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Survival Analysis Cure Fractions

VIEWS: 5 PAGES: 75

									                     Survival Analysis: Cure Fractions

                                     Rebecca Shanmugam




                                     Medical Research Council


                                     Journal Club 2008


Rebecca (Medical Research Council)         Cure Fractions       JC 2008   1 / 27
Outline


1   Data and Setup
      The Basic Question: Survival impact on Transplant Patients

2   Cure Fractions
      A Brief Word on Survival Analysis
      Mixture Cure Fraction Models
      Non-Mixture Cure Fractions
      Modelling Relative Survival

3   Application
      The models



Rebecca (Medical Research Council)   Cure Fractions           JC 2008   2 / 27
                                     Data and Setup   The Basic Question: Survival impact on Transplant Patients


Outline


1   Data and Setup
      The Basic Question: Survival impact on Transplant Patients

2   Cure Fractions
      A Brief Word on Survival Analysis
      Mixture Cure Fraction Models
      Non-Mixture Cure Fractions
      Modelling Relative Survival

3   Application
      The models



Rebecca (Medical Research Council)            Cure Fractions                                    JC 2008      3 / 27
                                     Data and Setup   The Basic Question: Survival impact on Transplant Patients


Data




       Kidney transplant study, Addington Hospital, KwaZulu-Natal, South
       Africa.


       221 patients; Jan 1990 Dec 2004.


       Analysis period 15 years.




Rebecca (Medical Research Council)            Cure Fractions                                    JC 2008      4 / 27
                                     Data and Setup   The Basic Question: Survival impact on Transplant Patients


Data




       Kidney transplant study, Addington Hospital, KwaZulu-Natal, South
       Africa.


       221 patients; Jan 1990 Dec 2004.


       Analysis period 15 years.




Rebecca (Medical Research Council)            Cure Fractions                                    JC 2008      4 / 27
                                     Data and Setup   The Basic Question: Survival impact on Transplant Patients


Data




       Kidney transplant study, Addington Hospital, KwaZulu-Natal, South
       Africa.


       221 patients; Jan 1990 Dec 2004.


       Analysis period 15 years.




Rebecca (Medical Research Council)            Cure Fractions                                    JC 2008      4 / 27
                                     Data and Setup   The Basic Question: Survival impact on Transplant Patients


Variables




Do the following variables:
       Age at transplant
       Living or cadaver donor
       Graft loss
       Gender of donor
influence survival after a kidney transplant.




Rebecca (Medical Research Council)            Cure Fractions                                    JC 2008      5 / 27
                                     Data and Setup   The Basic Question: Survival impact on Transplant Patients


Variables




Do the following variables:
       Age at transplant
       Living or cadaver donor
       Graft loss
       Gender of donor
influence survival after a kidney transplant.




Rebecca (Medical Research Council)            Cure Fractions                                    JC 2008      5 / 27
                                     Data and Setup   The Basic Question: Survival impact on Transplant Patients


Variables




Do the following variables:
       Age at transplant
       Living or cadaver donor
       Graft loss
       Gender of donor
influence survival after a kidney transplant.




Rebecca (Medical Research Council)            Cure Fractions                                    JC 2008      5 / 27
                                     Data and Setup   The Basic Question: Survival impact on Transplant Patients


Variables




Do the following variables:
       Age at transplant
       Living or cadaver donor
       Graft loss
       Gender of donor
influence survival after a kidney transplant.




Rebecca (Medical Research Council)            Cure Fractions                                    JC 2008      5 / 27
                                     Cure Fractions   A Brief Word on Survival Analysis


Outline


1   Data and Setup
      The Basic Question: Survival impact on Transplant Patients

2   Cure Fractions
      A Brief Word on Survival Analysis
      Mixture Cure Fraction Models
      Non-Mixture Cure Fractions
      Modelling Relative Survival

3   Application
      The models



Rebecca (Medical Research Council)           Cure Fractions                               JC 2008   6 / 27
                                     Cure Fractions   A Brief Word on Survival Analysis


What we know about Survival Analysis

       We assume some form of a distributional function for the time until
       the event of interest.
       And any such distribution tends to 1 as the time at risk becomes
       sufficiently large.
                                          limt→∞ F (t) = 1
       That is, every individual will eventually experience the event of
       interest.

                         In many instances this is just not true!

       Cure models are introduced to relax this assumption.
       There exist two types of cure models:
           1   Mixture Models
           2   Non-Mixture Models

Rebecca (Medical Research Council)           Cure Fractions                               JC 2008   7 / 27
                                     Cure Fractions   A Brief Word on Survival Analysis


What we know about Survival Analysis

       We assume some form of a distributional function for the time until
       the event of interest.
       And any such distribution tends to 1 as the time at risk becomes
       sufficiently large.
                                          limt→∞ F (t) = 1
       That is, every individual will eventually experience the event of
       interest.

                         In many instances this is just not true!

       Cure models are introduced to relax this assumption.
       There exist two types of cure models:
           1   Mixture Models
           2   Non-Mixture Models

Rebecca (Medical Research Council)           Cure Fractions                               JC 2008   7 / 27
                                     Cure Fractions   A Brief Word on Survival Analysis


What we know about Survival Analysis

       We assume some form of a distributional function for the time until
       the event of interest.
       And any such distribution tends to 1 as the time at risk becomes
       sufficiently large.
                                          limt→∞ F (t) = 1
       That is, every individual will eventually experience the event of
       interest.

