Transcript Does Socioeconomic Status Influence the Prospect of Cure

Cure models within the framework of flexible
parametric survival models
T.M-L. Andersson1,
S. Eloranta1,
P.W. Dickman1,
P.C. Lambert1,2
1
Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden
2 Department of Health Sciences, University of Leicester, UK
Relative survival
Cancer patient survival is often measured as 5-year relative
survival, R(t )
S (t )
R (t )  
S (t )

Expected survival, S (t ) , obtained from national population life
tables stratified by age, sex, calendar year and possibly other
covariates.
Estimate mortality associated with a disease without requiring
information on cause of death.
 (t )  h(t )  h (t )
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Definition of statistical cure
When the mortality rate observed in the patients eventually
returns to the same level as that in the general population
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Cure models
 Mixture cure model
S (t )  S  (t )(  (1   )Su (t ))
 Non-mixture cure model
S (t )  S  (t ) FZ (t )
 As well as the cure proportion, the survival of the
“uncured” can be estimated
 The commands strsmix and strsnmix in Stata1
1. P.C. Lambert. 2007. Modeling of the cure fraction in survival studies. Stata Journal 7:351-375.
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Cure models
 We need to choose a parametric form for Su (t ) or FZ (t ). For
many scenarios the Weibull distribution provides a good fit.
 Hard to fit survival functions flexible enough to capture high
excess hazard within a few months from diagnosis.
 Hard to fit high cure proportion.
 Flexible parametric approach for cure models would enable
inclusion of these patient groups.
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Flexible parametric survival model
 First introduced by Royston and Parmar2, stpm in Stata3
 Consider a Weibull survival curve
S (t )  exp(t  )
 Transforming to the log cumulative hazard scale gives
ln H (t )  ln( )   ln(t )
 Rather than assuming linearity with ln(t ) flexible
parametric models use restricted cubic splines
2. P. Royston and M. K. B. Parmar. 2002. Flexible proportional-hazards and proportional-odds models
for censored survival data, with application to prognostic modelling and estimation of treatment effects.
Statistics in Medicine 21:2175-2197.
3. P. Royston. 2001. Flexible alternatives to the Cox model, and more. The Stata Journal 1:1-28.
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Flexible parametric survival model
 Why model on log cumulative hazard scale?
 a generally stable function, easy to capture the shape
 easy to transform to the survival and hazard functions
 under the proportional hazards assumption covariate
effects are interpreted as hazard ratios
 Restricted cubic splines with k number of knots are used
to model the log baseline cumulative hazard
ln H (t )   0   1 z1  ...  K 1 zK 1
where
z j is a function of ln(t )
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Flexible parametric survival model
 When introducing covariates
ln H (t )   0   1z1  ...  K 1zK 1  X
 Possible to include time-dependant effects (nonproportional hazards)
 Extended to relative survival4, stpm2 in Stata5
4. C. P. Nelson, P. C. Lambert, I. B. Squire and D. R. Jones. 2007. Flexible parametric models for
relative survival, with application in coronary heart disease. Statistics in Medicine 26:5486–5498.
5. P. C. Lambert and P. Royston. 2009. Further development of flexible parametric models for survival
analysis. Stata Journal 9: 265-290.
Project presentation Leicester 29 April 2010
www.ki.se/research/thereseandersson
Flexible parametric cure model
 When cure is reached the excess hazard rate is zero, and
the cumulative excess hazard is constant.
 By incorporating an extra constraint on the log cumulative
excess hazard after the last knot, so that we force it not
only to be linear but also to have zero slope, we are able
to estimate the cure proportion.
 This is done by calculating the splines backwards and
introduce a constraint on the linear spline parameter in the
regression model.
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Flexible parametric cure model
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Comparing non-mixture and flexible
parametric cure model
 The FPCM looks like this:
R(t )  exp( exp( 0   1z1  ...  K 2 zK-2 ))
R(t )  exp( exp( 0 ))^exp( 1z1  ...  K 2 zK 2 )
which is a special case of a non-mixture model where
  exp( exp( 0 ))
FZ (t )  exp( 1 z1  ...   K 2 zK 2 )
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Comparing non-mixture and flexible
cure model
If we introduce covariates:
D
R(t )  exp( exp( 0  X ) exp( 1 z1  ...  K 2 z K 2   s( z ) x j ))
j 1
This means that the constant parameters are used to model
the cure proportion and the time-dependent parameters are
used to model the distribution function.
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Relative Survival
Relative Survival
Flexible parametric cure model
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Project presentation Leicester 29 April 2010
www.ki.se/research/thereseandersson
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Years from Diagnosis
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Years from Diagnosis
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Knots at centiles 0, 20, 40, 60, 80, 100
Knots at centiles 0, 50, 95, 100
Knots at centiles 0, 25, 50, 75, 95, 100
Knots at centiles 0, 33, 67, 95, 100
Knots at centiles 0, 35, 65, 80, 95, 100
Knots at centiles 0, 25, 50, 75, 95, 100
Knots at centiles 0, 35, 65, 75, 85, 95
Knots at centiles 0, 20, 40, 60, 80, 95, 100
Knots at centiles 0, 100 and year 3, 5, 7, 8
Knots at centiles 0, 17, 33, 50, 67, 83, 95, 100
Knots at year 0.5, 3.5, 6, 8, 9.5, 12
Knots at centiles 0, 14, 29, 43, 57, 71, 86, 95, 100
Ederer II life table estimates
Ederer II life table estimates
Ederer II 95% CI
Ederer II 95% CI
Comparing non-mixture and flexible
cure model
Comparison of cure models and life table estimates
Age group 80+ Period 1975-1984
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Relative Survival
Relative Survival
Age group 60-69 Period 1975-1984
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Unrestricted model: AIC 5805.46 BIC 5836.19
Restricted model: AIC 5807.68 BIC 5833.29
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Unrestricted model: AIC 2499.53 BIC 2529.81
Restricted model: AIC 2497.65 BIC 2522.89
Non-mixture cure model
Flexible parametric cure model
Ederer II life table estimates
Ederer II 95% CI
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Comparing non-mixture and flexible
cure model
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Cure Proportion
Cure Proportion
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Aged 60-69
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Year of Diagnosis
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Flexible cure model
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Median survival time of uncured
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Median survival time of uncured
Comparing non-mixture and flexible
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Year of Diagnosis
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Flexible cure model
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Thank you for listening!
.ssc install stpm2
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