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Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary Fitting smooth-in-time prognostic risk functions via logistic regression James A. Hanley 1 Olli S. Miettinen 1 1 Department of

  1. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FORM h ( x , t ) = e g ( x , t ,β ) log { h ( x , t ) } = g ( x , t , β ) ⇐ ⇒ • x is a realization of the covariate vector X , representing the patient profile P , and possible intervention I . • β : a vector of parameters with unknown values, • g () includes constant 1, variates for P , I and t ; • g () can have product terms involving P , I , and t . • g () must be ‘linear’ in parameters, in ‘ linear model’ sense. ————– • ‘proportional hazards’ if no product terms involving t & I • If t is represented by a linear term (so that ‘time to event’ ∼ Gompertz ), then � CI p , i ( t ) has a closed smooth form. • If t is replaced by log t , then ‘time to event’ ∼ Weibull .

  2. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FORM h ( x , t ) = e g ( x , t ,β ) log { h ( x , t ) } = g ( x , t , β ) ⇐ ⇒ • x is a realization of the covariate vector X , representing the patient profile P , and possible intervention I . • β : a vector of parameters with unknown values, • g () includes constant 1, variates for P , I and t ; • g () can have product terms involving P , I , and t . • g () must be ‘linear’ in parameters, in ‘ linear model’ sense. ————– • ‘proportional hazards’ if no product terms involving t & I • If t is represented by a linear term (so that ‘time to event’ ∼ Gompertz ), then � CI p , i ( t ) has a closed smooth form. • If t is replaced by log t , then ‘time to event’ ∼ Weibull .

  3. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FORM h ( x , t ) = e g ( x , t ,β ) log { h ( x , t ) } = g ( x , t , β ) ⇐ ⇒ • x is a realization of the covariate vector X , representing the patient profile P , and possible intervention I . • β : a vector of parameters with unknown values, • g () includes constant 1, variates for P , I and t ; • g () can have product terms involving P , I , and t . • g () must be ‘linear’ in parameters, in ‘ linear model’ sense. ————– • ‘proportional hazards’ if no product terms involving t & I • If t is represented by a linear term (so that ‘time to event’ ∼ Gompertz ), then � CI p , i ( t ) has a closed smooth form. • If t is replaced by log t , then ‘time to event’ ∼ Weibull .

  4. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FORM h ( x , t ) = e g ( x , t ,β ) log { h ( x , t ) } = g ( x , t , β ) ⇐ ⇒ • x is a realization of the covariate vector X , representing the patient profile P , and possible intervention I . • β : a vector of parameters with unknown values, • g () includes constant 1, variates for P , I and t ; • g () can have product terms involving P , I , and t . • g () must be ‘linear’ in parameters, in ‘ linear model’ sense. ————– • ‘proportional hazards’ if no product terms involving t & I • If t is represented by a linear term (so that ‘time to event’ ∼ Gompertz ), then � CI p , i ( t ) has a closed smooth form. • If t is replaced by log t , then ‘time to event’ ∼ Weibull .

  5. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FORM h ( x , t ) = e g ( x , t ,β ) log { h ( x , t ) } = g ( x , t , β ) ⇐ ⇒ • x is a realization of the covariate vector X , representing the patient profile P , and possible intervention I . • β : a vector of parameters with unknown values, • g () includes constant 1, variates for P , I and t ; • g () can have product terms involving P , I , and t . • g () must be ‘linear’ in parameters, in ‘ linear model’ sense. ————– • ‘proportional hazards’ if no product terms involving t & I • If t is represented by a linear term (so that ‘time to event’ ∼ Gompertz ), then � CI p , i ( t ) has a closed smooth form. • If t is replaced by log t , then ‘time to event’ ∼ Weibull .

  6. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FORM h ( x , t ) = e g ( x , t ,β ) log { h ( x , t ) } = g ( x , t , β ) ⇐ ⇒ • x is a realization of the covariate vector X , representing the patient profile P , and possible intervention I . • β : a vector of parameters with unknown values, • g () includes constant 1, variates for P , I and t ; • g () can have product terms involving P , I , and t . • g () must be ‘linear’ in parameters, in ‘ linear model’ sense. ————– • ‘proportional hazards’ if no product terms involving t & I • If t is represented by a linear term (so that ‘time to event’ ∼ Gompertz ), then � CI p , i ( t ) has a closed smooth form. • If t is replaced by log t , then ‘time to event’ ∼ Weibull .

