Cox proportional hazards model for survival data

115 Pages·2015·1.22 MB·English

J.-F.Dupuy
Cox proportional hazards model for survival data
Vietnam Institute for Advanced Study
in Mathematics
July 2015
Agnès LAGNOUX & Jean-François DUPUY
[email protected]
[email protected]
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Introduction J.-F.Dupuy
1 Introduction
Motivating examples
Probabilistic background for survival analysis
2 Cox proportional hazards model
The model
Estimation
Asymptotics
Hypothesis testing
3 Model validation in PH regression
Testing the PH assumption
Martingale residuals
Influence
4 Appendix
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Introduction J.-F.Dupuy
Motivatingexamples
Medicine
id time status age edema alb bili protime
1 400 1 58.7652 1 2.60 14.5 12.2
2 4500 0 56.4463 0 4.14 1.1 10.6
3 1012 1 70.0726 1 3.48 1.4 12.0
4 1925 1 54.7406 1 2.54 1.8 10.3
5 1504 0 38.1054 0 3.53 3.4 10.9
6 2503 1 66.2587 0 3.98 0.8 11.0
7 1832 0 55.5346 0 4.09 1.0 9.7
. . . . . . . .
. . . . . . . .
. . . . . . . .
Table1:Survivalduration(indays)betweenrandomizationanddeathfor418
patients with primary biliary cirrhosis (PBC, a liver disease) from a study
conducted between 1974 and 1984.
Source : Fleming T., Harrington D. "Counting Processes and Survival
Analysis", John Wiley & Sons, 1991.
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Introduction J.-F.Dupuy
Motivatingexamples
Public decision-making
id time censor demsnmaj deathrt lethal hoslength
(per 1000)
1 37.44 1 0 0.0995 1 3.00
2 84.29 1 0 0.0000 1 0.00
3 15.38 1 1 0.0995 0 3.04
4 46.15 1 1 0.0995 1 13.30
5 50.86 1 1 0.0579 0 33.90
6 3.024 0 1 0.0000 1 2.30
. . . . . . .
. . . . . . .
. . . . . . .
Table 2: Food and Drug Administration (FDA) approval time (in months) for
408 drugs.
Source : Carpenter, D.P. "Groups, the Media, Agency Waiting Costs, and FDA
Drug Approval". American Journal of Political Science 46(3) : 490-505, 2002.
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Introduction J.-F.Dupuy
Motivatingexamples
Public decision-making
id time event backlog eu10 formal voting thatcher
rule (qmv)
1 595 1 37 0 1 1
2 1246 1 37 0 1 0
3 341 1 162 1 0 0
4 335 1 115 0 1 0
5 522 1 172 0 0 0
6 1664 0 172 1 1 1
. . . . . . .
. . . . . . .
. . . . . . .
Table 3: Time (in days) until deliberation is stopped on a European Union
(EU) directive for 3001 deliberations.
Source : Golub J., Steunenberg B. "How Time Affects EU Decision Making".
European Union Politics 8(4) : 555-66, 2007.
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Introduction J.-F.Dupuy
Motivatingexamples
Customer relationship management
id time churn total day total day total day number customer
minutes calls charge service calls
1 128 1 265.1 110 45.07 1
2 107 0 161.6 123 27.47 1
3 137 1 243.4 114 41.38 0
4 84 1 299.4 71 50.90 2
5 75 1 166.7 113 28.34 3
6 118 0 223.4 98 37.98 0
. . . . . . .
. . . . . . .
. . . . . . .
Table4:Time(inweeks)untilcustomersofatelecommunicationcompany
switchtoacompetitor(churn :"changeandturn")
Source:LuJ."Predictingcustomerchurninthetelecommunicationsindustry-
anapplicationofsurvivalanalysismodelingusingSAS".Proceedingsofthe
SASConference,2002.
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Introduction J.-F.Dupuy
Motivatingexamples
...and many other examples in numerous fields
economy (unemployment duration, duration of economic
cycles...)
finance (time until repayment of a loan, time to default in
consumer credit risk)
marketing (churn), insurance (duration between the beginning
of long term care and death)
engineering and reliability (duration until failure of an
electronic device, of an engine...)
biology (time to migration of caterpillars from a plant to
another) and health (time to remission, time to development
of side-effects of a treatment...)
criminology (duration until recidivism)
...
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Introduction J.-F.Dupuy
Motivatingexamples
Censoring
One specific feature of survival data is censoring, which refers to
survival times being incompletely observed for some individuals.
The most common example is right censoring.
Definition 1 (Right-censoring)
A survival time X is said to be right censored if it is only known
that X is larger than an observed value C (called right censoring
time).
for example, in a clinical trial investigating time to death,
censoring may occur because a patient is still alive at the end
of the study1
censoring may also occur because the subject is lost for
follow-up during the course of the study
1because of time or cost constraints, it is not possible to follow individuals
until the event of interest is observed for all of them
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Introduction J.-F.Dupuy
Motivatingexamples
Censoring
Both situations might coexist :
if a patient is still alive at the end of the study, the observed
duration for this patient is equal to the length of the study
period ⇒ Type I censoring
a patient who leaves the study during the course of the trial is
said to be randomly right-censored
Many other types of incompleteness can also occur. There exist
various types of right censoring (progressive censoring, Type II
censoring...), left censoring, interval censoring, truncation...
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Introduction J.-F.Dupuy
Motivatingexamples
Censoring
0
1
8
l
nt 6
e
pati l
4 l
2 l
0
0 1000 2000 3000 4000
Survival time
Figure 1: Examples of Type I censoring and random censoring
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