Metlit 11-Survival Analysis - Kuntjoro

Center for Clinical Epidemiology and Evidence-Based Medicine (CEEBM)Faculty of Medicine, University of Indonesia – Cipto Mangunkusumo Hospital

Kuntjoro HarimurtiKuntjoro Harimurti

Center for Clinical Epidemiology and Evidence-Based Medicine (CEEBM)Cipto Mangunkusumo Hospital / Faculty of Medicine UI, Jakarta

[email protected]


Let we start with an example…

A cohort study was conducted to determine the survival of HIV(+) patients with CD4+ <100/L, treated with new

combination of antiretroviral (ARV). The determined event is death. The study was started at

January 1st 2001 and ended at December 31st 2005.

Results of the observation…


On study period, there were 15 HIV(+) patients enrolled:

ABCDEFGHIJKLMNO

1/1/01 1/1/02 1/1/03 1/1/04 1/1/05 31/12/05

34; died57; live at study end20; died47; died2; died38; died14; lost to follow-up23; lost to follow-up21; died23; died12; live at study end3; died1; lost to follow-up3; live at study end2; live at study end

Study period length observation (months)


Using usual methods of statistics on given data…

• Mean live survival– only calculate survival on subjects that experience the event

• Median live survival– needs 50% of subjects have experience the event

• Rate of survival– what the numerator and denominator?

• Survival at specific time– problem on determining the denominator: died?, alive?,

what about subjects that withdrawn and lost to follow- up?

Why use survival analysis?

• Usual methods of descriptive and analytic statistics cannot or unsatisfied for used in survival data, because:– subjects not enter the study at same time;– not all of the study subjects experience the event;– there were subjects that lost to follow-up or

withdrawn;– at the end of study, there were subjects still alive


What Is Survival Analysis?A collection of statistical procedures for data

analysis for which the outcome variable of interest is time until an event occurs

(Time to event analysis)

Start follow-up TIME Event

death

disease

relapse

recovery

days

weeks

months

years


Goals of survival analysis

• To estimate and interpret survivor and/or hazard functions from survival data

• To compare survivor and/or hazard function• To assess the relationship of explanatory

variables to survival time


Time as an outcome

• Survival time: – Leukemia patients/time in remission (weeks)– Diabetes patients/time until heart disease (years)– Elderly (60+) population/time until death (years)– Etc.

• From the beginning of follow-up until an event occur, age of individual when an event occur


Event• Any designated experience of interest that may

happened to an individual– Death – Disease incidence – Relapse from remission – Recovery– Etc.

• Typically refers to failure (negative event: e.g. death, relapse), but may be a positive event (e.g. recovery)

• Usually only one event is of designated interest; it could be >1 events competing risk


Censored data• A key analytical problem in survival analysis• Censoring occurs when there is some information

about individual survival time, but don’t know how the survival time exactly

• Censored data appears because we cannot follow every subjects until an event occurs

• Three reasons why censoring may occur:– The study end– Lost to follow up– Withdrawn from the study


Example: Leukemia patients in remission

Start remission: Beginning of

follow-up

Relapse: EventThe study end/ Lost to follow-up/ Withdrawal

Follow-up time

Relapse time

Follow-up time

Don’t know the relapse time exactly

?

Censored


A

B

C

D

E

F

2 4 6 8 10 12

X

X

Study start Study end

Event (relapse)

Event (relapse)

Censored

Censored

Censored

Censored

Withdrawn

Lost follow-up

S

U

B

J

E

C

T

S

W e e k s

Example: Leukemia patients in remission

T=5

T=12

T=3.5

T=8

T=6

T=3.5


Why censored data important?

• It can be used in analyzing survival data• Even though censored observations are

incomplete, we have the information on a censored person up to the time we lose track the person

• In survival analysis, every single information about the survival is important, so do not throw away the information by using the censored data


Common Techniques inSurvival Analysis

• Actuarial (Cutler-Ederer) method• Kaplan-Meier (product-limit) method• Log rank test• Cox’s proportional hazards model (Cox

regression)


Actuarial Method• Used to determine the survival on specific time interval• Time interval chosen depends on disease characteristic or

effect• Conditions and assumption in actuarial analysis:

– Beginning of the observation should be clearly defined– Effect studied should be clearly defined– Withdrawal and loss to follow-up should be independent to effect– Risk for experience the effect does not depends on calendar year– Risk for experience the effect in chosen interval should be equal– Censored patients assumed to experience ½ effect


Follow-up data of 15 HIV(+) patients; event=death


ABCDEFGHIJKLMNO

1/1/01 1/1/02 1/1/03 1/1/04 1/1/05 31/12/05


Study periodlength of observation

(months)

ABCDEFGHIJKL

MNO

ABCDEFGHIJKLMNO

We can rearrange the length of observation as if all observations started at the beginning of the study

1/1/ 01

1/1/ 02

1/1/ 03

1/1/ 04

1/1/ 05

31/12/ 05

0 1 2 3 4 5

Dates Years


Length of follow-up (yrs)


Study periodlength of observation (months)


Re-arranged data

ABCDEFGHIJKLMNO

0 1 2 3 4 5

Calculation the survival function on actuarial methods


1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Prob

abili

ty o

f sur

viva

l

Survival time (Years)

1 2 3 4 5

0.85

0.53

0.40

0.13 0.13

Survival Curve (Actuarial Method)


“Some people talks in their sleep.Lecturers talk while other people sleep.”