  7. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FORM h ( x , t ) = e g ( x , t ,β ) log { h ( x , t ) } = g ( x , t , β ) ⇐ ⇒ • x is a realization of the covariate vector X , representing the patient profile P , and possible intervention I . • β : a vector of parameters with unknown values, • g () includes constant 1, variates for P , I and t ; • g () can have product terms involving P , I , and t . • g () must be ‘linear’ in parameters, in ‘ linear model’ sense. ————– • ‘proportional hazards’ if no product terms involving t & I • If t is represented by a linear term (so that ‘time to event’ ∼ Gompertz ), then � CI p , i ( t ) has a closed smooth form. • If t is replaced by log t , then ‘time to event’ ∼ Weibull .

  8. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FORM h ( x , t ) = e g ( x , t ,β ) log { h ( x , t ) } = g ( x , t , β ) ⇐ ⇒ • x is a realization of the covariate vector X , representing the patient profile P , and possible intervention I . • β : a vector of parameters with unknown values, • g () includes constant 1, variates for P , I and t ; • g () can have product terms involving P , I , and t . • g () must be ‘linear’ in parameters, in ‘ linear model’ sense. ————– • ‘proportional hazards’ if no product terms involving t & I • If t is represented by a linear term (so that ‘time to event’ ∼ Gompertz ), then � CI p , i ( t ) has a closed smooth form. • If t is replaced by log t , then ‘time to event’ ∼ Weibull .

  9. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FORM h ( x , t ) = e g ( x , t ,β ) log { h ( x , t ) } = g ( x , t , β ) ⇐ ⇒ • x is a realization of the covariate vector X , representing the patient profile P , and possible intervention I . • β : a vector of parameters with unknown values, • g () includes constant 1, variates for P , I and t ; • g () can have product terms involving P , I , and t . • g () must be ‘linear’ in parameters, in ‘ linear model’ sense. ————– • ‘proportional hazards’ if no product terms involving t & I • If t is represented by a linear term (so that ‘time to event’ ∼ Gompertz ), then � CI p , i ( t ) has a closed smooth form. • If t is replaced by log t , then ‘time to event’ ∼ Weibull .

  10. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FITTING • Unable to find a ready-to-use ML procedure within the common statistical packages. • Likelihood becomes quite involved even if no censored observations. • Albertsen & Hanley(’98); Efron(’88, ’02); Carstensen(’00): - divide ‘survival time’ of each subject into time-slices; - treat # of events in each ∼ Binomial / Poisson.

  11. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FITTING • Unable to find a ready-to-use ML procedure within the common statistical packages. • Likelihood becomes quite involved even if no censored observations. • Albertsen & Hanley(’98); Efron(’88, ’02); Carstensen(’00): - divide ‘survival time’ of each subject into time-slices; - treat # of events in each ∼ Binomial / Poisson.

  12. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FITTING • Unable to find a ready-to-use ML procedure within the common statistical packages. • Likelihood becomes quite involved even if no censored observations. • Albertsen & Hanley(’98); Efron(’88, ’02); Carstensen(’00): - divide ‘survival time’ of each subject into time-slices; - treat # of events in each ∼ Binomial / Poisson.

  13. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FULLY-PARAMETRIC MODEL: FITTING • Unable to find a ready-to-use ML procedure within the common statistical packages. • Likelihood becomes quite involved even if no censored observations. • Albertsen & Hanley(’98); Efron(’88, ’02); Carstensen(’00): - divide ‘survival time’ of each subject into time-slices; - treat # of events in each ∼ Binomial / Poisson.

  14. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTING: OUR APPROACH • An extension of the method of Mantel (1973) to binary outcomes that deals with time dimension. • Mantel’s problem: • ( c =) 165 ‘cases’ of Y = 1 , • 4000 instances of Y = 0 . • Associated regressor vector X for each of the 4165 • A logistic model for Prob ( Y = 1 | X ) • A computer with limited capacity.

