# Survival Analysis Cure Fractions

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```					                     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
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
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
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
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
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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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