupsof
i ii
ii
E
EO
EOVar
EO
X
__# 2__# 2
2

19. Cox Proportional Hazards Regression
 Most interesting survival-analysis research examines the
relationship between survival — typically in the form of the
hazard function — and one or more explanatory variables (or
covariates).
 Most common are linear-like models for the log hazard.
 For example, a parametric regression model based on the
exponential distribution,
 Needed to assess effect of multiple covariates on survival
 Cox-proportional hazards is the most commonly used
multivariate survival method
 Easy to implement in SPSS, Stata, or SAS
 Parametric approaches are an alternative, but they require
stronger assumptions about h(t).

20.  Assumes multiplicative risk—this is the proportional hazard
assumption
 Conveniently separates baseline hazard function from
covariates
 Baseline hazard function over time
 Covariates are time independent
 Nonparametric
 Can handle both continuous and categorical predictor
variables (think: logistic, linear regression)
 Without knowing baseline hazard ho(t), can still calculate
coefficients for each covariate, and therefore hazard ratio
Multivariate methods:
Cox proportional hazards

21. Limitations of Cox PH model
 Covariates normally do not vary over time
 True with respect to gender, ethnicity, or congenital
condition
 One can program time-dependent variables
 Baseline hazard function, ho(t), is never specified, but
Cox PH models known hazard functions
 You can estimate ho(t) accurately if you need to estimate
S(t).

22. Hazard Ratio
 Interesting to interpret
For example, if HR = 0.70, we can deduce the following:
 Relative effect on survival is
or 30% reduction of the risk of death
 Absolute Difference in survival is given as
so, if S = 60%,
which represents a 10% difference.
 Difference in median survival is given as the difference between the
median/HR and the median. For example, if the median is
months, then the difference is given as
or 10.71 months increase in median survival.
30.070.011  HR
 
SSSe HRHRS
ln
10.060.060.0 70.0
AbsDiff
25
71.1025
70.0
25

V
EO
eHR