  15. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTING: OUR APPROACH • An extension of the method of Mantel (1973) to binary outcomes that deals with time dimension. • Mantel’s problem: • ( c =) 165 ‘cases’ of Y = 1 , • 4000 instances of Y = 0 . • Associated regressor vector X for each of the 4165 • A logistic model for Prob ( Y = 1 | X ) • A computer with limited capacity.

  16. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTING: OUR APPROACH • An extension of the method of Mantel (1973) to binary outcomes that deals with time dimension. • Mantel’s problem: • ( c =) 165 ‘cases’ of Y = 1 , • 4000 instances of Y = 0 . • Associated regressor vector X for each of the 4165 • A logistic model for Prob ( Y = 1 | X ) • A computer with limited capacity.

  17. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTING: OUR APPROACH • An extension of the method of Mantel (1973) to binary outcomes that deals with time dimension. • Mantel’s problem: • ( c =) 165 ‘cases’ of Y = 1 , • 4000 instances of Y = 0 . • Associated regressor vector X for each of the 4165 • A logistic model for Prob ( Y = 1 | X ) • A computer with limited capacity.

  18. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTING: OUR APPROACH • An extension of the method of Mantel (1973) to binary outcomes that deals with time dimension. • Mantel’s problem: • ( c =) 165 ‘cases’ of Y = 1 , • 4000 instances of Y = 0 . • Associated regressor vector X for each of the 4165 • A logistic model for Prob ( Y = 1 | X ) • A computer with limited capacity.

  19. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTING: OUR APPROACH • An extension of the method of Mantel (1973) to binary outcomes that deals with time dimension. • Mantel’s problem: • ( c =) 165 ‘cases’ of Y = 1 , • 4000 instances of Y = 0 . • Associated regressor vector X for each of the 4165 • A logistic model for Prob ( Y = 1 | X ) • A computer with limited capacity.

  20. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTING: OUR APPROACH • An extension of the method of Mantel (1973) to binary outcomes that deals with time dimension. • Mantel’s problem: • ( c =) 165 ‘cases’ of Y = 1 , • 4000 instances of Y = 0 . • Associated regressor vector X for each of the 4165 • A logistic model for Prob ( Y = 1 | X ) • A computer with limited capacity.

  21. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTING: OUR APPROACH • An extension of the method of Mantel (1973) to binary outcomes that deals with time dimension. • Mantel’s problem: • ( c =) 165 ‘cases’ of Y = 1 , • 4000 instances of Y = 0 . • Associated regressor vector X for each of the 4165 • A logistic model for Prob ( Y = 1 | X ) • A computer with limited capacity.

  22. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary MANTEL ’S SOLUTION • Form a reduced dataset containing... • All c instances (cases) of Y = 1 • Random sample of the Y = 0 observations • Fit the same logistic model to this reduced dataset. “Such sampling will tend to leave the dependence of the log odds on the variables unaffected except for an additive constant.” Anderson (Biometrika, 1972) had noted this too. • Outcome(Choice)-based sampling common in Epi, Marketing, etc...

  23. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary MANTEL ’S SOLUTION • Form a reduced dataset containing... • All c instances (cases) of Y = 1 • Random sample of the Y = 0 observations • Fit the same logistic model to this reduced dataset. “Such sampling will tend to leave the dependence of the log odds on the variables unaffected except for an additive constant.” Anderson (Biometrika, 1972) had noted this too. • Outcome(Choice)-based sampling common in Epi, Marketing, etc...

  24. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary MANTEL ’S SOLUTION • Form a reduced dataset containing... • All c instances (cases) of Y = 1 • Random sample of the Y = 0 observations • Fit the same logistic model to this reduced dataset. “Such sampling will tend to leave the dependence of the log odds on the variables unaffected except for an additive constant.” Anderson (Biometrika, 1972) had noted this too. • Outcome(Choice)-based sampling common in Epi, Marketing, etc...

  25. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary MANTEL ’S SOLUTION • Form a reduced dataset containing... • All c instances (cases) of Y = 1 • Random sample of the Y = 0 observations • Fit the same logistic model to this reduced dataset. “Such sampling will tend to leave the dependence of the log odds on the variables unaffected except for an additive constant.” Anderson (Biometrika, 1972) had noted this too. • Outcome(Choice)-based sampling common in Epi, Marketing, etc...

