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### Analysis of Survival Data

Time to Event outcomes

Censoring

Survival Function

Point estimation

Kaplan-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 summary

. tab gender

gender | Freq. Percent Cum.

------------+-----------------------------------

0 | 11 42.31 42.31

1 | 15 57.69 100.00

------------+-----------------------------------

Total | 26 100.00

gender age deathtime death

1 51 1 1

1 58 1 1

1 55 2 1

1 28 22 1

0 21 30 0

0 19 28 1

1 25 32 1

1 48 11 1

1 47 14 1

1 25 36 0

1 31 31 0

0 24 33 0

0 25 33 0

1 30 37 0

1 33 35 0

0 36 25 1

0 30 31 0

0 41 22 1

1 43 26 1

1 45 24 1

1 35 35 0

0 29 34 0

0 35 30 0

0 32 35 1

1 36 40 1

0 32 39 0

. sum age

Variable | Obs Mean Std. Dev. Min Max

-------------+--------------------------------------------------------

age | 26 35.15385 10.47928 19 58

Death time?

. sum deathtime

Variable | Obs Mean Std. Dev. Min Max

-------------+--------------------------------------------------------

deathtime | 26 26.42308 11.55915 1 40

Does that make sense?

. tab death

death | Freq. Percent Cum.

------------+-----------------------------------

0 | 12 46.15 46.15

1 | 14 53.85 100.00

------------+-----------------------------------

Total | 26 100.00

- 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?

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’

Additional issues

- Patient drop-out
- Loss to follow-up

How do we ‘treat” the data?

Shift everything

so each

patient time

represents time

on study

Time of

enrollment

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 some

other value

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 example

Adjusted 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,

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

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.

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:

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

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

Kaplan-Meier estimation

- Greenwood’s formula
- Most common variance estimator
- Point-wise

Example:

- Kim paper
- Event = time to relapse
- Data:
- 10, 20+, 35, 40+, 50+, 55, 70+, 71+, 80, 90+

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 KM

library(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.km

summary(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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