                         In many instances this is just not true!

       Cure models are introduced to relax this assumption.
       There exist two types of cure models:
           1   Mixture Models
           2   Non-Mixture Models

Rebecca (Medical Research Council)           Cure Fractions                               JC 2008   7 / 27
                                     Cure Fractions   A Brief Word on Survival Analysis


What we know about Survival Analysis

       We assume some form of a distributional function for the time until
       the event of interest.
       And any such distribution tends to 1 as the time at risk becomes
       sufficiently large.
                                          limt→∞ F (t) = 1
       That is, every individual will eventually experience the event of
       interest.

                         In many instances this is just not true!

       Cure models are introduced to relax this assumption.
       There exist two types of cure models:
           1   Mixture Models
           2   Non-Mixture Models

Rebecca (Medical Research Council)           Cure Fractions                               JC 2008   7 / 27
                                     Cure Fractions   A Brief Word on Survival Analysis


What we know about Survival Analysis

       We assume some form of a distributional function for the time until
       the event of interest.
       And any such distribution tends to 1 as the time at risk becomes
       sufficiently large.
                                          limt→∞ F (t) = 1
       That is, every individual will eventually experience the event of
       interest.

                         In many instances this is just not true!

       Cure models are introduced to relax this assumption.
       There exist two types of cure models:
           1   Mixture Models
           2   Non-Mixture Models

Rebecca (Medical Research Council)           Cure Fractions                               JC 2008   7 / 27
                                     Cure Fractions   A Brief Word on Survival Analysis


What we know about Survival Analysis

       We assume some form of a distributional function for the time until
       the event of interest.
       And any such distribution tends to 1 as the time at risk becomes
       sufficiently large.
                                          limt→∞ F (t) = 1
       That is, every individual will eventually experience the event of
       interest.

                         In many instances this is just not true!

       Cure models are introduced to relax this assumption.
       There exist two types of cure models:
           1   Mixture Models
           2   Non-Mixture Models

Rebecca (Medical Research Council)           Cure Fractions                               JC 2008   7 / 27
                                     Cure Fractions   Mixture Cure Fraction Models


Outline


1   Data and Setup
      The Basic Question: Survival impact on Transplant Patients

2   Cure Fractions
      A Brief Word on Survival Analysis
      Mixture Cure Fraction Models
      Non-Mixture Cure Fractions
      Modelling Relative Survival

3   Application
      The models



Rebecca (Medical Research Council)           Cure Fractions                          JC 2008   8 / 27
                                          Cure Fractions   Mixture Cure Fraction Models


Mixture Cure Fractions

Mixture Cure fractions are based on mixture models [1] and [2]. Basic idea
behind mixture models:

       Population broken up into sub-populations.
       Each sub-population generates its own density function, [fi (t)], thus
       the complete mixture density and distribution functions are,
                                      g                                        g
                           f (y ) =         πi fi (t) and F (t) =                   πi Fi (t),
                                      i=1                                     i=1

       respectively.

       The πi ’s are weights, estimated for each of the sub-populations.
         g
         i=1 πi = 1 and (0 < πi ≤ 1).


Rebecca (Medical Research Council)                Cure Fractions                                 JC 2008   9 / 27
                                          Cure Fractions   Mixture Cure Fraction Models


Mixture Cure Fractions

Mixture Cure fractions are based on mixture models [1] and [2]. Basic idea
behind mixture models:

       Population broken up into sub-populations.
       Each sub-population generates its own density function, [fi (t)], thus
       the complete mixture density and distribution functions are,
                                      g                                        g
                           f (y ) =         πi fi (t) and F (t) =                   πi Fi (t),
                                      i=1                                     i=1

       respectively.

       The πi ’s are weights, estimated for each of the sub-populations.
         g
         i=1 πi = 1 and (0 < πi ≤ 1).


Rebecca (Medical Research Council)                Cure Fractions                                 JC 2008   9 / 27
                                          Cure Fractions   Mixture Cure Fraction Models


Mixture Cure Fractions

Mixture Cure fractions are based on mixture models [1] and [2]. Basic idea
behind mixture models:

       Population broken up into sub-populations.
       Each sub-population generates its own density function, [fi (t)], thus
       the complete mixture density and distribution functions are,
                                      g                                        g
                           f (y ) =         πi fi (t) and F (t) =                   πi Fi (t),
                                      i=1                                     i=1

       respectively.

       The πi ’s are weights, estimated for each of the sub-populations.
         g
         i=1 πi = 1 and (0 < πi ≤ 1).


Rebecca (Medical Research Council)                Cure Fractions                                 JC 2008   9 / 27
                                          Cure Fractions   Mixture Cure Fraction Models


Mixture Cure Fractions

Mixture Cure fractions are based on mixture models [1] and [2]. Basic idea
behind mixture models:

       Population broken up into sub-populations.
       Each sub-population generates its own density function, [fi (t)], thus
       the complete mixture density and distribution functions are,
                                      g                                        g
                           f (y ) =         πi fi (t) and F (t) =                   πi Fi (t),
                                      i=1                                     i=1

       respectively.