  26. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary MANTEL ’S SOLUTION • Form a reduced dataset containing... • All c instances (cases) of Y = 1 • Random sample of the Y = 0 observations • Fit the same logistic model to this reduced dataset. “Such sampling will tend to leave the dependence of the log odds on the variables unaffected except for an additive constant.” Anderson (Biometrika, 1972) had noted this too. • Outcome(Choice)-based sampling common in Epi, Marketing, etc...

  27. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary MANTEL ’S SOLUTION • Form a reduced dataset containing... • All c instances (cases) of Y = 1 • Random sample of the Y = 0 observations • Fit the same logistic model to this reduced dataset. “Such sampling will tend to leave the dependence of the log odds on the variables unaffected except for an additive constant.” Anderson (Biometrika, 1972) had noted this too. • Outcome(Choice)-based sampling common in Epi, Marketing, etc...

  28. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary MANTEL ’S SOLUTION • Form a reduced dataset containing... • All c instances (cases) of Y = 1 • Random sample of the Y = 0 observations • Fit the same logistic model to this reduced dataset. “Such sampling will tend to leave the dependence of the log odds on the variables unaffected except for an additive constant.” Anderson (Biometrika, 1972) had noted this too. • Outcome(Choice)-based sampling common in Epi, Marketing, etc...

  29. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary DATA TO EXPLAIN OUR APPROACH Systolic Hypertension in Elderly Program (SHEP) .......................... SHEP Cooperative Research Group (1991). .......................... Journal of American Medical Association 265, 3255-3264. • 4,701 persons with complete data on P = { age, sex, race, and systolic blood pressure} and I = { active/placebo}. • Study base of B = 20 , 894 person-years of follow-up; c = 263 events ("cases") of stroke identified.

  30. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary DATA TO EXPLAIN OUR APPROACH Systolic Hypertension in Elderly Program (SHEP) .......................... SHEP Cooperative Research Group (1991). .......................... Journal of American Medical Association 265, 3255-3264. • 4,701 persons with complete data on P = { age, sex, race, and systolic blood pressure} and I = { active/placebo}. • Study base of B = 20 , 894 person-years of follow-up; c = 263 events ("cases") of stroke identified.

  31. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary DATA TO EXPLAIN OUR APPROACH Systolic Hypertension in Elderly Program (SHEP) .......................... SHEP Cooperative Research Group (1991). .......................... Journal of American Medical Association 265, 3255-3264. • 4,701 persons with complete data on P = { age, sex, race, and systolic blood pressure} and I = { active/placebo}. • Study base of B = 20 , 894 person-years of follow-up; c = 263 events ("cases") of stroke identified.

  32. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary DATA TO EXPLAIN OUR APPROACH Systolic Hypertension in Elderly Program (SHEP) .......................... SHEP Cooperative Research Group (1991). .......................... Journal of American Medical Association 265, 3255-3264. • 4,701 persons with complete data on P = { age, sex, race, and systolic blood pressure} and I = { active/placebo}. • Study base of B = 20 , 894 person-years of follow-up; c = 263 events ("cases") of stroke identified.

  33. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary DATA TO EXPLAIN OUR APPROACH Systolic Hypertension in Elderly Program (SHEP) .......................... SHEP Cooperative Research Group (1991). .......................... Journal of American Medical Association 265, 3255-3264. • 4,701 persons with complete data on P = { age, sex, race, and systolic blood pressure} and I = { active/placebo}. • Study base of B = 20 , 894 person-years of follow-up; c = 263 events ("cases") of stroke identified.

