1 Introduction to medical survival analysis John Pearson Biostatistics consultant University of...

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Introduction to medical survival analysis

John PearsonBiostatistics consultantUniversity of Otago Canterbury7 October 2008

2

Objectives

• Describe survival data

• Define survival analysis terms

• Compare survival of groups

• Describe study design

Acknowledgement:Thanks to Colm Fahy for providing the example data.

3

Omissions

• Not covered:– most methodology issues– mathematical justification

• See– Collett: Modelling Survival Data in Medical

Research– Hosmer & Lemeshow: Applied Survival

Analysis– Many other good texts.

4

Example: Metastatic Parotid SCC

• Disease risk factors:– >50 yo– Male– Exposure to sun– Caucasian ancestry

• 61 patients operated on since 1990

• Audit done 1/6/8

• 14 patients died from SCCMP, 20 died from other causes, 1 couldn’t be found

5

Example: Patient data

OpDate Died Status Preserved RadioTx ICOMP7/05/2002 ALIVE PARTIAL YES N

15/11/2007 ALIVE NO YES N12/10/2007 1/03/2008 DOC YES YES N17/04/1992 1/08/1993 DOD YES YES Y7/10/1996 1/04/1997 DOC NO YES N1/05/1991 LOST YES YES N

12/03/2003 1/05/2005 DOC YES YES Y

Only 7 patients shown.Dates have been confidentialized.

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1

2

3

4

5

6

7

1990 1995 2000 2005 6/2008

Parotidectomy patient medical records

Pa

tien

t

AliveDead OCDead OD

? Lost to follow up au

dit

7


1

2

3

4

5

6

7

1990 1995 2000 2005 6/2008


Pa

tien

t

AliveDead OCDead OD


dit

?

?

8

Example: Survival Data

1

2

3

4

5

6

7

0 5 10 15

Parotidectomy patient survival data

Pa

tien

t

?

AliveDead OCDead OD

Years post operation

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Date formats and manipulation can cause headaches. Check what happens when your software subtracts dates to get survival time.

1

2

3

4

5

6

7

0 5 10 15


Pa

tien

t

?

AliveDead OCDead OD


10

1

2

3

4

5

6

7

0 5 10 15


Pa

tien

t

?

AliveDead OCDead OD



censored

censored

Missing data

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1

2

3

4

5

6

7

0 5 10 15


Pa

tien

t

?

AliveDead OCDead OD



censored

censored

Missing data

censored

censored

censored

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1

2

3

4

5

6

7

0 5 10 15


Pa

tien

t

?

AliveDead OCDead OD



censored

censored

censored

Missing data

censored

Censored data is explicitly addressed by survivalanalysis, using simple linear regression is not recommended. Options:1. SPSS2. SAS3. R4. Other software

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1

2

3

4

5

6

7

0 5 10 15


Pa

tien

t

?

AliveDead OCDead OD



censored

censored

censored

Missing data

censored

Missing data can have a large effect on results, requires careful management.Options:1. Omit2. Impute3. Model

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What is survival analysis

• Time to event data– Continuous– Right skewed, ≥0, not normal– Censored– Analyse risk (hazard function)

• Examples– Time to death– Time to onset/relapse of disease– Length of stay in hospital

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What is survival analysis

• Time to event data– Continuous– Right skewed, ≥0, not normal– Censored– Analyse risk (hazard function)

• Examples– Time to death– Time to onset/relapse of disease– Length of stay in hospital

0

5

10

15

0 2 4 6 8 10

Post operative survival

Pa

tien

ts

Years

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Censoring

• Right censoring

• Left censoring

• Interval censoring

Censoring is also categorised by1. Fixed study length2. Fixed number of events3. Random entry to study

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Censoring

• Right censoring– observed survival time is less than actual– Study ends before event

• Left censoring


1

2

3

4

5

6

7

1990 1995 2000 2005 6/2008


Pa

tien

t

AliveDead OCDead OD


dit

?

?

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Censoring

• Right censoring

• Left censoring – Time to relapse

– Time to event is less than observed t < 3


Surgery

0

Recurrence

3 month examt

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Censoring

• Right censoring

• Left censoring

• Interval censoring– Time to relapse

– 3 < t < 6

Surgery

0

Free of disease

3 month exam t

Recurrence

6 month exam

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Censoring

Independent censoring

Survival time is independent of censoring process.