       The πi ’s are weights, estimated for each of the sub-populations.
         g
         i=1 πi = 1 and (0 < πi ≤ 1).


Rebecca (Medical Research Council)                Cure Fractions                                 JC 2008   9 / 27
                                          Cure Fractions   Mixture Cure Fraction Models


Mixture Cure Fractions

Mixture Cure fractions are based on mixture models [1] and [2]. Basic idea
behind mixture models:

       Population broken up into sub-populations.
       Each sub-population generates its own density function, [fi (t)], thus
       the complete mixture density and distribution functions are,
                                      g                                        g
                           f (y ) =         πi fi (t) and F (t) =                   πi Fi (t),
                                      i=1                                     i=1

       respectively.

       The πi ’s are weights, estimated for each of the sub-populations.
         g
         i=1 πi = 1 and (0 < πi ≤ 1).


Rebecca (Medical Research Council)                Cure Fractions                                 JC 2008   9 / 27
                                     Cure Fractions   Mixture Cure Fraction Models




Let,
       g =2
       and the mixture density and distribution functions have the properties
       of the density and distribution function in survival analysis, i.e.

                 f (t) = πc fc (t) + πu fu (t) and F (t) = πc Fc (t) + πu Fu (t).
Also define,
       πc is the probability of ‘cure’ or never experiencing the event of
       interest.
       πu = 1 − πc is the probability of eventual failure.




Rebecca (Medical Research Council)           Cure Fractions                          JC 2008   10 / 27
                                     Cure Fractions   Mixture Cure Fraction Models




Let,
       g =2
       and the mixture density and distribution functions have the properties
       of the density and distribution function in survival analysis, i.e.

                 f (t) = πc fc (t) + πu fu (t) and F (t) = πc Fc (t) + πu Fu (t).
Also define,
       πc is the probability of ‘cure’ or never experiencing the event of
       interest.
       πu = 1 − πc is the probability of eventual failure.




Rebecca (Medical Research Council)           Cure Fractions                          JC 2008   10 / 27
                                     Cure Fractions   Mixture Cure Fraction Models




Let,
       g =2
       and the mixture density and distribution functions have the properties
       of the density and distribution function in survival analysis, i.e.

                 f (t) = πc fc (t) + πu fu (t) and F (t) = πc Fc (t) + πu Fu (t).
Also define,
       πc is the probability of ‘cure’ or never experiencing the event of
       interest.
       πu = 1 − πc is the probability of eventual failure.




Rebecca (Medical Research Council)           Cure Fractions                          JC 2008   10 / 27
                                     Cure Fractions   Mixture Cure Fraction Models




Let,
       g =2
       and the mixture density and distribution functions have the properties
       of the density and distribution function in survival analysis, i.e.

                 f (t) = πc fc (t) + πu fu (t) and F (t) = πc Fc (t) + πu Fu (t).
Also define,
       πc is the probability of ‘cure’ or never experiencing the event of
       interest.
       πu = 1 − πc is the probability of eventual failure.




Rebecca (Medical Research Council)           Cure Fractions                          JC 2008   10 / 27
                                     Cure Fractions   Mixture Cure Fraction Models




Let,
       g =2
       and the mixture density and distribution functions have the properties
       of the density and distribution function in survival analysis, i.e.

                 f (t) = πc fc (t) + πu fu (t) and F (t) = πc Fc (t) + πu Fu (t).
Also define,
       πc is the probability of ‘cure’ or never experiencing the event of
       interest.
       πu = 1 − πc is the probability of eventual failure.




Rebecca (Medical Research Council)           Cure Fractions                          JC 2008   10 / 27
                                     Cure Fractions   Mixture Cure Fraction Models




Let,
       g =2
       and the mixture density and distribution functions have the properties
       of the density and distribution function in survival analysis, i.e.

                 f (t) = πc fc (t) + πu fu (t) and F (t) = πc Fc (t) + πu Fu (t).
Also define,
       πc is the probability of ‘cure’ or never experiencing the event of
       interest.
       πu = 1 − πc is the probability of eventual failure.




Rebecca (Medical Research Council)           Cure Fractions                          JC 2008   10 / 27
                                     Cure Fractions   Mixture Cure Fraction Models




Let,
       g =2
       and the mixture density and distribution functions have the properties
       of the density and distribution function in survival analysis, i.e.

                 f (t) = πc fc (t) + πu fu (t) and F (t) = πc Fc (t) + πu Fu (t).
Also define,
       πc is the probability of ‘cure’ or never experiencing the event of
       interest.
       πu = 1 − πc is the probability of eventual failure.




Rebecca (Medical Research Council)           Cure Fractions                          JC 2008   10 / 27
                                          Cure Fractions   Mixture Cure Fraction Models


The Survival Distribution Function


The survival distribution function can also be written in terms of the
mixture of the ‘cured’ part plus the ‘failure’ part.

                                      S(t) = πc Sc (t) + πu Su (t).

Since those who are ‘cured’ [πc Sc (t)] will never experience the event, thus
the limt→∞ S(t) = 1 and the survival distribution becomes.


