Basic Concepts in Survival
Analysis
Mark A. Weaver, PhD
Family Health International
Office of AIDS Research, NIH
ICSSC, FHI
Goa, India, September 2009
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“Survival Analysis”
• Seems to, unfortunately, imply studying
deaths
– Indeed, this is what prompted development of
these methods
– Insurance companies had great interest in
estimating the probability that a new client
would soon die or get really sick (“actuarial
methods”)
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Other Names
• Some other, less morbid, names for survival
analysis methods:
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Time-to-event analysis
Failure-time analysis
Event-history analysis
Duration analysis
Transition analysis
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Survival Outcomes
• Outcome variable: time until an event occurs
– Note that this is like a composite outcome with 2
components
• time – usually continuous
• event – a transition from one well-defined state to
another well-defined state.
– alive / dead
– no disease / disease
– not married / married
– unemployed / employed
• “Survival analysis was designed for longitudinal data on the
occurrence of events.” Allison 1995
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Survival Data - Time
• All subjects must have a precisely defined time
origin (Time 0)
– Typically date of randomization in RCT
– Must be able to exclude potential subjects who have event
prior to time 0 (e.g., use screening test with high
sensitivity)
• Typically follow subjects prospectively until they
have an event or until they leave the study
– In an RCT, time is often calculated in days
– Date of event or date left study – date of randomization
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Survival Data – End-Points
• Must have clearly defined case definition, and
must be able to identify subjects who have an
event
– Typically simply a 0/1 variable in dataset
• Typically, assume at most 1 event / subject
– If multiple events are possible, might use “time to first
event”
– Methods for repeated events are available, but not
often used for various reasons
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Censoring
• Ok, so what’s the big deal? Why do we need
special analysis methods?
• Not all subjects will have an event during the
study period (or possibly ever) AND
• Not all subjects will be followed for the same
amount of time.
• This combination makes “standard” analysis
methods (e.g., t-tests, ANOVA, Chi-Squared
tests, logistic regression, etc.) inappropriate
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Censoring
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Subjects for whom no event is observed are referred to
as censored.
Some possible reasons for censoring:
1. Study ends before subject has an event
2. Subject withdrawals from study
3. Subject is lost to follow-up
4. Subject has another kind of event that precludes
observing event of interest (e.g., subject dies, or
event is incidence of preterm birth but subject has
abortion)
Survival analysis has built-in methods for dealing with
these types of missing data
– As long as censoring is not related to event (i.e.,
requires “non-informative” censoring).
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Simple Example
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Clinical trial for preventing secondary strokes among sickle
cell patients over 5 years
2 treatments: New Drug (ND), Standard Transfusions (ST)
ND: 3+, 3, 4, 5, 5+
ST: 1, 2+, 3, 4+, 5+
Naïve analysis methods
1. Ignore +’s and compare mean survival times
ND: 20/5 = 4
ST: 15/5 = 3
ND better?
2. Compare “raw” proportions of subjects with event
ND: 60%
ST: 40%
ST better?
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Right Censoring
• Assume each subject will have some (possibly
unobserved) event time T
• If you observe that a subject has the event at time T, then
subject is uncensored.
• If all you know is that T is greater than some value c, then
the subject is right censored at c.
– E.g., suppose a subject completes a 1-year study
without event interest, all you know is that T > 1 year.
– By far, the most common type of censoring!
– All survival analysis methods easily handle right
censoring by default.
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Subjects
Right-Censored Data
A
Uncensored
B
Censored
C
Uncensored
D
Censored
E
Censored
0
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Months in Study
End of Study
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3 Types of Right-Censoring
• Type I censoring – study ends when a certain
time point is reached
– E.g., each subject is intended to be followed for 1
year
• Type II censoring – study ends when a certain
number of events occur
– E.g., study ends after 82 events have occurred
• Random censoring – observation terminated for
reasons not under control of investigator
– E.g., participant dropout
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Other Kinds of Censoring
• Interval Censoring: you don’t observe actual
time of event, all you know is that a < T < b for
some values a and b.
– Testing for STI at 0, 6, and 12 months. If negative at
6 months but positive at 12 months, then you know
that 6 < T < 12.
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Other Kinds of Censoring
• Left Censoring: you don’t observe actual time of
event, all you know is that T happened prior to
time of observation.
– Special case of interval censoring when a = 0.
– E.g., studying menarche and begin following girls at
age 12 – some will already have started
menstruating.
– Typically not an issue in RCTs
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Two Key Functions
• Two functions are fundamental and of central
interest in survival analysis:
– S(t) = survivor function
– h(t) = hazard function
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Survivor Function
• S(t) = Pr{ T > t }
= Prob. of surviving longer than time t
• Estimating survival probabilities for different,
meaningful values of t provides crucial summary
information for survival data.
– E.g., in studying breast cancer, one might be
particularly interested in the 5- and 10-year survival
rates.
• Note: Probability of having an event by time t
(failure probability) is just 1 – S(t).
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Theoretical Properties of Survivor Function
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S(t)
0
t
→∞
• S(0) = 1
• S(t) is non-increasing, smooth function
• S(∞) = 0
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Hazard Function
⎧ P(t ≤ T < t + Δt | T ≥ t )⎫
h(t ) = lim ⎨
⎬
Δt →0 ⎩
Δt
⎭
• h(t) = instantaneous event (or failure) rate at time t,
given that event hasn’t occurred up until time t
• Sometimes called “instantaneous failure
probability”, but it’s really not a probability
– “Potential”? “Propensity”?
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Theoretical Properties of Hazard Functions
h(t)
0
• h(t) ≥ 0
• no upper bound
t
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Special Case - Exponential Failure Times
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h(t)
S(t)
0
t
• h(t) = λ
• S(t) = exp{ -λt }
• Common assumption for sample size calculations
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