nasser davarzani department of knowledge engineering maastricht university, 6200 maastricht, the...

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)




Model

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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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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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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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Examples


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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




Model

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?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


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Table of Survival Time


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Right Censored Data


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Contents




Model

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Terminology and Notation

T = Survival time (T ≥ 0)t = Specific value for TSurvives > 5 years?T > t = 5

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Survival Function16


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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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?


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Contents




Model

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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


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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

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TreatmentRx

Placebo

Contents




Model

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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


Semi parametric Model

is unspecified

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Proportional Hazard40


bb

h1

h2


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nasser davarzani department of knowledge engineering maastricht university, 6200 maastricht, the...

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