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Statistical Analysis:Time to Event Endpoints

Marta FioccoUtrecht, April 3, 2014

LUMC & SKION

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What is survival analysis? Survival analysis is the study of the distribution of

life times, i.e. the times from an initiating event (birth, diagnosis, start of treatment) to some terminal event (relapse, death).

It is most prominently (but not only) used in the biomedical sciences.

Special feature of survival data: need time to observe the event of interest.

As a result for a number of subjects the event is not observed, it is only known that the event has not taken place yet.

This phenomenon is called censoring and it requires special statistical methods.

Censoring means: observation ends before occurrence of event of interest

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Examples of events Death Cure Response to treatment Relapse of disease

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What do we wish to know? Probability distribution of the survival time Effect of treatment increase on survival Which patient characteristics determine the

survival prognosis Make predictions about survival based on

the clinical history of the patient

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Causes of Censoring

End of study Loss to follow-up (drop-out, removal,

emigration, etc.) censoring due to another terminating event

(e.g. death due to other cause) Competing risk

Censoring occurs when the value of an observation is only partially known.

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Example (in “calendar time”)

0123456

0 3 6 9 12 15 18 21 24 27 30 33 36

Months since start study

Patie

nt

The study has a duration of 36 months:Patient 1 was included at the start and died after

4 monthsPatient 2 was included at the start and died after

9 monthsPatient 3 was included after 4 months and died

after 24 monthsPatients 4 and 6 were included after 24 months

and were alive when the study stoppedPatient 5 was included at the start and was lost-

to-follow-up at 33 months

†††

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Example in follow-up time

0123456

0 3 6 9 12 15 18 21 24 27 30 33 36

Months since start study

patient Survival time

status

1 4 12 9 16 12 04 12 03 20 15 33 0

††

†

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The survival function S(t)

S(t) is the probability that the event has not yet occurred at time t

Example S(30)=0.45

follow-up time

706050403020100

prob

abilit

y of

bei

ng a

live

1.0

.9

.8

.7

.6

.5

.4

.3

.2

.1

.0

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How to estimate this curve? Observations:

12, 15, 16+, 30, 35, 41+, 43+, 50, 52, 60+ (+ : censored)

• Censored observations cannot be left out: they contain information (for example, the last person here lives until at least 60 months!)

• Assumption: the reason of censoring does not depend on survival• NOT: the patient does not show up since he/she is too sick or

too well !!

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Kaplan-Meier curveObservations: 12, 15, 16+, 30, 35, 41+, 43+, 50, 52, 60+

Time Number at risk

Number dead

Risk of dying Fraction survivors

Probability of survival

0 10 1

12 10 1 1/10 9/10 9/10=0.9

15 9 1 1/9 8/9 8/9*0.9=0.8

16 8 0 0 1 0.830 7 1 1/7 6/7 6/7*0.8=0.69

35 6 1 1/6 5/6 5/6*0.69=0.5841 5 0 0 1 0.58

43 4 0 0 1 0.58

50 3 1 1/3 2/3 2/3*0.58=0.39

52 2 1 1/2 1/2 1/2*0.39-0.20

60 1 0 0 `1 0.20

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Kaplan-Meier curvepicture obtained in SPSS

Survival Function

TIME

70605040302010

Cum

Sur

viva

l

1.0

.8

.6

.4

.2

0.0

Survival Function

Censored

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Confidence intervals (pointwise)Part of SPSS-outputTime Status Cumulative

Survival Standard Error

12 1 0.9000 0.0949

15 1 0.8000 0.1265

16 0

30 1 0.6857 0.1515

35 1 0.5714 0.1638

41 0

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Estimate and 95% Confidence Interval at each time

A picture with pointwise confidence intervals

time

6050403020100

surv

ival

pro

babi

lity

1.0

.8

.6

.4

.2

0.0

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Median survival time

0.5In example: median survival time is 50 months

Survival Function

time

706050403020100

prob

abilt

y of

sur

viva

l 1.0

.8

.6

.4

.2

0.0

Survival Function

Censored

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Comparing two survival curves In a clinical trial, we want to know whether

new treatment is better than control/placebo

Better means that patients live longer, have longer time to treatment failure

So we want to compare two (or more) survival curves

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Example Dutch Gastric Cancer Trial 711 patients with gastric cancer Randomized between limited (D1-)

dissection and extensive (D2-) dissection Expected: more postoperative mortality in

D2 group, but fewer cancer related deaths in long term

Endpoint: overall survival

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Results over first two years

0.00 0.50 1.00 1.50 2.00Years since surgery

0.0

0.2

0.4

0.6

0.8

1.0

Cum

Sur

viva

l

randomisatie groepD1D2D1-censoredD2-censored

Survival Functions Overall survival in D1 seems to be higher than in D2

Is this difference statistically significant?

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Logrank test Null hypothesis: no difference in survival Test uses ordering like Wilcoxon test Test is two-sided by construction Reject if P-value too small In example P=0.057 With alpha-level of 0.05, null hypothesis is not

rejected Not sufficient evidence of difference in overall

survival between D1- and D2-dissection

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Risk of death: the hazard Instantaneous rate of death Probability of dying in next (small) interval,

conditional on being alive Formally

Higher hazard means individuals die quicker (at higher rate)

ttTttTtPth

t

)|(lim)(0

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Example

Hazard is higher for D2-dissectionEspecially in the beginning patients in the D2-group have a higher overall death rate

0.0 0.5 1.0 1.5

0.00

50.

