Michael Schemper, Samo Wakounig

1
Weighted Estimation in Cox Regression Revisited

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

2
Contents

• Average hazard ratios in a population
• Weighted estimation in Cox regression
• Time - dependent effects and residuals for
  weighted estimation
• Comparison of standard and weighted estimation
  in Monte Carlo study
• Comparative statistical analyses of PBC study

3
Motivation

• Analysis by the proportional hazards (Cox)
  model if hazards are not proportional leads to
• biased estimates of the (average) hazard ratio
  (AHR)
• possible loss of power of tests of the AHR
• Methods by additional parameters available
• Explicit estimation of the AHR avoids
• the disadvantages of the standard Cox model
• and the need for additional parameters
• good choice with small samples and/or many
  covariates

4
AHR in Population (1)

• Possible definitions of a HR on continuous
  time

versus
where and denote the
hazards of groups and , respectively

intuitive versus pragmatic (used in
Cox model and by Mantel-Haenszel est.)
5
AHR in Population (2)

• Definition of a HR in Coxs philosophy

where denotes the frequency
(density) of events (i.e., tables in the model)
at t.
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AHR in Population (3)

• Definition of the AHR within Coxs
  philosophy

where denotes the overall survival
function, symbolizing the number of individuals
affected by the hazard ratio at time t. This AHR
is estimated by the weighted Cox estimation we
propose.
7
AHR in Population (4)

• Note that the AHR

is mathematically close to the hazard ratio
definition which also does not require
proportional hazards and gives equal weight to
all individuals (by pairwise comparisons of all
times and ).
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Weighted estimation in Cox regression (1)

• Sample of individuals
• uncensored survival times among
  possibly censored survival times survival
  status risk sets
• covariate values for each individual
• Then for each of the covariates, say the
  th, the following estimating equation
  is defined

weight at
observed expected covariate value
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Weighted estimation in Cox regression (2)

• parameters obtained as solutions to
  this equation.
• If standard Cox
  model.
• Other choices for possible
• generalisation of
  Breslow (1974) test
• generalisation of
  Prentice (1978) - test
• to multiple regression models.

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Weighted estimation in Cox regression (3)

• The choices of reflect the relative
  importance attached to hazard ratios at
  different times and
  weight by the number of individuals
  actually or likely affected by the log hazard
  ratio at .
• This weighting gives equal weight to all
  individuals in the case of no censoring and the
  resulting does not rely on the assumption of
  proportional hazards.

11
Weighted estimation in Cox regression (4)

• Score, Wald and Likelihood Ratio tests and
  confidence intervals are available under
  weighted estimation, as will be presented by
  Georg Heinze in the next talk of this session.

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Empirical Investigations (Weibull-Populations)
A
B
C

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Empirical Investigations (Study of Bias)

A
B
C

Simulated samples 10 000 n1 n2 40
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Empirical Investigations (Study of Precision)

A
B
C

Simulated samples 10 000 n1 n2 40
15
Empirical Investigations (Study of Power)

A
B
C

Simulated samples 10 000 n1 n2 40
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Example Primary Biliary Cirrhosis Trial

• Study of survival of n312 patients of the Mayo
  Clinic, (60 censored survival times)
• Five prognostic factors used
• Age (in years)
• Edema (no / yes)
• log (Bilirubin)
• log (Prothrombin time)
• Albumin
• Edema now studied in more detail

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Primary Biliary Cirrhosis Trial

KM - survival functions and analysis by Cox
regression
without edema
with edema
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Survival functions for Edema based on weighted
versus unweighted estimation

Explained variation 10 vs. 11
KM- functions
weighted estimation of
unweighted estimation of
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Time-dependent Survival functions for Edema
based on weighted versus unweighted estimation

KM- functions (black)
Time-dependent survival functions (using
)
are virtually identical under
unweighted and weighted estimation of !
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dfbetas for Edema based on weighted versus
unweighted estimation

weighted estimation
unweighted estimation
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Conclusions on the role of weighted estimation

• Disadvantages of weighted estimation
• slightly larger
• Equal performance
• Time-dependent effects modelling
• Explained variation
• Advantages of weighted estimation
• Hazard ratio estimates always interpretable
• Decision - theoretic
• Robustness (dfbetas)