  34. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary STUDY BASE, and the 263 cases 6000 − 20,894 person−years [B=20,894 PY] STUDY BASE − 10,982,000,000 person−minutes (approx) 5000 No. of Persons − infinite number of person−moments ↑ Being Followed c = 263 events (Y=1) ● ● ● ● ● ● ● in this infinite number ● ● ● 4000 ● ● ● ● ● ● ● ● ● ● ● ● ● of person−moments ● ● ● ● ● ● ● ● ● ● ● ● ● Persons ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● infinite number ● ● ● ● ● ● ● ● 3000 ● ● ● ● ● ● ● ● ● ● of person−moments ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● with Y=0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● 0 1 2 3 4 5 6 7 t: Years since Randomization

  35. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary OUR APPROACH • Base series: representative (unstratified) sample of base. • b : size of base series • B : amount of population-time constituting study base. • B ( x , t ) : population-time element in study base Pr ( Y = 1 | x , t ) Pr ( Y = 0 | x , t ) = h ( x , t ) × B ( x , t ) b × [ B ( x , t ) / B ] = h ( x , t ) × ( B / b ) , • log ( B / b ) is an offset [a regression term with known coefficient of 1] . → logistic model, with t having same status as x , and offset, directly yields � ID x , t = exp { � h ( x , t ) = � g ( x , t ) } .

  36. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary How large should b be on relation to c ? b : no. of instances of Y = 0 ; c : no. of instances of Y = 1 • Mantel (1973)... little to be gained by letting the size of one series, b, become arbitrarily large if the size of the other series, c , must remain fixed. • With 2008 computing, we can use a b / c ratio as high as 100, and thereby extract virtually all of the information in the base.

  37. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary How large should b be on relation to c ? b : no. of instances of Y = 0 ; c : no. of instances of Y = 1 • Mantel (1973)... little to be gained by letting the size of one series, b, become arbitrarily large if the size of the other series, c , must remain fixed. • With 2008 computing, we can use a b / c ratio as high as 100, and thereby extract virtually all of the information in the base.

  38. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary How large should b be on relation to c ? b : no. of instances of Y = 0 ; c : no. of instances of Y = 1 • Mantel (1973)... little to be gained by letting the size of one series, b, become arbitrarily large if the size of the other series, c , must remain fixed. • With 2008 computing, we can use a b / c ratio as high as 100, and thereby extract virtually all of the information in the base.

  39. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary How large should b be on relation to c ? b : no. of instances of Y = 0 ; c : no. of instances of Y = 1 • Mantel (1973)... little to be gained by letting the size of one series, b, become arbitrarily large if the size of the other series, c , must remain fixed. • With 2008 computing, we can use a b / c ratio as high as 100, and thereby extract virtually all of the information in the base.

  40. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary OUR HAZARD MODEL FOR SHEP DATA log [ h ] = Σ β k X k , where X 1 = Age (in yrs) - 60 X 2 = Indicator of male gender X 3 = Indicator of Black race X 4 = Systolic BP (in mmHg) - 140 ...................................................................... X 5 = Indicator of active treatment ...................................................................... X 6 = T ...................................................................... X 7 = X 5 × X 6 . (non-proportional hazards)

  41. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary OUR HAZARD MODEL FOR SHEP DATA log [ h ] = Σ β k X k , where X 1 = Age (in yrs) - 60 X 2 = Indicator of male gender X 3 = Indicator of Black race X 4 = Systolic BP (in mmHg) - 140 ...................................................................... X 5 = Indicator of active treatment ...................................................................... X 6 = T ...................................................................... X 7 = X 5 × X 6 . (non-proportional hazards)

  42. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary PARAMETER ESTIMATION • Formed person-moments dataset pertaining to: • case series of size c = 263 ( Y = 1) and • (randomly-selected) base series of size b = 26 , 300 ( Y = 0 ) . • Each of 26,563 rows contained realizations of • X 1 , . . . , X 7 • Y • offset = log ( 20 , 894 / 26 , 300 ) . • Logistic model fitted to data in the two series.

  43. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary PARAMETER ESTIMATION • Formed person-moments dataset pertaining to: • case series of size c = 263 ( Y = 1) and • (randomly-selected) base series of size b = 26 , 300 ( Y = 0 ) . • Each of 26,563 rows contained realizations of • X 1 , . . . , X 7 • Y • offset = log ( 20 , 894 / 26 , 300 ) . • Logistic model fitted to data in the two series.

