essentials of survival analysis how to practice evidence based oncology european school of oncology...
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Essentials of survival analysis How to practice evidence based
oncology European School of OncologyJuly 2004Antwerp, Belgium
Dr. Iztok HozoProfessor of MathematicsIndiana University NWwww.iun.edu/~mathiho
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
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.
Example:
X means an event occurred; O means that the subject was censored.
Example 2 (from Kleinbaum: “Survival Analysis”)
Patient Time (t) Censor ()
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)
Survival Curve - Calculus S(t) = cumulative survival function = proportion that survive until time t
f(t) = frequency distribution of age at deathh(t) = hazard function (i.e. death rate at age t) = event rate
Relationships:
t
duuht
eduuftTPtS 0
dt
tdStf
tS
dt
d
tS
tf
TtPt
ttTtPt
TtttTtPth
t
t
lnlim
|lim
0
0
Distribution Function, Survival Function and Density Function
)(1)Pr()( tFtTtS
Probability Distribution function
Probability Density function
Survival function t
tFtf
)(
)(
)Pr()( tTtF
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:
Kaplan-Meier Product Limit EstimateConsider 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 nj 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
t181
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
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
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
WarfASA
No Rx
Age 76 Years and Older (N = 394)
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
eventstotalgroupExpected
__#
__##
12
21
NN
MNMnnVar
ExpectedObservedEO
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 expectedThis 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
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.
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 0.4882 21 0 19 2 1.05 -1.05 0.4863 21 0 17 1 0.55 -0.55 0.2474 21 0 16 2 1.14 -1.14 0.477 Log-rank Statistic5 21 0 14 2 1.20 -1.20 0.4666 21 3 12 0 1.91 1.09 0.6517 17 1 12 0 0.59 0.41 0.2438 16 0 12 4 2.29 -2.29 0.87110 15 1 8 0 0.65 0.35 0.22711 13 0 8 2 1.24 -1.24 0.448 Chi-square p-value12 12 0 6 2 1.33 -1.33 0.41813 12 1 4 0 0.75 0.25 0.18815 11 0 4 1 0.73 -0.73 0.19616 11 1 3 0 0.79 0.21 0.16817 10 0 3 1 0.77 -0.77 0.17822 7 1 2 1 1.56 -0.56 0.30223 6 1 1 1 1.71 -0.71 0.204
Total: 9 21 -10.25 6.257
16.7929
0.00004
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0 10 20 30
Survival data vs. two-by-two table = differentTimen1 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 TotalRx1 9 12 21Rx2 21 0 21Total 30 12 42
Surv. Rx1 =12/21= 57.1%
Surv. Rx2 =0/21= 0.0%
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
iigroupsof
i ii
ii
E
EO
EOVar
EOX
__# 2__# 22
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).
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
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).
Hazard Ratio
Interesting to interpretFor 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
71.102570.0
25
V
EO
eHR