010

0.01

50.

020

0.02

50.

030

Years since surgery

Haz

ard

D1-dissectionD2-dissection

Hazards for D1- and D2- dissection

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Cox proportional hazards model In previous example, we have two hazard rates,

h1(t) and h2(t) Simple model: ratio of these hazards is

constant This ratio is called hazard ratio Model: Or also Beta is regression coefficient, exp(beta) is hazard ratio,

h0(t) is baseline hazard Implemented in most statistical packages, SAS, SPSS,

R/S-plus etc

)(/)( 12 thth

)exp()(/)( 12 thth)exp()()|( 0 ZthZth

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Typical outputVariables in the Equation

.238 .125 3.602 1 .058 1.268 .992 1.621randgrB SE Wald df Sig. Exp(B) Lower Upper

95.0% CI for Exp(B)

Hazard ratio (Exp(B)) is estimated to be 1.27P-value=0.058 indicates that hazard ratio is not significantly different from 195%-confidence interval for HR: 0.99 – 1.62This hazard ratio is of D2-dissection with respect to D1-dissection; this should be clear from context

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Proportional hazards regression Cox’ proportional hazards model can be extended to

more explanatory variables than treatment alone Model:

We now have a hazard ratio for each explanatory variables

This can be used to adjust treatment effect for possible confounding factors, or to build prognostic models

)...exp()()|( 22110 ppZZZthZth

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Example Nodal status (N0, N1, N2, N3) and treatment Three hazard ratios for nodal status, first is N1

wrt N0 (2.67), second is N2 wrt N0 (5.22), third is N3 wrt N0 (8.04)

Nodal status far more important determinant of death rate than D1- or D2-dissection

Variables in the Equation

.155 .126 1.510 1 .219 1.168 .912 1.495119.417 3 .000

.981 .170 33.257 1 .000 2.666 1.911 3.7211.653 .187 77.886 1 .000 5.220 3.617 7.5352.084 .214 95.081 1 .000 8.036 5.286 12.216

randgrn0123n0123(1)n0123(2)n0123(3)

B SE Wald df Sig. Exp(B) Lower Upper95.0% CI for Exp(B)

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Non-standard situations Non-proportional hazards Time-dependent covariates Informative censoring

Just to make you aware of these issues

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Non-proportional hazards The important assumption of the Cox

proportional hazards model is that the hazard ratio h2(t)/h1(t) is a constant

It should not vary over time Especially with long follow-up this

assumption is not always fulfilled

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Dutch Gastric Cancer trial I only showed the first

two years, look what happens over 10 years

Hazard ratio is above 1 for the first two years, below 1 after 2 years

Points to a short-term benefit of D1, long-term benefit of D2

0 2 4 6 8 10

0.00

50.

010

0.01

50.

020

0.02

50.

030

Years since surgery

Haz

ard

D1-dissectionD2-dissection

Hazards for D1- and D2- dissection

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The corresponding survival curves

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00survival since surgery (yrs)

0.0

0.2

0.4

0.6

0.8

1.0

Cum

Sur

viva

l

randomisatie groepD1D2D1-censoredD2-censored

Survival Functions

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Cox regression output

Variables in the Equation

-.057 .092 .386 1 .535 .944 .789 1.131randgrB SE Wald df Sig. Exp(B) Lower Upper

95.0% CI for Exp(B)

Nothing here indicates violation of proportional hazardsPositive effects in the beginning and negative effects later on cancel out, result: a hazard ratio of practically 1But this is not reality….

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Non-proportional hazards Often leads to crossing survival curves It is possible to test for evidence of non-

proportional hazards (by including interaction with time)

Hazard ratio varies over time (can be modeled)

Consult a statistician

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Informative censoring All procedures for survival data that we have seen

(Kaplan-Meier, logrank test, Cox regression) assume that censoring is independent of time to event

Individual who is censored has same risk of dying as individual who is not censored

Important to think through whether this is a reasonable assumption

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Consider the following reasons for censoring End of study Patient is too ill to come to the hospital for routine follow-

up Patient no longer needs treatment medication or care that

trial provides Patient has died from cause other than the cause of

interest

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Consider the following reasons for censoring End of study

Usually OK Patient is too ill to come to the hospital for routine follow-

up Survival estimates are based on remaining “healthier” patients, so

too optimistic Patient no longer needs treatment medication or care that

trial provides Survival estimates based on remaining “sicker” patients, so too

pessimistic Patient has died from cause other than the cause of

interest Competing risks

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References Kaplan EL, Meier P (1958). Nonparametric estimation from

incomplete observations. J Am Stat Assoc 53: 457-481 Cox DR (1972). Regression models and life tables. J Roy

Stat Soc B: 187-220 Klein JP, Moeschberger ML. Survival Analysis. Springer,

New York Kleinbaum DG, Klein M. Survival Analysis: A Self-learning

Text. Springer, New York Clark TG, Bradburn MJ, Love SB, Altman DG (2003).

Survival analysis part I: basic concepts and first analyses. Br J Cancer 89: 232-238. (three more parts in later issues)