A censored patient is representative of those at risk at censoring time.

The methods described here assume independent censoring

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Censoring

Independent censoring

Survival time is independent of censoring process.

Informative censoring

Patients removed from study if condition deteriorates.

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Censoring example

How are the SCCMP patients censored?

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Censoring example

How are the SCCMP patients censored?

• Enter study on surgery date

• Last known status is at audit

Random right censoring.

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Survival function

The survival function S(t) is the probability of surviving longer than time t.

S(t) = P(T>t)

Where T is the survival time.

patients of number total

than longer surviving patients of Number tS(t)

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Hazard function

The hazard function λ(t) is the probability of dying “at” time t.

Also called the instantaneous failure rate and force of mortality.

S(t)

f(t)(t)

)(log tS(t) Usually plotted is the cumulative hazard function, that is the accumulated hazard until time t.

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Survival function

For censored data the survival function can only be estimated.

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2

3

4

5

6

7

0.0 0.5 1.0 1.5 2.0 2.5 3.0


Pa

tien

t

AliveDead OCDead OD


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Survival function

Life table estimates

WHO, StatsNZ

All causes mortality

0

20

40

60

80

100

0 10 20 30 40 50 60 70 80 90 100

Age

Per

cen

t su

rviv

ing

NZ

Australia

Chad

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Survival function

Kaplan Meier estimatesMonths n d (n-d)/n S(t)

1 2.2 57 1 0.982 0.9822 6.12 51 1 0.980 0.9633 10.32 46 1 0.978 0.9424 10.78 45 1 0.978 0.9215 10.88 44 1 0.977 0.96 13.08 41 1 0.976 0.8787 13.35 39 1 0.974 0.8568 16.11 37 1 0.973 0.8339 26.2 34 1 0.971 0.808

10 29.42 31 1 0.968 0.78211 37.48 26 1 0.962 0.75212 45.86 23 1 0.957 0.71913 59.08 19 1 0.947 0.68214 65.33 14 1 0.929 0.633

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Survival function

Kaplan Meier estimatesMonths n d (n-d)/n S(t)

1 2.2 57 1 0.982 0.9822 6.12 51 1 0.980 0.9633 10.32 46 1 0.978 0.9424 10.78 45 1 0.978 0.9215 10.88 44 1 0.977 0.96 13.08 41 1 0.976 0.8787 13.35 39 1 0.974 0.8568 16.11 37 1 0.973 0.8339 26.2 34 1 0.971 0.808

10 29.42 31 1 0.968 0.78211 37.48 26 1 0.962 0.75212 45.86 23 1 0.957 0.71913 59.08 19 1 0.947 0.68214 65.33 14 1 0.929 0.633

1. Order data by time to event (death) 2. Number at risk of

event is number surviving less number censored.

3. Estimate of probability of surviving to next event

4. Multiply probabilities to estimate survival

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Kaplan Meier plot

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120

Kaplan Meier estimate

Est

ima

ted

su

rviv

or

fun

ctio

n

Months

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Kaplan Meier plot SCCMP

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120


Est

ima

ted

su

rviv

or

fun

ctio

n

Standard errors and 95% CI’s calculated by most software (SPSS, R, SAS)

Usually use Greenwood’s or Tsiatis’ formula, software dependent.

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Cumulative Hazard SCCMP

0.0

0.1

0.2

0.3

0.4

0 20 40 60 80 100 120

Cumulative Hazard Function

Cu

mu

lativ

e h

aza

rd

Months

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Summary statistics

1. Median survival: time when S(t) = 0.5• Must have enough data

2. Mean survival: area under the survival curve

3. 5 year survival is survival rate at 5 years

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KM and lifetables are non-parametric methods: no assumptions are made about the distribution on the survival times.

Typical distributions are exponential and Weibull. More powerful but can be sensitive to getting the distribution right.

