Analysis of Survival Data

Time to Event outcomesCensoringSurvival FunctionPoint estimationKaplan-Meier

Introduction to survival analysis

What makes it different? Three main variable types

Continuous Categorical Time-to-event

Examples of each

Example: Death Times of Psychiatric Patients (K&M 1.15)

Dataset reported on by Woolson (1981) 26 inpatient psychiatric patients admitted

to U of Iowa between 1935-1948. Part of larger study Variables included:

Age at first admission to hospital Gender Time from first admission to death (years)

Data summarygender age deathtime death1 51 1 11 58 1 11 55 2 11 28 22 10 21 30 00 19 28 11 25 32 11 48 11 11 47 14 11 25 36 01 31 31 00 24 33 00 25 33 01 30 37 01 33 35 00 36 25 10 30 31 00 41 22 11 43 26 11 45 24 11 35 35 00 29 34 00 35 30 00 32 35 11 36 40 10 32 39 0

. tab gender

gender | Freq. Percent Cum.------------+----------------------------------- 0 | 11 42.31 42.31 1 | 15 57.69 100.00------------+----------------------------------- Total | 26 100.00

0.0

1.0

2.0

3.0

4D

ensi

ty

20 30 40 50 60age

. sum age

Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- age | 26 35.15385 10.47928 19 58

Death time?0

.01

.02

.03

.04

.05

Den

sity

0 10 20 30 40deathtime

. sum deathtime

Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- deathtime | 26 26.42308 11.55915 1 40

Does that make sense?

Only 14 patients died The rest were still alive at the end of the study Does it make sense to estimate mean? Median? How can we interpret the histogram? What if all had died? What if none had died?

. tab death

death | Freq. Percent Cum.------------+----------------------------------- 0 | 12 46.15 46.15 1 | 14 53.85 100.00------------+----------------------------------- Total | 26 100.00

CENSORING

Different types Right Left Interval

Each leads to a different likelihood function

Most common is right censored

Right censored data

“Type I censoring” Event is observed if it occurs before

some prespecified time Mouse study Clock starts: at first day of

treatment Clock ends: at death Always be thinking about ‘the clock’

Simple example: Type I censoring

Time 0

Introduce “administrative” censoring

Time 0 STUDY END

Introduce “administrative” censoring

Time 0 STUDY END

More realistic: clinical trial

Time 0 STUDY END

“Generalized Type I censoring”

More realistic: clinical trial

Time 0 STUDY END

“Generalized Type I censoring”

Additional issues

Patient drop-out Loss to follow-up

Drop-out or LTFU

Time 0 STUDY END

How do we ‘treat” the data?

Time of enrollment

Shift everythingso each patient timerepresents timeon study

Another type of censoring:Competing Risks

Patient can have either event of interest or another event prior to it

Event types ‘compete’ with one another Example of competers:

Death from lung cancer Death from heart disease

Common issue not commonly addressed, but gaining more recognition

Left Censoring

The event has occurred prior to the start of the study

OR the true survival time is less than the person’s observed survival time

We know the event occurred, but unsure when prior to observation

In this kind of study, exact time would be known if it occurred after the study started

Example: Survey question: when did you first smoke? Alzheimers disease: onset generally hard to

determine HPV: infection time

Interval censoring

Due to discrete observation times, actual times not observed

Example: progression-free survival Progression of cancer defined by change in

tumor size Measure in 3-6 month intervals If increase occurs, it is known to be within

interval, but not exactly when. Times are biased to longer values Challenging issue when intervals are long

Key components

Event: must have clear definition of what constitutes the ‘event’ Death Disease Recurrence Response

Need to know when the clock starts Age at event? Time from study initiation? Time from randomization? time since response?

Can event occur more than once?

Time to event outcomes

Modeled using “survival analysis” Define T = time to event

T is a random variable Realizations of T are denoted t T 0

Key characterizing functions: Survival function Hazard rate (or function)

Survival Function

S(t) = The probability of an individual surviving to time t

Basic properties Monotonic non-increasing S(0)=1 S(∞)=0*

* debatable: cure-rate distributions allow plateau at someother value

Example: exponential

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lambda=0.1lambda=0.05lambda=0.01

Weibull example

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lam=0.05,a=0.5lam=0.05,a=1lam=0.01,a=0.5lam=0.01,a=1

Applied example

Van Spall, H. G. C., A. Chong, et al. (2007). "Inpatient smoking-cessation counseling and all-cause mortality in patients with acute myocardial infarction." American Heart Journal 154(2): 213-220.

Background Smoking cessation is associated with improved health outcomes, but the prevalence, predictors, and mortality benefit of inpatient smoking-cessation counseling after acute myocardial infarction (AMI) have not been described in detail.

Methods The study was a retrospective, cohort analysis of a population-based clinical AMI database involving 9041 inpatients discharged from 83 hospital corporations in Ontario, Canada. The prevalence and predictors of inpatient smoking-cessation counseling were determined.

