VIEWS: 5 PAGES: 75 POSTED ON: 4/29/2011
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 Berkson, J. and Gage, R.P. (1952), Survival curve for cancer patients 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