                                     S(t) = πc + πu Su (t),
                                           = πc + (1 − πc )Su (t).                                  (1)




Rebecca (Medical Research Council)                Cure Fractions                          JC 2008   11 / 27
                                          Cure Fractions   Mixture Cure Fraction Models


The Survival Distribution Function


The survival distribution function can also be written in terms of the
mixture of the ‘cured’ part plus the ‘failure’ part.

                                      S(t) = πc Sc (t) + πu Su (t).

Since those who are ‘cured’ [πc Sc (t)] will never experience the event, thus
the limt→∞ S(t) = 1 and the survival distribution becomes.


                                     S(t) = πc + πu Su (t),
                                           = πc + (1 − πc )Su (t).                                  (1)




Rebecca (Medical Research Council)                Cure Fractions                          JC 2008   11 / 27
                                          Cure Fractions   Mixture Cure Fraction Models


The Survival Distribution Function


The survival distribution function can also be written in terms of the
mixture of the ‘cured’ part plus the ‘failure’ part.

                                      S(t) = πc Sc (t) + πu Su (t).

Since those who are ‘cured’ [πc Sc (t)] will never experience the event, thus
the limt→∞ S(t) = 1 and the survival distribution becomes.


                                     S(t) = πc + πu Su (t),
                                           = πc + (1 − πc )Su (t).                                  (1)




Rebecca (Medical Research Council)                Cure Fractions                          JC 2008   11 / 27
                                      Cure Fractions   Mixture Cure Fraction Models


The Failure Distribution Function



The corresponding failure distribution function is,

                              F (t) = 1 − S(t)
                                     = 1 − [πc + (1 − πc )Su (t)]
                                     = (1 − πc )Fu (t).                                         (2)

Therefore, limt→∞ Fu (t) = 1, implies that the limt→∞ = 1 − πc . Thus, at
least some of the individuals experience cure.




Rebecca (Medical Research Council)            Cure Fractions                          JC 2008   12 / 27
                                         Cure Fractions   Mixture Cure Fraction Models


The log-likelihood function


Substitution of the mixture density and survival functions into the
standard likelihood,
                                        N                         N
                    ln L(ti ) =              δi ln f (ti ) +            (1 − δi ) ln S(ti ),
                                       i=1                        i=1

 yields,
                          N                                  N
     ln L(ti )    =           δi ln [(1 − πc )fu (ti )] +         (1 − δi ) ln [πc + (1 − πc )Su (t)].
                        i=1                                 i=1




Rebecca (Medical Research Council)               Cure Fractions                                JC 2008   13 / 27
                                         Cure Fractions   Mixture Cure Fraction Models


The log-likelihood function


Substitution of the mixture density and survival functions into the
standard likelihood,
                                        N                         N
                    ln L(ti ) =              δi ln f (ti ) +            (1 − δi ) ln S(ti ),
                                       i=1                        i=1

 yields,
                          N                                  N
     ln L(ti )    =           δi ln [(1 − πc )fu (ti )] +         (1 − δi ) ln [πc + (1 − πc )Su (t)].
                        i=1                                 i=1




Rebecca (Medical Research Council)               Cure Fractions                                JC 2008   13 / 27
                                     Cure Fractions   Non-Mixture Cure Fractions


Outline


1   Data and Setup
      The Basic Question: Survival impact on Transplant Patients

2   Cure Fractions
      A Brief Word on Survival Analysis
      Mixture Cure Fraction Models
      Non-Mixture Cure Fractions
      Modelling Relative Survival

3   Application
      The models



Rebecca (Medical Research Council)           Cure Fractions                        JC 2008   14 / 27
                                         Cure Fractions   Non-Mixture Cure Fractions


Non-Mixture Cure fractions
Non-Mixture [5] models are of the form:

                                          S(t) = π (1−Sz (t)) .

The corresponding density function is:

                                     f (t) = −π Fz (t) ln (π)fz (t).

       This model was derived under the threshold model for tumor
       resistance. Where, Fz (t) refers to the distribution of division time for
       each cell in a homogenous clone of cells [4].
       The hazard function for this model takes the form,
                              λ(t) = − ln (π)fz (t).

However, these models can be considered a useful mathematical function, with an
asymptote that can be applied to estimate the cure fraction in any cure model
where it is reasonable to assume cure.
Rebecca (Medical Research Council)               Cure Fractions                        JC 2008   15 / 27
                                         Cure Fractions   Non-Mixture Cure Fractions


Non-Mixture Cure fractions
Non-Mixture [5] models are of the form:

                                          S(t) = π (1−Sz (t)) .

The corresponding density function is:

                                     f (t) = −π Fz (t) ln (π)fz (t).

       This model was derived under the threshold model for tumor
       resistance. Where, Fz (t) refers to the distribution of division time for
       each cell in a homogenous clone of cells [4].
       The hazard function for this model takes the form,
                              λ(t) = − ln (π)fz (t).

