Lecture 16: Survival Analysis I Kaplan Meier and Log-rank test Ani - - PowerPoint PPT Presentation

Lecture 16: Survival Analysis I – Kaplan Meier and Log-rank test

Ani Manichaikul amanicha@jhsph.edu 11 May 2007

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Survival Analysis

n Statistical methods for the study of time

to an event

n Accounts for:

n Time that events occur n Different follow-up times

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Survival Analysis

n Survival analysis methods allow us to

incorporate information about both frequency of event occurrence and time to event information

n Subjects are followed until they have an

“event,” or the study ends

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Endpoint

n The endpoint doesn’t have to be

‘death’; it can be any well-defined event

n Death n Disease onset n Menopause n Pregnancy n Relapse

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Time Scale

n When do you start the clock?

n Time from diagnosis of disease to death n Time from HIV infection to AIDS n Time from birth (chronological age) n Time from randomization in clinical trial

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Why Is Survival Analysis Tricky?

n We need a method which can

incorporate information about censored data into an analysis

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S(t) Time

S(t) is an estimate of the proportion of individuals still alive (have not had the event) at time t

The Survival Curve

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The Survival Curve

n The survival curve as an important and

complete summary

n Time 0: “start of clock”

( ) ( )

at time alive # t time followup at alive # ) ( = t S

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Survival Curve Facts:

n The curve starts at 1 and decreases n Estimating these curves and comparing them

among groups constitutes a “survival analysis”

n Need to decide on what summary is

important

n Mean survival time n Median survival time n Height at a specific time: One, two year survival rates n Difference of curves: S1(12) - S2(12)

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.50

Estimating Median Survival

S(t) Time

m

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S(t) Time

Caveat—Medians Do Not Describe Whole Curve

.50 m

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Survival Function

n The survival function, denoted S(t), is a

better way to represent the probability distribution of the survival time T, when some

• f the observed times are censored

n only know that T> t, rather than T= t

n S(t) = Pr(T > t) = Pr(No event by time t) n S(t) is the probability of surviving beyond t

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n Uncensored data: The event has

• ccurred

n Censored data: The event has yet to

• ccur

n Event-free at the current followup time n A competing event that is not an endpoint

stops followup

n Death (if not part of the endpoint) n Clinical event that requires treatment, etc.

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n Important issue: If no events are

reported in the interval from last follow- up to “now”, need to choose between:

n No news is good news? n No news is no news

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n Ignore the incomplete cases; drop them

n Produces bias in the estimated curve n Unbalanced censoring produces biased

comparisons

n Impute an event time

n Depends on a model

n Use the available information on each

participant

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n Example: 5 events in 600 person months

n

5/600 = 1/120 events per month = 0.1 events per year = 10 events per 100 person-years

n Gives an average event rate over the follow-

up period

n For a finer time resolution, do the above for

small intervals

( )

n time

• bservatio

total events # Rate Event =

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Quantities of Interest

n The survivor function S(t)

S(t)= P(T> t)= P(No event by time t)

n Hazard function (t)

(t) “= ” P(T= t)/ P(T> t)

= risk of event occurring at time t

The above form is true for discrete time, but involves more complicated calculus-based notation for continuous time.

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Quantities of Interest

n Often, we are interested in comparing

the hazard between groups, for example, the relative hazard of relapse comparing those on chemo to those not

• n chemo

n Relative Risk n Hazard Ratio n Risk Ratio

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Estimation

n Kaplan-Meier survivor function

estimator

n Cox proportional hazards model (PHM)

for hazard ratio

n We’ll start with Kaplan-Meier (K-M)

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Central Problem

n Estimation of the survival curve n S(t) = Proportion surviving at least to

time t or beyond

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S(t) Time

S(0) always equals 1 All subjects are alive at beginning of the study 1.0

The Survival Curve

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S(t) Time

Curve can only remain at same value or decrease as time progresses 1.0

The Survival Curve

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S(t) Time

If all the subjects do not experience the event by the end of the study window, the curve may never reach zero 1.0

The Survival Curve

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Example

n Consider a clinical trial in patients with

acute myelogenous leukemia (AML) comparing two groups of patients: no maintenance treatment with chemotherapy (X= 0) -vs- maintenance chemotherapy treatment (X= 1)

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Example: Data

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Why Survival Methods?

n We are interested in estimating the

relationship between chemotherapy and the time to AML relapse in weeks.

n We need some tools because:

n Data are censored, so linear regression is

not appropriate

n We are interested in time to relapse, not

just relapse (yes/no), so logistic regression is not appropriate

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Kaplan-Meier Estimate

n Curve can be estimated at each event,

but not at censoring times

n S(t) = proportion of individuals

surviving beyond time t

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) _ _ (Pr ) ( ) ( ) ( ) ( Time Event evious t n t y t n t S S ×         − =

