A relative survival regression model to analyze competing risks data A' Belot1, 2, L' Remontet1, G'

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A relative survival regression model to analyze
competing risks dataA. Belot(1, 2), L.
Remontet(1), G. Launoy(3), M. Abrahamowicz(4),
R. Giorgi(5)
Email aurelien.belot_at_chu-lyon.fr

• Hospices Civils de Lyon, Service de
  Biostatistique, Lyon, F-69424, France
  Université de Lyon Université Lyon I,
  Villeurbanne, F-69622, France CNRS UMR 5558,
  Laboratoire Biostatistique Santé, Pierre-Bénite,
  F-69495, France.
• Institut de Veille Sanitaire, DMCT, Saint
  Maurice, France
• ERI3 INSERM cancers et populations, CHU Caen,
  France FRANCIM, French network of cancer
  registries.
• Department of Epidemiology and Biostatistics,
  McGill University, Montreal, Canada
• LERTIM, Université de la méditerranée, Marseille,
  France

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Objectives

• Epidemiological cancer context
• Estimate hazard functions and covariates effect
  associated with relapses, metastasis and death
  due to the disease
• Relative survival permits to assess prognostic
  factor effect on the disease-specific mortality
  hazard function
• Competing risks framework permits to study
  multiple events
• ? To combine these two approaches

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Outline

• Background
• Relative survival concept
• Competing risks situation (with Lunn McNeil
  approach)
• Method
• Simulation and application
• Conclusion/discussion

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Background Relative survival (1)
?d disease-specific mortality hazard
Hazard ratio for covariate xi on the
disease-specific mortality hazard
?e expected mortality hazard due to other causes
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Background Relative survival (2)

• Provides a measure of the proportion of patients
  dying from the disease
• Possibility to model the effect of prognostic
  factors on the disease-specific mortality hazard
• Useful when cause of death is unknown or
  ambiguous (like in cancer registries)

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Background Competing risks

• Study of multiple events
• Analyze the first event that occurs, based on the
  Event-Specific Hazard Function
• Estimate the effect of some covariates associated
  with the risk of a specific event

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Background Lunn McNeil approach (1)
Based on duplication of data Example in the case
of 2 different event types (relapse, death)
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Background Lunn McNeil approach (2)

• The model (in case of J different types of events)

with dk1 when jk and dk0 otherwise

• ?1(t) baseline hazard function for event of
  type 1
• exp (bk) hazard ratio between event-specific
  baseline hazard functions (k vs 1)

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Background Lunn McNeil approach (3)

• The model (in case of J different types of events)

with dk1 when jk and dk0 otherwise

• ?1(t) baseline hazard function for event of
  type 1
• exp (bk) hazard ratio between event-specific
  baseline hazard functions (k vs 1)

exp (a) hazard ratio for covariates x for event
of type 1 exp (a bk) hazard ratio for
covariates x for event of type k
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Background Main limit of these methods

• Relative survival method studies only one event
• Competing risks method doesnt permit to estimate
  disease specific mortality without the cause of
  death
• We propose a relative survival regression model
  to overcome this two limits

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The proposed model
exp( bk(t) ) Time-dependent hazard ratio between
baseline hazard functions of event k and event 1
?e expected mortality hazard due to other causes
bk(t) (and l1(t)) are modelled with regression
spline
Maximum likelihood estimates are obtained with
IRLS algorithm
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Evaluation of estimators properties by
simulation (1)

• Design of simulation
• 400 samples
• Number of patients by sample 400, 1000
• Censoring 0, 15, 30
• 3 independent prognostic factors with an effect
  on each event
• Sex , X qualitative covariates
• Age quantitative covariate

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Evaluation of estimators properties by
simulation (2)

• Algorithm of simulations
• 3 events of interest
• Death due to cancer
• Type 1 event
• Type 2 event
• Simulation of 3 times T(death cancer), T1 , T2
  (using independent generalized Weibull
  distribution)
• Simulation of the expected time to death Texp
  (exponential distribution)
• Simulation of a censoring time C (uniform
  distribution)
• Time to first event T min(T(death cancer) ,
  Texp , T1 , T2 , C )
• Type of this event (or censor) death, Type 1
  event, Type 2 event

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Evaluation of estimators properties by
simulation (3)

• Statistical criteria used
• Graphical representation of the true and the
  estimated hazard function
• Relative bias
• Coverage rate

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Results of simulations (1)
Baseline hazard function for event of type 1
(N1000, 15 censoring)
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Results of simulations (2)
Time-dependent hazard ratio b2(t)
Time-dependent hazard ratio b3(t)
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Results of simulations (3)
Parameter for the effect of X is 0 on event of
type 3
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Results of simulations (4)
Parameter for the effect of X is 0 on event of
type 3
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Application on real data

• Material
• 1,000 patients with colon cancer
• 9 French cancer registries
• Information on 4 events relapse, metastasis,
  diagnosis of second cancer, death
• Method
• Baseline hazard function for the event of type 1
  (relapse) modeled with regression cubic spline
  with 1 knot at 1 year, selected with Akaike
  criterion
• Time-dependent hazard ratio for metastasis,
  second cancer and death

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Results (1)Baseline hazard functions
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Results (2)Estimated parameters bk
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Conclusion

• The proposed model permits to estimate
• Flexible baseline hazard function for each event
  type (trough Time-dependent hazard ratio)
• Covariates effects associated with each event
  type, including death due to the disease
• Simulations show good properties of estimators
• Small relative bias
• Coverage rate close to nominal values