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Disease specific survival

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120

SCCMP survival

Est

ima

ted

su

rviv

or

fun

ctio

n

Months

Disease specificAll causes

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Comparing 2 groups

Log rank test

• Computed in SPSS, SAS, R

• Most popular – (Bland Altman BMJ 2004;328:1073 (1 May)

• Limitations– No estimate of size– Unlikely to detect a difference when risk is not

consistent

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Immuno compromised

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120 140

SCCMP survival: Immuno Compromised

Est

ima

ted

su

rviv

or

fun

ctio

n

Months

No

Yes

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Immuno compromised

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120 140


Est

ima

ted

su

rviv

or

fun

ctio

n

Months

No

Yes

Case Processing Summary

53 9 44 83.0%

7 5 2 28.6%

60 14 46 76.7%

ICOMPN

Y

Overall

Total N N of Events N Percent

Censored

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Immuno compromised

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120 140


Est

ima

ted

su

rviv

or

fun

ctio

n

Months

No

Yes

Means and Medians for Survival Time

101.048 7.616 . .

22.978 7.653 16.110 3.293

91.761 7.842 . .

ICOMPN

Y

Overall

Estimate Std. Error Estimate Std. Error

Meana

Median

Estimation is limited to the largest survival time if itis censored.

a.

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Immuno compromised

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120 140


Est

ima

ted

su

rviv

or

fun

ctio

n

Months

No

Yes

Overall Comparisons

19.579 1 .000Log Rank (Mantel-Cox)Chi-Square df Sig.

Test of equality of survival distributions for the different levels ofICOMP.

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Age group

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120 140

SCCMP survival: Age group

Est

ima

ted

su

rviv

or

fun

ctio

n

Months

75+<75

Call:survdiff(formula = Surv(mths,Status == "DOD") ~ ICOMP)

N Observed Expected (O-E)^2/E (O-E)^2/VAge75=<75 24 7 5.63 0.332 0.557Age75=75+ 36 7 8.37 0.224 0.557

Chisq= 0.6 on 1 degrees of freedom, p= 0.455

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Facial Nerve

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120 140

SCCMP survival: Facial Nerve Preserved

Est

ima

ted

su

rviv

or

fun

ctio

n

Months

NO

PARTIAL

YES

Log rank p value: 0.09

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Multiple independent variables

Cox proportional hazards model• Most common model• Linear model for the log of the hazard ratio

• Baseline hazard unspecified

2211

)(

)(

0

1 ZBZBeth

th

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SCCMP example

CPH model:Survival ~ Preserved + Age + ICOMP

Preserved and ICOMP categoricalAge continuous

Plot survival for patients with each of /Y/N/partial nerve preservation adjusted for age and immuno compromised status

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SCCMP example - SPSS

Analyze > Survival > Cox Regression

COXREG

Months /STATUS=Status('DEAD')

/PATTERN BY Preserved

/CONTRAST (Preserved)=Indicator /CONTRAST (ICOMP)=Indicator(1)

/METHOD=ENTER Preserved Age ICOMP

/PLOT SURVIVAL

/SAVE=PRESID XBETA

/PRINT=CI(95) CORR SUMMARY BASELINE

/CRITERIA=PIN(.05) POUT(.10) ITERATE(20) .

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SCCMP example - SPSS

Patients with their facial nerve preserved have 12.6 times less hazard ratio, (95% CI 2-70) .

Preserving the facial nerve significantly reduces patients risk, (p value <0.001 CPH model).

Variables in the Equation

8.493 2 .014

2.535 .871 8.470 1 .004 12.617 2.288 69.564

2.091 1.110 3.549 1 .060 8.093 .919 71.279

3.588 .918 15.274 1 .000 36.166 5.981 218.676

-.011 .028 .149 1 .700 .989 .936 1.046

Preserved

No

Partial

ICOMP

Age

B SE Wald df Sig. Exp(B) Lower Upper

95.0% CI for Exp(B)

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SCCMP CPH model

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 70

SCCMP survival: Facial nerve preserved

Est

ima

ted

su

rviv

or

fun

ctio

n

Months

NO

PARTIAL

YES

Adjusted for age and immuno compromised patients

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Next Steps:

• Check proportional hazards assumption– Residual plots for groups

• Time dependent covariates

• More complex models

• we also didn’t do power calculations

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Summary

• Survival analysis accounts for censoring in time to event data

• Log rank test: difference in survival between 2 groups

• Cox proportional hazard model

• More complex/powerful models available

• SPSS, R, SAS, Stata

1 Introduction to medical survival analysis John Pearson Biostatistics consultant University of...

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