Cox Regression Model - Upload

8/9/2019 Cox Regression Model - Upload

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Cox Regression [2]

(

Survival Rate:

Chances that the su')ect will stay at time t

Hazard Rate:

%isk of failure

*ime+de!endent

#se Co$ %e ression to identify the chances of survival and failure

Pro'a'ility of the event ha!!enin at time t iven that theindividual is at risk at time t


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Cox Regression [2]

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,ow can you use Co$ %e ression"

Step 1: -ecide on what you want to know. -ecide on the event totime you want to analy/e.

0$: ,ow lon does it take you to nd a )o'" What s the chance thatyou will nd a )o'" ,ow lon until a com!any ado!ts a new technolo y"

,ow lon until a student dro!s out from the #niversity"Step 2: *rack su')ects and wait for a the event to ha!!en

Step 3: Com!ute for Survival %ate3 Cumulative Survival %ate3,a/ard %ate3 Cumulative ,a/ard %ate

Survival Rate: ,ow lon the su')ect stays in the sam!le.Hazard Rate: %isk of failure. Chance of an event ha!!enin .


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Censored Data

*hese are sam!les that cannot 'e tracked anymore.0$. We do not know if they found a )o'.

We do not know if they survived their 'attle a ainst cancer.

Wh !

*hey did not re!ly to the survey anymore. *hey could no lon er 'e contacted.

Do ou still use this data!

5es it is still data while they are still in the study.


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Survival "nal sis [2]

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0$amines the len th of time toa critical event

-ata may have censoreddata . Some entries are notyet conclusive.c

0$. Customers who have notleft yet as of writin 3 do nothave a de nite futuresurvival


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What is Survival "nal sis! #$ontinued% [2]

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Censored data arethose su')ects that areno lon er tracked

n.d.82919 . Predictive ;odelin with I


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Censored Data

>

CancerDeath

Censored Explanation

0 1 Subject has not died yet (Censored)

1 0 Subject has died

0 1 Subject has not died yet (Censored)

1 0 Subject has died

&vent: -eath from Cancer

Censored: Su')ects who survived until the end of the study 'ut we do not kwhat

ha!!ened after the studycSu')ects who dro!!ed out from the study while the study was ooin


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Time Censored if 0 Age

59 1 72

115 1 74156 1 66

268 1 74

329 1 43

353 0 63

365 0 64377 0 58

421 0 53

431 1 50

448 0 56

464 0 56475 0 59

477 0 64

563 0 55

638 1 56


744 0 50

769 0 59770 0 57

803 0 39

855 0 43

1040 0 38

1106 0 441129 0 53

1206 0 44

1227 0 59

Data of Can$er 'atients(otal 'atients: 2)


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59 1 72

115 1 74156 1 66

268 1 74

329 1 43

353 0 63

365 0 64377 0 58

421 0 53

431 1 50

448 0 56

464 0 56475 0 59

477 0 64

563 0 55

638 1 56


744 0 50

769 0 59770 0 57

803 0 39

855 0 43

1040 0 38

1106 0 441129 0 53

1206 0 44

1227 0 59

At time ?3 how many have died #1

At time 11 3 how many have died2


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59 1 72

115 1 74156 1 66

268 1 74

329 1 43

353 0 63

365 0 64377 0 58

421 0 53

431 1 50

448 0 56

464 0 56475 0 59

477 0 64

563 0 55

638 1 56


744 0 50

769 0 59770 0 57

803 0 39

855 0 43

1040 0 38

1106 0 441129 0 53

1206 0 44

1227 0 59

At time ?3 how many are at ris* "2)

At time 11 3 how many are at ris* "2+,-e$ause 1 patient has alrea


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59 1 72

115 1 74156 1 66

268 1 74

329 1 43

353 0 63

365 0 64377 0 58

421 0 53

431 1 50

448 0 56

464 0 56475 0 59

477 0 64

563 0 55

638 1 56


744 0 50

769 0 59770 0 57

803 0 39

855 0 43

1040 0 38

1106 0 441129 0 53

1206 0 44

1227 0 59

At time 11 3 how many are at ris* "2+,ow many event "2,a/ard %ate"2.2+ / !


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59 1 72

115 1 74156 1 66

268 1 74

329 1 43

353 0 63

365 0 64377 0 58

421 0 53

431 1 50

448 0 56

464 0 56475 0 59

477 0 64

563 0 55

638 1 56


744 0 50

769 0 59770 0 57

803 0 39

855 0 43

1040 0 38

1106 0 441129 0 53

1206 0 44

1227 0 59

At time 11 3 how many are at ris* "2+,ow many event "2Survival %ate"#2+02%.2+ / !


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otations:

t j time

n j num'er at risk

d j num'er of events

S j num'er of censored

j ha/ard function

(t j ) cumulative ha/ard function

S(t j ) Survival function

Hazard un$tion:

Cu ulative Hazard un$tion:

4aplan05eier Survival un$tion:


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Hazard Rate of the Can$er 'atients

Com!ute for the ha/ard rate.

Re$all:n j num'er at risk

d j num'er of events

@um'er at risk at a iven time j @um'er of events at a iven time j


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4aplan 5eier Survival Rate of the Can$er 'atients

Com!ute for the survival rate.

Re$all:


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'ara etri$ Survival un$tion

ook at the sha!e of the ha/ard function

Para etric !ode" #a$ard %unction Sur&i&a" %unction

' onentia"

*eibu""

+o ert$

,o-."o-istic

0$!onential model assumes that the !ro'a'ility of the eventha!!enin does not chan e over time.


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Cox Regression for Survival Data [2]

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What can you say a'out this ra!h"

After 69 months3 9B of the customers left


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What is Survival "nal sis! [2]

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0$amines the len th of time to a critical event

n.d.82919 . Predictive ;odelin with I


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Cox Regression or the Cox 'roportional Hazard 5odel[2]

2(

Inter!retin Coe cients

If Coe cient is positive 3ower duration3 hi her ha/ard rates 8more likely toha!!enAs an independent varia-le in$reases 3 ti e0to0event de$reases 8the sooner the event will ha!!en

If Coe cient is negative 3,i her duration3 lower ha/ard rate 8less likely toha!!enAs an independent varia-le in$reases, ti e0to0event in$reases


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Cox Regression Model - Upload

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