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NASSER DAVARZANI
DEPARTMENT OF KNOWLEDGE ENGINEERING MAASTRICHT UNIVERSITY, 6200 MAASTRICHT, THE NETHERLANDS
22 OCTOBER 2012
Introduction to Survival Analysis
Contents Introduction to Survival Analysis
Censored DataTerminologies and NotationsKaplan-Meier Method Modeling in Survival DataParametric Regression
ModelsCox Proportional Regression
Model
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April 19, 2023Survival Analysis (N. Davarzani)
Contents Introduction to Survival Analysis
Censored DataTerminologies and NotationsKaplan-Meier Method Modeling in Survival DataParametric Regression
ModelsCox Proportional Regression
Model
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April 19, 2023Survival Analysis (N. Davarzani)
What is survival analysis?
Outcome variable: Time until an event occurs
Time: years, months, weeks, or days
Start follow-up Event
Event: death, disease, relapse, recovery
TimeTime
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April 19, 2023Survival Analysis (N. Davarzani)
Time ≡ survival time
It gives the time that an individual has survived over some follow up period
Event ≡ failureThe event of interest usually is death, disease or any other negative individual experience.Maybe failure is a positive event
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April 19, 2023Survival Analysis (N. Davarzani)
Examples
1) Leukemia patients/time in remission (weeks) Event: Going out of remission Outcome: Time in weeks until a person goes out of remission
2) Disease-free cohort/time until heart disease (years) Event: Developing heart disease Outcome: time in years until a person develops heart disease.
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April 19, 2023Survival Analysis (N. Davarzani)
Examples
April 19, 2023Survival Analysis (N. Davarzani)
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3) Elderly (60+) population/time until death (years) Event: Death Outcome: Time in years until death
4) Parolees (recidivism study)/time until get rearrested
Event: Getting rearrested Outcome: Time in weeks until rearrest
Contents Introduction to Survival Analysis
Censored DataTerminologies and NotationsKaplan-Meier Method Modeling in Survival DataParametric Regression
ModelsCox Proportional Regression
Model
8
?April 19, 2023Survival Analysis (N. Davarzani)
Censored data
We don’t know the survival time exactly.
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Why censoring may occur?
1) A person does not experience the event before the study ends.
2) A person is lost to follow-up during the study period.
3) A person withdraws from the study because of death (if death is not the event of interest)
or some other reason.
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Examples
April 19, 2023Survival Analysis (N. Davarzani)
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Table of Survival Time
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Right Censored Data
April 19, 2023Survival Analysis (N. Davarzani)
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Contents
Introduction to Survival Analysis
Censored DataTerminologies and NotationsKaplan-Meier Method Modeling in Survival DataParametric Regression
ModelsCox Proportional Regression
Model
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April 19, 2023Survival Analysis (N. Davarzani)
Terminology and Notation
T = Survival time (T ≥ 0)t = Specific value for TSurvives > 5 years?T > t = 5
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Survival Function16
April 19, 2023Survival Analysis (N. Davarzani)
Survival Curve
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Hazard function
The hazard function gives the instantaneous potential per unit time for the event to occur, given that the individual has survived up to time t
Hazard function focuses on failing
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Different type of hazard functions
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Relationship of S(t) and h(t)
If you know one, you can determine the other
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April 19, 2023Survival Analysis (N. Davarzani)
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Appropriate Distributions
Some popular distributions for estimating survival curves are:
WeibullExponentiallog-normallog-logistic
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2: To compare survivor and/or hazard functions.
1: To estimate and interpret survivor and/or hazard functions from survival data.
3: To assess the relationship of explanatory variables to survival time.
How to Draw the Survival Curves?
April 19, 2023Survival Analysis (N. Davarzani)
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Contents
Introduction to Survival Analysis
Censored DataTerminologies and NotationsKaplan-Meier Method Modeling in Survival DataParametric Regression
ModelsCox Proportional Regression
Model
25
April 19, 2023Survival Analysis (N. Davarzani)
Kaplan – Meier Method
The Kaplan – Meier (KM) estimator is the most widely used for estimating survival function
product-limit estimatornonparametric maximum likelihood
estimator.When there are no censored data, the KM
estimator is simple estimator
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Practical Example
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How to estimate Survival Curve?
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Suppose there are k distinct event times
At each time there are individuals who are said to be at risk of an event.At risk, means they have not experienced an event nor have they been censored prior to time .
Lifetime Table
April 19, 2023Survival Analysis (N. Davarzani)
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KM Estimator
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The KM estimator is defined as
Where number of failures at time jm
Lifetime Table
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Are KM curves statistically equavalent?
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H: Survival functions of all groups are the same
Log-Rank testWald testScore testLR testWilcoxon testPeto testTaron-Ware test
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Weakness of KM method
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Teatment V.S Placebo
Continuous variablesEffect of several variables
0
1
TreatmentRx
Placebo
Contents
Introduction to Survival Analysis
Censored DataTerminologies and NotationsKaplan-Meier Method Modeling in Survival DataParametric Regression
ModelsCox Proportional Regression
Model
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April 19, 2023Survival Analysis (N. Davarzani)
Modeling Survival Data with Regression
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Parametric Regression Models
are the regression coeffcients of interest.
is a scale parameter
is the random disturbance term
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The Cox Proportional Hazard Model38
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Semi parametric Model
is unspecified
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Proportional Hazard40
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bb
h1
h2
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