  44. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary PARAMETER ESTIMATION • Formed person-moments dataset pertaining to: • case series of size c = 263 ( Y = 1) and • (randomly-selected) base series of size b = 26 , 300 ( Y = 0 ) . • Each of 26,563 rows contained realizations of • X 1 , . . . , X 7 • Y • offset = log ( 20 , 894 / 26 , 300 ) . • Logistic model fitted to data in the two series.

  45. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary PARAMETER ESTIMATION • Formed person-moments dataset pertaining to: • case series of size c = 263 ( Y = 1) and • (randomly-selected) base series of size b = 26 , 300 ( Y = 0 ) . • Each of 26,563 rows contained realizations of • X 1 , . . . , X 7 • Y • offset = log ( 20 , 894 / 26 , 300 ) . • Logistic model fitted to data in the two series.

  46. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary PARAMETER ESTIMATION • Formed person-moments dataset pertaining to: • case series of size c = 263 ( Y = 1) and • (randomly-selected) base series of size b = 26 , 300 ( Y = 0 ) . • Each of 26,563 rows contained realizations of • X 1 , . . . , X 7 • Y • offset = log ( 20 , 894 / 26 , 300 ) . • Logistic model fitted to data in the two series.

  47. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary PARAMETER ESTIMATION • Formed person-moments dataset pertaining to: • case series of size c = 263 ( Y = 1) and • (randomly-selected) base series of size b = 26 , 300 ( Y = 0 ) . • Each of 26,563 rows contained realizations of • X 1 , . . . , X 7 • Y • offset = log ( 20 , 894 / 26 , 300 ) . • Logistic model fitted to data in the two series.

  48. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary PARAMETER ESTIMATION • Formed person-moments dataset pertaining to: • case series of size c = 263 ( Y = 1) and • (randomly-selected) base series of size b = 26 , 300 ( Y = 0 ) . • Each of 26,563 rows contained realizations of • X 1 , . . . , X 7 • Y • offset = log ( 20 , 894 / 26 , 300 ) . • Logistic model fitted to data in the two series.

  49. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary PARAMETER ESTIMATION • Formed person-moments dataset pertaining to: • case series of size c = 263 ( Y = 1) and • (randomly-selected) base series of size b = 26 , 300 ( Y = 0 ) . • Each of 26,563 rows contained realizations of • X 1 , . . . , X 7 • Y • offset = log ( 20 , 894 / 26 , 300 ) . • Logistic model fitted to data in the two series.

  50. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary DATASET: c = 263 ; b = 10 × 263 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Persons ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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  51. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTED VALUES Proposed Cox logistic regression regression β age − 60 0.041 0.041 0.041 β I male 0.257 0.258 0.259 β I black 0.302 0.301 0.303 β SBP − 140 0.017 0.017 0.017 .................... β I Active treatment -0.200 -0.435 -0.435 .................... β 0 -5.390 -5.295 β t -0.014 -0.057 β t × I Active treatment -0.107 • Fitted logistic function represents log [ h x ( t )] • → cumulative hazard H X ( t ) , and, thus, X -specific risk.

  52. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTED VALUES Proposed Cox logistic regression regression β age − 60 0.041 0.041 0.041 β I male 0.257 0.258 0.259 β I black 0.302 0.301 0.303 β SBP − 140 0.017 0.017 0.017 .................... β I Active treatment -0.200 -0.435 -0.435 .................... β 0 -5.390 -5.295 β t -0.014 -0.057 β t × I Active treatment -0.107 • Fitted logistic function represents log [ h x ( t )] • → cumulative hazard H X ( t ) , and, thus, X -specific risk.

  53. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTED VALUES Proposed Cox logistic regression regression β age − 60 0.041 0.041 0.041 β I male 0.257 0.258 0.259 β I black 0.302 0.301 0.303 β SBP − 140 0.017 0.017 0.017 .................... β I Active treatment -0.200 -0.435 -0.435 .................... β 0 -5.390 -5.295 β t -0.014 -0.057 β t × I Active treatment -0.107 • Fitted logistic function represents log [ h x ( t )] • → cumulative hazard H X ( t ) , and, thus, X -specific risk.

