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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 LambinMaastro 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 treatmentChemo/radiotherapy dosage

Different patient characteristicsAge/gender/health

Survival time

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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2001TIME

Start of the study Data collected at this time

Patient 1 Death

What is censored data?

2005

End of study

Censored Data

At the end of the study a lot of patients may still survive.

Some patients die during the study period.

Patient unavailable for follow-up

The exact survival time may be longer than the observation period

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Censoring provides only partial information

Censored Data

Observed Data

Su

rviv

al Tim

e

Typically a large portion of the data is censored.

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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.

Baseline hazard function

relativehazard function

covariate

unknown regression parameters

• 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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1

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1

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4Survival time

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covariate

C=1 perfect prediction accuracyC=0.5 as good as a random predictor

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Concordance Index-with censoring

Censored

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4Survival time

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No arrow can go above a censored point

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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 differentiableconcave lower bound

Related to the PH model

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Maximize lower bounds on the CI

Linear ranking functions

RegularizationUse gradient based methods to maximize this

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Connection to the PH model

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.

Log-likelihood for correct ranking

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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 ?