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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! #$ontinued% [2]
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Censored data arethose su')ects that areno lon er tracked
n.d.82919 . Predictive ;odelin with I
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Censored Data
>
CancerDeath
Censored Explanation
0 1 Subject has not died yet (Censored)
1 0 Subject has died
0 1 Subject has not died yet (Censored)
1 0 Subject has died
&vent: -eath from Cancer
Censored: Su')ects who survived until the end of the study 'ut we do not kwhat
ha!!ened after the studycSu')ects who dro!!ed out from the study while the study was ooin
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Time Censored if 0 Age
59 1 72
115 1 74156 1 66
268 1 74
329 1 43
353 0 63
365 0 64377 0 58
421 0 53
431 1 50
448 0 56
464 0 56475 0 59
477 0 64
563 0 55
638 1 56
Time Censored if 0 Age
744 0 50
769 0 59770 0 57
803 0 39
855 0 43
1040 0 38
1106 0 441129 0 53
1206 0 44
1227 0 59
Data of Can$er 'atients(otal 'atients: 2)
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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
Time Censored if 0 Age
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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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
Time Censored if 0 Age
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$ause 1 patient has alrea
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Time Censored if 0 Age
59 1 72
115 1 74156 1 66
268 1 74
329 1 43
353 0 63
365 0 64377 0 58
421 0 53
431 1 50
448 0 56
464 0 56475 0 59
477 0 64
563 0 55
638 1 56
Time Censored if 0 Age
744 0 50
769 0 59770 0 57
803 0 39
855 0 43
1040 0 38
1106 0 441129 0 53
1206 0 44
1227 0 59
At time 11 3 how many are at ris* "2+,ow many event "2,a/ard %ate"2.2+ / !
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Time Censored if 0 Age
59 1 72
115 1 74156 1 66
268 1 74
329 1 43
353 0 63
365 0 64377 0 58
421 0 53
431 1 50
448 0 56
464 0 56475 0 59
477 0 64
563 0 55
638 1 56
Time Censored if 0 Age
744 0 50
769 0 59770 0 57
803 0 39
855 0 43
1040 0 38
1106 0 441129 0 53
1206 0 44
1227 0 59
At time 11 3 how many are at ris* "2+,ow many event "2Survival %ate"#2+02%.2+ / !
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otations:
t j time
n j num'er at risk
d j num'er of events
S j num'er of censored
j ha/ard function
(t j ) cumulative ha/ard function
S(t j ) Survival function
Hazard un$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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