Implementation and application of weighted Cox regression

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Implementation and application of weighted Cox
regression

• Georg Heinze, Daniela Dunkler,
• Samo Wakounig and Michael Schemper
• Section of Clinical Biometrics
• Core Unit of Medical Statistics and Informatics
• Medical University of Vienna, Austria
• Project sponsored by the Austrian Research Fund

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Weighted estimation in Cox regression

• The standard Cox model weights each risk set
  equally
• log-rank test
• For estimating br , weighted Cox regression
  assigns weights wr (tj) to the risk sets j
• Breslow (1974) test weight by Rj
• Prentice (1978) test weight by Sj
• In practice, weighting may be required for some
  but not for all covariates in a model

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Outline

• Investigate applicability of inference methods in
  weighted estimation
• Wald
• Score
• Likelihood ratio
• Implementation of weighted Cox regression
• Application lung cancer data

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Applicability of inference methods

• Are Wald, score, LR methods applicable to
  weighted Cox regression with
• equal weighting
• mixed weighting?
• Do these methods depend on the scaling of the
  weights?
• Prentice weights Sj ? 0, 1
• Breslow weights Rj ? 1, N
• Assume a factor c such that

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The Wald statistic

• The Wald test statistic
• Divide all weights by c
• but hence
• Thus, the Wald test requires properly normalized
  weights.
• Use of the robust variance circumvents the
  normalization problem.

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The score statistic

• The score statistic
• For inference about b1
• Dividing weights by a factor c

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Score confidence interval

• The score test can be inverted to obtain
  confidence intervals
• Choose b L (b U) such that
• Estimation of b L (b U) by separate binary
  searches
• Requires multiple of 2k additional iterative
  estimations

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Inference based on likelihood

• r th score function
• Log likelihood
• Requires w(tj) w1 (tj) wk(tj)
• (equal weighting for all covariates)
• Dividing weights by c

r ?
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Inference summary

• Wald
• Requires proper normalization of weights
• Robust (sandwich) estimate is independent of
  normalization
• Score
• Available, but confidence limits numerically
  intensive
• Independent of normalization of weights
• Likelihood ratio
• Unavailable for mixed weighting
• Requires proper normalization of weights

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Proper normalization of weights

• Cox regression
• Weighted Cox regression
• Proper normalization

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Comparison of score, Wald and robust confidence
intervals

• Simulation study compare coverage of confidence
  intervals by
• Score
• Wald (normalized weights)
• Robust standard error (only for equal weighting)
• All CI methods approximately equivalent
• Suggestion use simplest method
• (Wald with normalized weights)

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Implementation of WCR

• No standard software available for WCR
• In the special case of equal type of weighting
  for all covariates
• Some data transformation, then use
• SAS/PROC PHREG
• R/coxph

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Implementation example

• A simple data set

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Implementation example

• A simple data set

Wrong naive weighting of observations
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Implementation example

• Original data set

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Implementation example

• Original data set Transformed data set

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Implementation example

• Original data set Transformed data set

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Implementation example

• Original data set Transformed data set

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Implementation example

• Original data set Transformed data set

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Implementation example

• Original data set Transformed data set

normalized weights
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Implementation example

• Original data set Transformed data set

The same individual is given different weights in
different risk sets Patient 4 weights are 4, 3,
2, 1
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Implementation of WCR

• Transformation approach
• cannot be applied with mixed weighting
• We produced specialized software based on FORTRAN
  90
• SAS macro WCM
• R package coxphw
• Weights can be set to Rj, Sj or 1 (no weighting)
  for each covariate separately
• Inference (test and confidence intervals) based
  on normalized Wald or on score statistic
• Fixed and time-dependent effects
• Optional counting-process style input

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Implementation of WCR

• Our programs are available at
• http//www.muw.ac.at/msi/biometrie

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Application of WCR

• Now that we have efficient software, we can apply
  WCR to a large-scale data set
• Lung cancer study of Battarcharjee et al. (PNAS,
  2001)
• Gene expression of 12 600 genes
• Clinical data (survival, TNM-classification)
• N125
• Gene expressions were standardized using IQR

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Application of WCR example

• Application of WCR and CR in 12 600 univariate
  models
• Compare estimates obtained by WCR and CR
• Compare DFBETA residuals by WCR and CR

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Compare by WCR and CR
bCR - bWCR
(bCR bWCR)/2
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Compare by WCR and CR
HR increases with time
bCR - bWCR
HR decreases with time
(bCR bWCR)/2
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abs(DFBETA) residuals vs. time
abs(DFBETA)
CR WCR
Survival time (months)
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Slopes of abs(DFBETA) vs. time
Weighted Cox regression Slopes are centered at
0 Equal influence of short and long survival
times
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Slopes of abs(DFBETA) vs. time
Standard Cox regression Slopes tend to be
positive Overweights long survival
times Problematic if paired with outliers in gene
expression
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Conclusions from example

• WCR and CR estimates differ mainly if
  non-proportional hazards are present
• WCR provides unbiased estimates also in case of
  non-proportionality
• WCR provides good balance of influence of long
  and short survival times on estimate
• CR overweights long survival times

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Thank you!

• Daniela Dunkler
• Samo Wakounig
• Michael Schemper

http//www.muw.ac.at/msi/biometrie