INTRODUCTION TO CLINICAL RESEARCH Survival Analysis – Getting Started Karen Bandeen-Roche, Ph.D.
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INTRODUCTION TO CLINICAL RESEARCH
Survival Analysis – Getting Started
Karen Bandeen-Roche, Ph.D.
July 20, 2010
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Acknowledgements
• Scott Zeger
• Marie Diener-West
• ICTR Leadership / Team
July 2010 JHU Intro to Clinical Research 2
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Introduction to Survival Analysis
1. Thinking about times to events; contending with “censoring”
2. Counting process view of times to events
3. Hazard and survival functions
4. Kaplan-Meier estimate of the survival function
5. Future topics: log-rank test; Cox proportional hazards model
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“Survival Analysis”
• Approach and methods for analyzing times to events
• Events not necessarily deaths (“survival” is historical term)
• Need special methods to deal with “censoring”
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Typical Clinical Study with Time to Event Outcome
Start End Enrollment End Study
0 2 4 6 8 10
Calendar time
Loss to Follow-up
Event
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Switching from Calendar to Follow-up Time
0 2 4 6 8 10
Follow-up time
>3 5
>8
1
>6
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The Problem with Standard Analyses of Times to Events
• Mean: (1 + 3 + 5 + 6 + 8)/5 = 4.6 - right?
• Median: 5 – right?
• Histogram
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Censoring
> 3 is not 3, it may be 33
Mean is not 4.6, it may be (1 + 33 + 5 + 6 + 8)/5 = 10.6
Or any value greater than 4.6
> 3 is a right “censored value” – we only know the value exceeds 3
> x is often written “x+”
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• Uncensored data: The event has occurred– Event occurrence is observed
• Censored data: The event has yet to occur– Event-free at the current follow-up time– A competing event that is not an endpoint stops
follow-up– Death (if not part of the endpoint)– Clinical event that requires treatment, etc.– Our ability to observe ends before event happens
Censoring
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Contending with Censored Data
Standard statistical methods do not work for censored data
We need to think of times to events as a natural history in time, not just a single number
Issue: If no events are reported in the interval from last follow-up to “now”, need to choose between:
No news is good news?No news is no news?
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One Option: Overall Event Rate
• Example: 2 events in 23 person months = 1 event per 11.5 months = 1.04 events per year = 104 events per 100 person-years
• Gives an average event rate over the follow-up period; actual event rate may vary over time
• For a finer time resolution, do the above for small intervals
# eventsEvent rate = total observation time
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Switching from Calendar to Follow-up Time
0 2 4 6 8 10
Follow-up time
>3 5
>8
1
>6
3+5+8+1+6 person months of observation; 2 actual events
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Second Option: Natural history“One day at a time”
0 2 4 6 8 10
Follow-up time
>3 5
>8
1>6
0 0 00 0 0 0 1
0 0 0 0 0 0 0 0
1
0 0 0 0 0 0
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Thinking about Times to EventsInterval of follow-up
Event Times 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8
1 13+ 0 0 05 0 0 0 0 16+ 0 0 0 0 0 08+ 0 0 0 0 0 0 0 0No. at risk 5 4 4 3 3 2 1 1Fraction of events=“hazard”
0.2 0.0 0.0 0.0 .33 0.0 0.0 0.0
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Survival Function
“Survival function”, S(t), is defined to be the probability a person survives beyond time t
S(0) = 1.0
S(t+1) S(t)
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Hazard Function• Hazard at time t, h(t), is the probability per unit time of
having the event in a small interval around time t
• Force of mortality
• ~ Pr{event in (t,t+dt)}/dt
• Need not be between 0 and 1 because it is per unit time
• h(t) ~ {S(t)-S(t+dt)}/{S(t) dt}
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• Basic idea: Live your life one interval (day, month, or year) at a time
• Example:S(3) = Pr(survive for 3 months)
= Pr(survive 1st month) × Pr(survive 2nd month | survive 1st month) × Pr(survive 3rd month | survive 2nd month)
• Thus, S(2) S(3)S(3) = S(1)S(1) S(2)
= Pr(survive for 1st month & 2nd & 3rd)
Hazard Function