Results…..Conclusions Post-MI inpatient smoking-cessation counseling is an

underused intervention, but is independently associated with a significant mortality benefit. Given the minimal cost and potential benefit of inpatient counseling, we recommend that it receive greater emphasis as a routine part of post-MI management.

Applied exampleAdjusted 1-year survival curves of counseled smokers, noncounseled smokers, and never-smokers admitted with AMI (N = 3511). Survival curves have been adjusted for age, income quintile, Killip class, systolic blood pressure, heart rate, creatinine level, cardiac arrest, ST-segment deviation or elevated cardiac biomarkers, history of CHF; specialty of admitting physician; size of hospital of admission; hospital clustering; inhospital administration of aspirin and β-blockers; reperfusion during index hospitalization; and discharge medications.

Hazard Function

A little harder to conceptualize Instantaneous failure rate or conditional failure rate

Interpretation: approximate probability that a person at time t experiences the event in the next instant.

Only constraint: h(t)0 For continuous time,

t

tTttTtPth

t

)|(lim)(

0

)(ln)(/)()( tStStfth dtd

Hazard Function

Useful for conceptualizing how chance of event changes over time

That is, consider hazard ‘relative’ over time Examples:

Treatment related mortality Early on, high risk of death Later on, risk of death decreases

Aging Early on, low risk of death Later on, higher risk of death

Shapes of hazard functions

Increasing Natural aging and wear

Decreasing Early failures due to device or transplant

failures Bathtub

Populations followed from birth Hump-shaped

Initial risk of event, followed by decreasing chance of event

Examples

0 1 2 3 4 5 6

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zard

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Median

Very/most common way to express the ‘center’ of the distribution

Rarely see another quantile expressed Find t such that

Complication: in some applications, median is not reached empirically

Reported median based on model seems like an extrapolation

Often just state ‘median not reached’ and give alternative point estimate.

5.0)( tS

X-year survival rate

Many applications have ‘landmark’ times that historically used to quantify survival

Examples: Breast cancer: 5 year relapse-free survival Pancreatic cancer: 6 month survival Acute myeloid leukemia (AML): 12 month

relapse-free survival Solve for S(t) given t

Competing Risks

Used to be somewhat ignored. Not so much anymore Idea:

Each subject can fail due to one of K causes (K>1)

Occurrence of one event precludes us from observing the other event.

Usually, quantity of interest is the cause-specific hazard

Overall hazard equals sum of each hazard:

K

kkT thth

1

)()(

Example Myeloablative Allogeneic Bone Marrow

Transplant Using T Cell Depleted Allografts Followed by Post-Transplant GM-CSF in High Risk Myelodysplastic Syndromes

Interest is in RELAPSE Need to account for treatment related

mortality (TRM)? Should we censor TRM?

No. that would make things look more optimistic

Should we exclude them? No. That would also bias the

results Solution:

Treat it as a competing risk Estimate the incidence of both

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Time from BMT (Months)

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RelapseTRM

Estimating the Survival Function

Most common approach abandons parametric assumptions

Why? Not one ‘catch-all’ distribution No central limit theorem for large

samples

Censoring

Assumption: Potential censoring time is unrelated to the

potential event time Reasonable?

Estimation approaches are biased when this is violated

Violation examples Sick patients tend to miss clinical visits more

often High school drop-out. Kids who move may be

more likely to drop-out.

Terminology

D distinct event times t1 < t2 < t3 < …. < tD

ties allowed at time ti, there are di deaths Yi is the number of individuals at risk at ti

Yi is all the people who have event times ti

di/Yi is an estimate of the conditional probability of an event at ti, given survival to ti

Kaplan-Meier estimation

AKA ‘product-limit’ estimator

Step-function Size of steps depends on

Number of events at t Pattern of censoring before t

tt

Yd

i

i

i tt

tttS

1

1

if ]1[

if 1)(ˆ

Kaplan-Meier estimation

Greenwood’s formula Most common variance estimator Point-wise

tt iii

i

idYY

dtStSV

)()(ˆ)](ˆ[ˆ 2

Example:

Kim paper Event = time to relapse Data:

10, 20+, 35, 40+, 50+, 55, 70+, 71+, 80, 90+

Plot it:

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Time to relapse (months)

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Interpreting S(t)

General philosophy: bad to extrapolate

In survival: bad to put a lot of stock in estimates at late time points

Fernandes et al: A Prospective Follow Up of Alcohol Septal Ablation For Symptomatic Hypertrophic Obstructive Cardiomyopathy The Ten-Year Baylor and MUSC Experience (1996-2007)”

R for KMlibrary(survival)library(help=survival)

t <- c(10,20,35,40,50,55,70,71,80,90)d <- c(1,0,1,0,0,1,0,0,1,0)cbind(t,d)

st <- Surv(t,d)st

help(survfit)fit.km <- survfit(st)fit.kmsummary(fit.km)attributes(fit.km)

plot(fit.km, conf.int=F, xlab="time to relapse (months)",ylab="Survival Function“, lwd=2)

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Kaplan-Meier Curve