However, these models can be considered a useful mathematical function, with an
asymptote that can be applied to estimate the cure fraction in any cure model
where it is reasonable to assume cure.
Rebecca (Medical Research Council)               Cure Fractions                        JC 2008   15 / 27
                                         Cure Fractions   Non-Mixture Cure Fractions


Non-Mixture Cure fractions
Non-Mixture [5] models are of the form:

                                          S(t) = π (1−Sz (t)) .

The corresponding density function is:

                                     f (t) = −π Fz (t) ln (π)fz (t).

       This model was derived under the threshold model for tumor
       resistance. Where, Fz (t) refers to the distribution of division time for
       each cell in a homogenous clone of cells [4].
       The hazard function for this model takes the form,
                              λ(t) = − ln (π)fz (t).

However, these models can be considered a useful mathematical function, with an
asymptote that can be applied to estimate the cure fraction in any cure model
where it is reasonable to assume cure.
Rebecca (Medical Research Council)               Cure Fractions                        JC 2008   15 / 27
                                         Cure Fractions   Non-Mixture Cure Fractions


Non-Mixture Cure fractions
Non-Mixture [5] models are of the form:

                                          S(t) = π (1−Sz (t)) .

The corresponding density function is:

                                     f (t) = −π Fz (t) ln (π)fz (t).

       This model was derived under the threshold model for tumor
       resistance. Where, Fz (t) refers to the distribution of division time for
       each cell in a homogenous clone of cells [4].
       The hazard function for this model takes the form,
                              λ(t) = − ln (π)fz (t).

However, these models can be considered a useful mathematical function, with an
asymptote that can be applied to estimate the cure fraction in any cure model
where it is reasonable to assume cure.
Rebecca (Medical Research Council)               Cure Fractions                        JC 2008   15 / 27
                                         Cure Fractions   Non-Mixture Cure Fractions


Non-Mixture Cure fractions
Non-Mixture [5] models are of the form:
                                          S(t) = π (1−Sz (t)) .
The corresponding density function is:
                                     f (t) = −π Fz (t) ln (π)fz (t).

       This model was derived under the threshold model for tumor
       resistance. Where, Fz (t) refers to the distribution of division time for
       each cell in a homogenous clone of cells [4].
       The hazard function for this model takes the form,

                                        λ(t) = − ln (π)fz (t).

However, these models can be considered a useful mathematical function, with an
asymptote that can be applied to estimate the cure fraction in any cure model
where it is reasonable to assume cure.
Rebecca (Medical Research Council)               Cure Fractions                        JC 2008   15 / 27
                                         Cure Fractions   Non-Mixture Cure Fractions


Non-Mixture Cure fractions
Non-Mixture [5] models are of the form:
                                          S(t) = π (1−Sz (t)) .
The corresponding density function is:
                                     f (t) = −π Fz (t) ln (π)fz (t).

       This model was derived under the threshold model for tumor
       resistance. Where, Fz (t) refers to the distribution of division time for
       each cell in a homogenous clone of cells [4].
       The hazard function for this model takes the form,

                                        λ(t) = − ln (π)fz (t).

However, these models can be considered a useful mathematical function, with an
asymptote that can be applied to estimate the cure fraction in any cure model
where it is reasonable to assume cure.
Rebecca (Medical Research Council)               Cure Fractions                        JC 2008   15 / 27
                                          Cure Fractions     Non-Mixture Cure Fractions


The log-likelihood and the Desirability of the Non-Mixture
Model


The log-likelihood function for the non-mixture model is given by,
                    N                             N                                   N
ln L(ti )    =           δi Fz (ti ) ln (πi ) +         δi ln [ln (πi )fz (ti )] +         (1 − δi )Fz (ti ) ln (πi ).
                   i=1                            i=1                                i=1



       When the parameters in Fz (t) do not not depend on covariates then
       the non-mixture cure model is a proportional hazards model.
       The model converges more easily than the mixture model does in
       many in many instances.




Rebecca (Medical Research Council)                 Cure Fractions                                    JC 2008    16 / 27
                                          Cure Fractions     Non-Mixture Cure Fractions


The log-likelihood and the Desirability of the Non-Mixture
Model


The log-likelihood function for the non-mixture model is given by,
                    N                             N                                   N
ln L(ti )    =           δi Fz (ti ) ln (πi ) +         δi ln [ln (πi )fz (ti )] +         (1 − δi )Fz (ti ) ln (πi ).
                   i=1                            i=1                                i=1



       When the parameters in Fz (t) do not not depend on covariates then
       the non-mixture cure model is a proportional hazards model.
       The model converges more easily than the mixture model does in
       many in many instances.




Rebecca (Medical Research Council)                 Cure Fractions                                    JC 2008    16 / 27
                                          Cure Fractions     Non-Mixture Cure Fractions


The log-likelihood and the Desirability of the Non-Mixture
Model


The log-likelihood function for the non-mixture model is given by,
                    N                             N                                   N
ln L(ti )    =           δi Fz (ti ) ln (πi ) +         δi ln [ln (πi )fz (ti )] +         (1 − δi )Fz (ti ) ln (πi ).
                   i=1                            i=1                                i=1



       When the parameters in Fz (t) do not not depend on covariates then
       the non-mixture cure model is a proportional hazards model.
       The model converges more easily than the mixture model does in
       many in many instances.