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Estimating the Survival Function: Kaplan-Meier Method
Interval of Follow-Up
Times 0 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8
No. at risk 5 5 4 4 3 3 2 1 1No of events 0 1 0 0 0 1 0 0 0Fraction of events=“hazard”
- 0.2 0.0 0.0 0.0 .33 0.0 0.0 0.0
Fraction without event in interval
- 0.8 1.0 1.0 1.0 0.67 1.0 1.0 1.0
Fraction without event since start
1.0 0.8 0.8 0.8 0.8 0.53 0.53 0.53 0.53
Pr(survive past 5) = Pr(survive past 5|survive past 4) *Pr(survive past 4)
[ = Pr(survive past 5 and survive past 4) ]
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Displaying the Survival Function
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Notes on Estimating Survival Function• Estimate only changes in intervals where an event
occurs
• Censored observations contribute to denominators, but never to numerators
• Intervals are arbitrary; want narrow ones
• Kaplan-Meier estimate results from using infinitesimal interval widths
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Acute Myelogenous Leukemia ExampleData: 5,5,8,8,12,16+, 23, 27, 30+, 33, 43,45
55
88
1216+
2327
30+33
4345
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Kaplan-Meier Estimate of S(t) – AML Data
Event Times
At risk # of Events
# Survive Fraction Survive
Estimate of S(t)
0 12 - 12 - 1.05 12 2 10 0.83 0.838 10 2 8 0.80 0.66
12 8 1 7 0.88 0.5823 6 1 5 0.83 0.4827 5 1 4 0.80 0.3833 3 1 2 0.67 0.2543 2 1 1 0.50 0.1345 1 1 0 0.00 0.00
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Graph of K-M Estimate of Survival Curve for AML Data
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K-M Estimate for Risp/Halo Trial
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Comparing Survival Functions
• Suppose we want to test the hypothesis that two survival curves, S1(t) and S2(t) are the same
• Common approach is the “log-rank” test
• It is effective when we can assume the hazard rates in the two groups are roughly proportional over time
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Logrank test: “Drug trial” data0.
000.
250.
500.
751.
00
0 10 20 30analysis time
A B
Kaplan-Meier survival estimates, by drug
Logrank: 1.72
p-value: .19
Conclusion: We lack strong support for a drug effect on survival
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Comparing Survival Functions
• Suppose we want to test the hypothesis that two survival curves, S1(t) and S2(t) are the same
• Common approach is the “log-rank” test
• It is effective when we can assume the hazard rates in the two groups are roughly proportional over time
• Regression analysis—“Cox” model: more to come
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Regression Analysis for Times to Events
• Cox proportional hazards model
• Hazard of an event is the product of two terms– Baseline hazard, h(t), that depends on time, t– Relative risk, rr(x) that depends on predictor variables,
x, but not time
• Each person’s hazard varies over time in the same way, but can be higher or lower depending on their predictor variables x
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Cox Proportional Hazards Model
(t,x) = hazard for people at risk with predictor values x = (x1,x2, …..xp)
(t,x) =
• ln[(t,x)] =
pp xxxet ......221
0)(
pp2211 x.......xx)]t(ln[0
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Cox Proportional Hazards Model
• Relative Hazard (hazard ratio) interpretation of the ’s
= relative risk for one unit difference in x1 with same values for x2, …. xp (at any fixed time t)
)x,........x,x,t()x,........x,1x,t(
ep21
p211
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Cox Proportional Hazards Model
• Proportional hazards over time:
(t;x)
0 t
x1 =1
x1 =0
)x,........x,0x,t()x,........x,1x,t(
ep21
p211
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Main Points Once Again• Time to event data can be censored because every
person does not necessarily have the event during the study
• Think of time to event as a natural history, that is 0 before the event and then switches to 1 when the event occurs; analysis counts the events
• Survival function, S(t), is the probability a person’s event occurs after each time t
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Main Points Once Again
• Kaplan-Meier estimator of the survival function is a product of interval-specific survival probabilities
• Hazard function, h(t), is the risk per unit time of
having the event for a person who is at risk (not previously had event)
• Logrank tests evaluate differences among survival in population subgroups
• Cox model used for regression for survival data