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

dttdStf

tS

dtd

tStf

TtPtttTtPt

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 ttFtf

)()(

)Pr()( tTtF

Creating a Kaplan-Meier curve

j

jj

ndn

211 | jjj tTtTPt

1

11

1

11

0132

21

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|

ndn

ndn

ndn

tTtTPtTtTP

tTtTPtTPtS

j

jj

j

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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

ndn

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

EEO

EOVarEOX

__# 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 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

71.102570.0

25

VEO

eHR