(Albert Camus)


http://rds.yahoo.com/_ylt=A0S0202tEoVLLG4AONKjzbkF/SIG=125lvco8o/EXP=1267098669/**http:/www.flickr.com/photos/austinjp/3277270090/

Introduction toKaplan Meier


Kaplan-Meier Method• The most common method for survival analysis is

Kaplan-Meier (product limit) estimation • This technique measures the hazard every time there

is an event• The rates are based on the number of individuals

living at the start of the time interval • These counts of living people at risk vary with the

number of censored records and number of events• Used to estimate the survival curve from observed

survival times without the assumption of an underlying probability distribution


Kaplan-Meier Method• Probability of surviving k or more periods from entering the study

is a product of the k observed survival rates for each period (i.e. the cumulative proportion surviving):

S(k) = p1 x p2 x p3 x … x pk

S = survival function p = proportion surviving in given period

• Proportion surviving period i having survived up to period i:

pi = proportion surviving in a periodri = number alive at the beginning of the period

di = number of deaths within the period

ri - di

ri

pi =


ABCDEFGHIJKLMNO


Study periodlength of observation (months)

Length of follow-up (months)

Re-arranged data from HIV(+) study


0 12 24 36 48 60

Study periodlength of observation (days)

Length of follow-up (days)

31; lost to follow-up 60; died 62+; live at study end 86; died92; live at study end356; live at study end410; lost to follow-up590; died 610; died680; lost to follow-up700; died1050; died1130; died 1400; died 1704; live at study end

MEOLNKGCIHJAFDB

Ordered data


0 365 730 1095 1440 1825


Patient name

Survival time (days)

NO known to be alive (ri)

Deaths (di)

Proportion surviving (pi=[ri-di]/ri)

Cumulative proportion surviving (S[t])

0M 31+ 14 1.000E 60 14 1 (14-1)/14=0.929 0.929O 62+L 86 12 1 (12-1)/12= 0.917 0.917*0.929=0.852N 92+K 356+G 410+C 590 8 1 (8-1)/8=0.875 0.875*0.852=0.746I 610 7 1 (7-1)/7=0.857 0.857*0.746=0.640H 680+J 700 5 1 (5-1)/5=0.800 0.800*0.640=0.512A 1050 4 1 (4-1)/4=0.750 0.750*0.512=0.384F 1130 3 1 (3-1)/3=0.667 0.667*0.384=0.256D 1400 2 1 (2-1)/2=0.500 0.500*0.256=0.128B 1704+

0

Prob

abili

ty o

f sur

viva

l

Survival time (Years)1 2 3 4 5

**

**

***

*

60 86590

610700

10501130 1400

Kaplan-Meier Curve 1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Example


Calculation for the Kaplan-Meier estimate of the survival function for the treatment 1


1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Prob

abili

ty o

f sur

viva

l

Survival time (days)20 40 60

Plot of the survival curve for treatment 1


Calculation for the Kaplan-Meier estimate of the survival function for the treatment 2


1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Prob

abili

ty o

f sur

viva

l


Plot of the survival curve for treatment 2


Estimating and comparing survival curve for the two treatment group using the Kaplan-Meier method

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Prob

abili

ty o

f sur

viva

l


Treatment 1

Treatment 2

Median survival time for Treatment Group 1 = 37 days

Median survival time for Treatment Group 2 = 5 days


Comparing survival curves of two groups using the log rank test

• Log rank test: a statistical hypothesis test to compare two survival curves

• Null hypothesis: no difference between the population survival curves

• It can be calculated manually or by statistical packages computer program


Calculation of log-rank test

O1 and O2 = total numbers of observed events in

groups 1 and 2E1 and E2 = total numbers of expected events


Group 1

Group 2

Yes No

a b

c d

Event

E (a) = (a+b)(a+c)/(a+b+c+d)E (b) = (a+b)(b+d)/(a+b+c+d), etc

P Value


Cox’s proportional hazards model

• Enables the difference between survival times of particular groups of patients to be tested while allowing for other factors handles >1 variables

• The response (dependent) variable is the ‘hazard’ probability of dying

• Hazard ratio does not depend on time (same at any other time)

h

s


Cox’s proportional hazards model


An example from the literature


Survival of patients with bronchiectasis after the first ICU stay for respiratory failure Dupont et al. Chest 2004;125:1815-20

• Objectives of the study: to assess the long term outcomes and to identify the factors associated with a reduced survival on patients with bilateral bronchiectasis admitted for the first time to the ICU for respiratory failure

• Study period: 10 years (January 1990 to March 2000) – retrospectively

• Time variable: days after ICU admission• Event: death


The Kaplan–Meier estimates of survival for (a) age > 65 years or ≤65 years, and (b) long-term oxygen therapy (LTOT) before intensive care unit admission (yes/no). The P values are for the log rank test.





Conclusions• Survival analysis provides special techniques that are

required to compare risks for event associated with different treatment groups, where the risk change over time

• In measuring survival time, the start and end-points must be clearly defined and the censored observation noted

• Actuarial method and Kaplan-Meier provide a method for estimating the survival curve

• The log rank test provides a statistical comparison of two groups

• Cox’s proportional hazards model allow additional covariates to be included



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Metlit 11-Survival Analysis - Kuntjoro

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