On ranking in survival analysis: Bounds on the concordance index Harald Steck

On ranking in survival analysis: Bounds
on the concordance index
Vikas C. Raykar | Harald Steck | Balaji Krishnapuram
CAD & Knowledge Solutions (IKM CKS), Siemens Medical Solutions USA, Inc., Malvern, USA
Cary Dehing-Oberije | Philippe Lambin
Maastro clinic, University Hospital Maastricht, University Maastricht-GROW, The Netherlands
NIPS 2007
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Organization
•
•
•
•
•
•
Motivation
Brief review of survival analysis
Concordance index
Our proposed ranking approach
Connections to survival analysis
Results
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Motivation: Personalized medicine
Predict survival time of lung cancer
patients.
Different kinds of treatment
Chemo/radiotherapy dosage
Survival time
Different patient characteristics
Age/gender/health
Dataset available from MAASTRO
hospital our collaborator.
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Why not use regression?
• Not amenable to standard statistical/
machine learning methods due to
censored data.
• Well studied in statistics as survival
analysis.
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Review: Survival Analysis
Branch of statistics that deals with
time until the occurrence of a event
 When did a patient die ?
 When did the disease manifest?
 When did the machine fail?
Widely used in medical statistics, epidemiology,
reliability engineering, economics, sociology,
marketing, insurance, etc.
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What is censored data?
Start of the study
Patient unavailable
for follow-up
Some
patients die
during the
study period.
Patient 1
2001
At the end of the
study a lot of
patients may
still survive.
Data collected
at this time
End of study
Death
TIME
Censored Data
2005
The exact survival time may be longer than the observation period
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Censoring provides only partial information
Typically a large portion of the data is censored.
Survival Time
Observed Data
Censored Data
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Notation: Survival analysis
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Proportional Hazard (PH) Model
•
Has become a standard model for studying the effect of
covariates on survival time distributions.
unknown regression
parameters
relative
hazard function
Baseline hazard function
covariate
• Parameter estimates for PH model are obtained by maximizing
Cox’s partial likelihood.
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Concordance Index or c-index
• Standard performance measure for model
assessment in survival analysis.
• Generalization of the area under the ROC
curve to regression problems/censored
data.
• Fraction of all pairs of subjects who's
survival times can be ordered such that the
subject with higher predicted survival is the
one who actually survived longer.
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Concordance Index-no censoring
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5
4
Survival time
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3
2
3
1
covariate
2
1
C=1 perfect prediction accuracy
C=0.5 as good as a random predictor
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Concordance Index-with censoring
5
5
4
Survival time
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3
3
No arrow can
go above a
censored point
1
2
2
1
Censored
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Proposed approach:
Maximize CI directly
•
While CI is widely used to evaluate a learnt
model, it is not generally used as an objective
function for training.
• CI is invariant to monotone transformation of the
survival times.
• Hence the model learnt by maximizing the CI is a
ranking function. (N-partite ranking problem)
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Lower bounds on the CI
Discrete optimization problem
Use a differentiable
concave lower bound
Related to the PH model
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Maximize lower bounds on the CI
Linear ranking functions
Regularization
Use gradient
based methods to
maximize this
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Connection to the PH model
Log-likelihood for correct ranking
For a proportional hazard model we can
show that
This is a common assumption made in ranking
literature. We have shown that if we use PH
models this is exactly the case.
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Penalized log-likelihood
Compare this with the objective function
using the lower bound approach
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Cox partial likelihood
• Our proposed method explicitly maximizes
a lower bound.
• Cox method maximizes partial likelihood.
• Experimental results indicate that both do
well.
• Conjecture: Is Cox’s partial likelihood also
a lower bound on the CI?
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Cox partial likelihood (cont.)
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Results
Proposed method slightly
better than Cox-PH.
However differences not
significant.
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Thank You ! | Questions ?
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