Rebecca (Medical Research Council)                 Cure Fractions                                    JC 2008    16 / 27
                                     Cure Fractions   Modelling Relative Survival


Outline


1   Data and Setup
      The Basic Question: Survival impact on Transplant Patients

2   Cure Fractions
      A Brief Word on Survival Analysis
      Mixture Cure Fraction Models
      Non-Mixture Cure Fractions
      Modelling Relative Survival

3   Application
      The models



Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   17 / 27
                                     Cure Fractions   Modelling Relative Survival


Relative Survival and Excess Hazards


Why use Background Mortality:
       A subject may die of the disease they are diagnosed with or they may
       die of something else.
       Often interest lies in mortality due to the disease of interest and not
       other causes.
       There exist problems with cause-specific survival/death due to
       inaccuracy of death certificates.
       An alternative to cause-specific survival is to model relative survival
       or its converse, excess mortality.




Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   18 / 27
                                     Cure Fractions   Modelling Relative Survival


Relative Survival and Excess Hazards


Why use Background Mortality:
       A subject may die of the disease they are diagnosed with or they may
       die of something else.
       Often interest lies in mortality due to the disease of interest and not
       other causes.
       There exist problems with cause-specific survival/death due to
       inaccuracy of death certificates.
       An alternative to cause-specific survival is to model relative survival
       or its converse, excess mortality.




Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   18 / 27
                                     Cure Fractions   Modelling Relative Survival


Relative Survival and Excess Hazards


Why use Background Mortality:
       A subject may die of the disease they are diagnosed with or they may
       die of something else.
       Often interest lies in mortality due to the disease of interest and not
       other causes.
       There exist problems with cause-specific survival/death due to
       inaccuracy of death certificates.
       An alternative to cause-specific survival is to model relative survival
       or its converse, excess mortality.




Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   18 / 27
                                     Cure Fractions   Modelling Relative Survival


Relative Survival and Excess Hazards


Why use Background Mortality:
       A subject may die of the disease they are diagnosed with or they may
       die of something else.
       Often interest lies in mortality due to the disease of interest and not
       other causes.
       There exist problems with cause-specific survival/death due to
       inaccuracy of death certificates.
       An alternative to cause-specific survival is to model relative survival
       or its converse, excess mortality.




Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   18 / 27
                                     Cure Fractions   Modelling Relative Survival


Background Mortality

Suppose that,
           S ∗ (t) is the expected survival.
           λ∗ (t) is the expected mortality rate.
Then the total survival S(t), can be written as the product of the relative
survival, R(t), and the expected survival S ∗ (t),

                                      S(t) = S ∗ (t)R(t).

Expected survival obtained from national population life tables stratified by:
       Age
       Sex
       Year of Diagnosis
       Other covariates if possible

Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   19 / 27
                                     Cure Fractions   Modelling Relative Survival


Background Mortality

Suppose that,
           S ∗ (t) is the expected survival.
           λ∗ (t) is the expected mortality rate.
Then the total survival S(t), can be written as the product of the relative
survival, R(t), and the expected survival S ∗ (t),

                                      S(t) = S ∗ (t)R(t).

Expected survival obtained from national population life tables stratified by:
       Age
       Sex
       Year of Diagnosis
       Other covariates if possible

Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   19 / 27
                                     Cure Fractions   Modelling Relative Survival


Background Mortality

Suppose that,
           S ∗ (t) is the expected survival.
           λ∗ (t) is the expected mortality rate.
Then the total survival S(t), can be written as the product of the relative
survival, R(t), and the expected survival S ∗ (t),

                                      S(t) = S ∗ (t)R(t).

Expected survival obtained from national population life tables stratified by:
       Age
       Sex
       Year of Diagnosis
       Other covariates if possible

Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   19 / 27
                                     Cure Fractions   Modelling Relative Survival


Background Mortality

Suppose that,
           S ∗ (t) is the expected survival.
           λ∗ (t) is the expected mortality rate.
Then the total survival S(t), can be written as the product of the relative
survival, R(t), and the expected survival S ∗ (t),

                                      S(t) = S ∗ (t)R(t).

Expected survival obtained from national population life tables stratified by:
       Age
       Sex
       Year of Diagnosis
       Other covariates if possible

Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   19 / 27
                                     Cure Fractions   Modelling Relative Survival


Background Mortality for Mixture Cure

Survival function: S(t) = S ∗ (t)R(t)

                              S(t) = S ∗ (t)[π + (1 − π)Su (t)].


The hazard function is: λ(t) = λ∗ (t) + λE (t)

                                                        (1 − π)fu (t)
                             λ(t) = λ∗ (t) +                            .
                                                      π + (1 − π)Su (t)

The corresponding density function is

               f (t) = f ∗ (t)[π + (1 − π)Su (t)] + S ∗ (t)(1 − π)fu (t).



Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   20 / 27
                                     Cure Fractions   Modelling Relative Survival


Background Mortality for Mixture Cure

Survival function: S(t) = S ∗ (t)R(t)

                              S(t) = S ∗ (t)[π + (1 − π)Su (t)].


