INTRODUCTION TO CLINICAL RESEARCH

Survival Analysis – Getting Started

Karen Bandeen-Roche, Ph.D.

July 20, 2010

Acknowledgements

• Scott Zeger

• Marie Diener-West

• ICTR Leadership / Team

July 2010 JHU Intro to Clinical Research 2

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

“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”

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

Switching from Calendar to Follow-up Time

0 2 4 6 8 10

Follow-up time

>3 5

>8

1

>6

The Problem with Standard Analyses of Times to Events

• Mean: (1 + 3 + 5 + 6 + 8)/5 = 4.6 - right?

• Median: 5 – right?

• Histogram

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+”

9

• 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

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

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

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

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

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)

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) ]

Displaying the Survival Function

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

Acute Myelogenous Leukemia ExampleData: 5,5,8,8,12,16+, 23, 27, 30+, 33, 43,45

55

88

1216+

2327

30+33

4345

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

Graph of K-M Estimate of Survival Curve for AML Data

K-M Estimate for Risp/Halo Trial

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

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

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

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