Kaplan-Meier Estimate

n Curve can be estimated at each event,

but not at censoring times

n y(t) = # events at time t n n(t) = # subjects at risk for event at

time t

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) _ _ (Pr ) ( ) ( ) ( ) ( Time Event evious t n t y t n t S S ×         − =

Proportion of original sample making it to time t

Kaplan-Meier Estimate

n Curve can be estimated at each event,

but not at censoring times

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Proportion surviving to time t who survive beyond time t

Kaplan-Meier Estimate

n Curve can be estimated at each event,

but not at censoring times

) _ _ (Pr ) ( ) ( ) ( ) ( Time Event evious t n t y t n t S S ×         − =

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n Start estimate at first event time

n No Chemotherapy Group: Time = 5

833 . 12 10 12 2 12 ) 5 ( ) 5 ( ) 5 ( ) 5 ( = = − =         − = n y n S

Kaplan-Meier Estimate

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n No Chemotherapy group: Time= 8

n 2nd event time

666 . 833 . 10 8 ) 833 (. 10 2 10 ) 5 ( ) 8 ( ) 8 ( ) 8 ( ) 8 ( = × = ×       − = ×         − = S n y n S

Kaplan-Meier Estimate

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Kaplan-Meier Estimate

n Skip over censoring times: Remove

from number at risk for next event time

n Continue through final event time

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Alternative Notation

1 ) ( ˆ ) ( ˆ

: :

=         − = ∏

≤

S n y n t S

t t i i i i

i

(by convention)

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Notice

n Time 16 was not included in the table,

yet 2 people were subtracted from the risk set at time 23

n The estimated survivor function does not

change at censoring times when no event

• ccurs

n Censored individuals are subtracted from

the risk set at subsequent times because they are “lost to follow-up”

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Kaplan-Meier Estimate

n Graph is a step function n “Jumps” at each observed event time n Nothing is assumed about curved shape

between each observed event time

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Kaplan-Meier Estimate

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Kaplan-Meier Estimate

n Product limit estimate

n Order survival times n Computed at observed events n Multiplying conditional probabilities

n Next time we’ll discuss Confidence

Intervals for S(t)!

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Big Assumption

n Independence of censoring and survival n Those censored at time t have the same

prognosis as those not censored at t

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Comparing Survival Curves

n Common statistical tests:

n Generalized Wilcoxon

(Breslow, Gehan)

n Logrank

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Comparing Survival Curves

n Both compare survival curves across

multiple time points to answer the question: “Is overall survival different between any of the groups?”

n Ho: No difference in S(t) n Ha: Difference in S(t)

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0.00 0.25 0.50 0.75 1.00 100 200 300 400 analysis time

Kaplan Meier Curve, by Group

Comparing Survival Curves

n Wilcoxon (Breslow, Gehan) more sensitive to

early survival differences

Group 1 Group 2

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Comparing Survival Curves

n Logrank more sensitive to later survival

differences

0.00 0.25 0.50 0.75 1.00 100 200 300 400 analysis time

Kaplan Meier Curve, by Group

Group 1 Group 2

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Comparing Survival Curves

n Neither test very good if curves “crossover”

0.00 0.25 0.50 0.75 1.00 100 200 300 400 analysis time

Kaplan Meier Curve, by Group

Group 1 Group 2

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Logrank Test

n Answers the Quesiton:

Are two survivor curves the same?

n Use the times of events: t1, t2, ...

(do not include censoring times)

n Treat each event and its “set of persons

still at risk” (i.e., risk set) at each time tj as an independent table

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Logrank Test: Recipe

n Make a 2×2 table at each tj

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Logrank test

n At each event time tj, under assumption of

equal survival (SA(t) = SB(t)) the expected number of events in Group A

• ut of the total events (dj= aj+ cj) is in

proportion to the numbers at risk in group A to the total at risk at time tj: E(aj)= dj* njA/nj

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Logrank Test: Formula

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Logrank Test

n Uses the Cochran Mantel-Haenszel idea

• f pooling over events j to get the log-

rank statistic

n This Chi-square statistic has 1 degree of

freedom (use to get p-value)

n Small p-value; Reject H0; n Conclusion: Survival Curves ARE different!

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Logrank Test: Our Example

Chi2=2.61 pval = .1061

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Conclusion

n Fail to reject the null hypothesis.

Cannot conclude that there is a difference between the survival (time to relapse) of those on maintenance chemotherapy and those not on maintenance chemotherapy.

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Conclusion

n What if we want to adjust for other

factors?

n Cox Proportional Hazards Model! n Next time…