The hazard function is: λ(t) = λ∗ (t) + λE (t)

                                                        (1 − π)fu (t)
                             λ(t) = λ∗ (t) +                            .
                                                      π + (1 − π)Su (t)

The corresponding density function is

               f (t) = f ∗ (t)[π + (1 − π)Su (t)] + S ∗ (t)(1 − π)fu (t).



Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   20 / 27
                                     Cure Fractions   Modelling Relative Survival


Background Mortality for Mixture Cure

Survival function: S(t) = S ∗ (t)R(t)

                              S(t) = S ∗ (t)[π + (1 − π)Su (t)].


The hazard function is: λ(t) = λ∗ (t) + λE (t)

                                                        (1 − π)fu (t)
                             λ(t) = λ∗ (t) +                            .
                                                      π + (1 − π)Su (t)

The corresponding density function is

               f (t) = f ∗ (t)[π + (1 − π)Su (t)] + S ∗ (t)(1 − π)fu (t).



Rebecca (Medical Research Council)           Cure Fractions                         JC 2008   20 / 27
                                          Cure Fractions   Modelling Relative Survival


Background Mortality for Non-Mixture Cure

Survival function: S(t) = S ∗ (t)R(t)

                                      S(t) = S ∗ (t)[π 1−Sz (t) ].


The hazard function is: λ(t) = λ∗ (t) + λE (t)

                                     λ(t) = λ∗ (t) − fz (t) ln (λ).


The corresponding density function is

                     f (t) = Fz (t) ln(π)[f ∗ (t) − S ∗ (t)fz (t) ln (π)].

See De Angelis et al 1999 for a discussion [3]

Rebecca (Medical Research Council)                Cure Fractions                         JC 2008   21 / 27
                                          Cure Fractions   Modelling Relative Survival


Background Mortality for Non-Mixture Cure

Survival function: S(t) = S ∗ (t)R(t)

                                      S(t) = S ∗ (t)[π 1−Sz (t) ].


The hazard function is: λ(t) = λ∗ (t) + λE (t)

                                     λ(t) = λ∗ (t) − fz (t) ln (λ).


The corresponding density function is

                     f (t) = Fz (t) ln(π)[f ∗ (t) − S ∗ (t)fz (t) ln (π)].

See De Angelis et al 1999 for a discussion [3]

Rebecca (Medical Research Council)                Cure Fractions                         JC 2008   21 / 27
                                          Cure Fractions   Modelling Relative Survival


Background Mortality for Non-Mixture Cure

Survival function: S(t) = S ∗ (t)R(t)

                                      S(t) = S ∗ (t)[π 1−Sz (t) ].


The hazard function is: λ(t) = λ∗ (t) + λE (t)

                                     λ(t) = λ∗ (t) − fz (t) ln (λ).


The corresponding density function is

                     f (t) = Fz (t) ln(π)[f ∗ (t) − S ∗ (t)fz (t) ln (π)].

See De Angelis et al 1999 for a discussion [3]

Rebecca (Medical Research Council)                Cure Fractions                         JC 2008   21 / 27
                                     Application   The models


Outline


1   Data and Setup
      The Basic Question: Survival impact on Transplant Patients

2   Cure Fractions
      A Brief Word on Survival Analysis
      Mixture Cure Fraction Models
      Non-Mixture Cure Fractions
      Modelling Relative Survival

3   Application
      The models



Rebecca (Medical Research Council)        Cure Fractions        JC 2008   22 / 27
                                       Application   The models




                 Figure 1: Estimated relative survival and excess hazard functions




Rebecca (Medical Research Council)          Cure Fractions                           JC 2008   23 / 27
                                                   Application    The models




Model 1 Cure fraction model (background mortality) Distribution - Weibull. Link - loglog
Model 2 Model 2 = Model 1, but with the graftlos variable removed
Model 3 Cure fraction model (no background mortality) Distribution - Weibull. Link - loglog
Model 4 Standard, survival model. Distribution - Weibull. Link - NA



  Rebecca (Medical Research Council)                     Cure Fractions                       JC 2008   24 / 27
                                                   Application    The models




Model 1 Cure fraction model (background mortality) Distribution - Weibull. Link - loglog
Model 2 Model 2 = Model 1, but with the graftlos variable removed
Model 3 Cure fraction model (no background mortality) Distribution - Weibull. Link - loglog
Model 4 Standard, survival model. Distribution - Weibull. Link - NA



  Rebecca (Medical Research Council)                     Cure Fractions                       JC 2008   24 / 27
                                                   Application    The models




Model 1 Cure fraction model (background mortality) Distribution - Weibull. Link - loglog
Model 2 Model 2 = Model 1, but with the graftlos variable removed
Model 3 Cure fraction model (no background mortality) Distribution - Weibull. Link - loglog
Model 4 Standard, survival model. Distribution - Weibull. Link - NA



  Rebecca (Medical Research Council)                     Cure Fractions                       JC 2008   24 / 27
                                                   Application    The models




Model 1 Cure fraction model (background mortality) Distribution - Weibull. Link - loglog
Model 2 Model 2 = Model 1, but with the graftlos variable removed
Model 3 Cure fraction model (no background mortality) Distribution - Weibull. Link - loglog
Model 4 Standard, survival model. Distribution - Weibull. Link - NA



  Rebecca (Medical Research Council)                     Cure Fractions                       JC 2008   24 / 27
                                     Application   The models




       Impact of cure models with background mortality differed only
       marginally when compared to other models.
       Removing the graftlos variable showed that these models had more of
       an impact.
       Theoretically these models make more sense when modelling
       transplant mortality rates, since there is high mortality in the period
       immediately after the transplant.
       Clearly for a larger sample the effect of the background rates would
       certainly have been more pronounced.




Rebecca (Medical Research Council)        Cure Fractions            JC 2008   25 / 27
                                     Application   The models




       Most authors, warn that these models are to be used under very
       specific conditions and assumptions
       The fact that the four models above, had similar effects, implies that
       the conditions and assumptions underlying these cure models were
       not violated. Implies that cure models are useful in modelling time to
       failure in transplant analysis.
       Although, differences in effects were minimal, they were present,
       especially when correct procedures were followed, as in the case of
       model 2.
       It is clear that once all assumptions regarding the cure fraction model,
       (and its underlying population data), have been fulfilled, it is an
       important model in estimating survival times for transplant analysis.
       The model gives a good indication of trends and differences in
       survival and cure rates, over time.


Rebecca (Medical Research Council)        Cure Fractions           JC 2008   26 / 27
                                     Application   The models




       Most authors, warn that these models are to be used under very
       specific conditions and assumptions
       The fact that the four models above, had similar effects, implies that
       the conditions and assumptions underlying these cure models were
       not violated. Implies that cure models are useful in modelling time to
       failure in transplant analysis.
       Although, differences in effects were minimal, they were present,
       especially when correct procedures were followed, as in the case of
       model 2.
       It is clear that once all assumptions regarding the cure fraction model,
       (and its underlying population data), have been fulfilled, it is an
       important model in estimating survival times for transplant analysis.
       The model gives a good indication of trends and differences in
       survival and cure rates, over time.


Rebecca (Medical Research Council)        Cure Fractions           JC 2008   26 / 27
                                     Application   The models




       Most authors, warn that these models are to be used under very
       specific conditions and assumptions
       The fact that the four models above, had similar effects, implies that
       the conditions and assumptions underlying these cure models were
       not violated. Implies that cure models are useful in modelling time to
       failure in transplant analysis.
       Although, differences in effects were minimal, they were present,
       especially when correct procedures were followed, as in the case of
       model 2.
       It is clear that once all assumptions regarding the cure fraction model,
       (and its underlying population data), have been fulfilled, it is an
       important model in estimating survival times for transplant analysis.
       The model gives a good indication of trends and differences in
       survival and cure rates, over time.


Rebecca (Medical Research Council)        Cure Fractions           JC 2008   26 / 27
                                     Application   The models




       Most authors, warn that these models are to be used under very
       specific conditions and assumptions
       The fact that the four models above, had similar effects, implies that
       the conditions and assumptions underlying these cure models were
       not violated. Implies that cure models are useful in modelling time to
       failure in transplant analysis.
       Although, differences in effects were minimal, they were present,
       especially when correct procedures were followed, as in the case of
       model 2.
       It is clear that once all assumptions regarding the cure fraction model,
       (and its underlying population data), have been fulfilled, it is an
       important model in estimating survival times for transplant analysis.
       The model gives a good indication of trends and differences in
       survival and cure rates, over time.


Rebecca (Medical Research Council)        Cure Fractions           JC 2008   26 / 27
                                     Application   The models




       Most authors, warn that these models are to be used under very
       specific conditions and assumptions
       The fact that the four models above, had similar effects, implies that
       the conditions and assumptions underlying these cure models were
       not violated. Implies that cure models are useful in modelling time to
       failure in transplant analysis.
       Although, differences in effects were minimal, they were present,
       especially when correct procedures were followed, as in the case of
       model 2.
       It is clear that once all assumptions regarding the cure fraction model,
       (and its underlying population data), have been fulfilled, it is an
       important model in estimating survival times for transplant analysis.
       The model gives a good indication of trends and differences in
       survival and cure rates, over time.


Rebecca (Medical Research Council)        Cure Fractions           JC 2008   26 / 27
                                     Application   The models


THE END




Rebecca (Medical Research Council)        Cure Fractions        JC 2008   27 / 27
                                     Application   The models


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      following treatment. Journal of the American Statistical
      Association,Vol. 47, 501-515.
      Boag, J. (1949), Maximum likelihood estimates of the proportion of
      patients cured by cancer therapy. Journal of the Royal Statistical
      Society (B), Vol. 11, 15-44.
      De Angelis, R., Capocaccia, R., Hakulinen, T., Soderman, B. and
      Verdecchia, A. (1999), Mixture models for cancer survival analysis:
      application to population-based data with covariates, Statistics in
      Medicine, Vol. 18, 441-54.
      Yakovlev, A.Y. (1996), Threshold models for tumor recurrence,
      Mathematical and computer modelling, Vol. 23(6), 153-164.
      Yin, G. and Ibrahim, J.G. (2005), Cure rate models: a unified
      approach. The Canadian Journal of Statistics (In Press), Vol. 33.



Rebecca (Medical Research Council)        Cure Fractions         JC 2008    27 / 27

								
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