  54. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary FITTED VALUES Proposed Cox logistic regression regression β age − 60 0.041 0.041 0.041 β I male 0.257 0.258 0.259 β I black 0.302 0.301 0.303 β SBP − 140 0.017 0.017 0.017 .................... β I Active treatment -0.200 -0.435 -0.435 .................... β 0 -5.390 -5.295 β t -0.014 -0.057 β t × I Active treatment -0.107 • Fitted logistic function represents log [ h x ( t )] • → cumulative hazard H X ( t ) , and, thus, X -specific risk.

  55. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary ESTIMATED 5-YEAR RISK OF STROKE Risk I h ( t ) H ( 5 ) CI ( 5 ) ∆ � 5 [ 1 − e − H ( 5 ) ] [ ID(t) ] [ 0 h x ( t ) dt ] e − 4 . 86 − 0 . 014 t Low 0 0.037 0.036 e − 5 . 06 − 0 . 124 t 1 0.024 0.024 1.2% High 0 0.16 1 0.10 6% Overall 0 0.076 1 0.049 2.7% Low: 65 year old white female with a SBP of 160 mmHg. High: 80 year old black male with a SBP of 180 mmHg

  56. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary ESTIMATED 5-YEAR RISK OF STROKE Risk I h ( t ) H ( 5 ) CI ( 5 ) ∆ � 5 [ 1 − e − H ( 5 ) ] [ ID(t) ] [ 0 h x ( t ) dt ] e − 4 . 86 − 0 . 014 t Low 0 0.037 0.036 e − 5 . 06 − 0 . 124 t 1 0.024 0.024 1.2% High 0 0.16 1 0.10 6% Overall 0 0.076 1 0.049 2.7% Low: 65 year old white female with a SBP of 160 mmHg. High: 80 year old black male with a SBP of 180 mmHg

  57. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary ESTIMATED 5-YEAR RISK OF STROKE Risk I h ( t ) H ( 5 ) CI ( 5 ) ∆ � 5 [ 1 − e − H ( 5 ) ] [ ID(t) ] [ 0 h x ( t ) dt ] e − 4 . 86 − 0 . 014 t Low 0 0.037 0.036 e − 5 . 06 − 0 . 124 t 1 0.024 0.024 1.2% High 0 0.16 1 0.10 6% Overall 0 0.076 1 0.049 2.7% Low: 65 year old white female with a SBP of 160 mmHg. High: 80 year old black male with a SBP of 180 mmHg

  58. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary ESTIMATED 5-YEAR RISK OF STROKE Risk I h ( t ) H ( 5 ) CI ( 5 ) ∆ � 5 [ 1 − e − H ( 5 ) ] [ ID(t) ] [ 0 h x ( t ) dt ] e − 4 . 86 − 0 . 014 t Low 0 0.037 0.036 e − 5 . 06 − 0 . 124 t 1 0.024 0.024 1.2% High 0 0.16 1 0.10 6% Overall 0 0.076 1 0.049 2.7% Low: 65 year old white female with a SBP of 160 mmHg. High: 80 year old black male with a SBP of 180 mmHg

  59. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary ESTIMATED 5-YEAR RISK OF STROKE Risk I h ( t ) H ( 5 ) CI ( 5 ) ∆ � 5 [ 1 − e − H ( 5 ) ] [ ID(t) ] [ 0 h x ( t ) dt ] e − 4 . 86 − 0 . 014 t Low 0 0.037 0.036 e − 5 . 06 − 0 . 124 t 1 0.024 0.024 1.2% High 0 0.16 1 0.10 6% Overall 0 0.076 1 0.049 2.7% Low: 65 year old white female with a SBP of 160 mmHg. High: 80 year old black male with a SBP of 180 mmHg

  60. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary ESTIMATED 5-YEAR RISK OF STROKE Risk I h ( t ) H ( 5 ) CI ( 5 ) ∆ � 5 [ 1 − e − H ( 5 ) ] [ ID(t) ] [ 0 h x ( t ) dt ] e − 4 . 86 − 0 . 014 t Low 0 0.037 0.036 e − 5 . 06 − 0 . 124 t 1 0.024 0.024 1.2% High 0 0.16 1 0.10 6% Overall 0 0.076 1 0.049 2.7% Low: 65 year old white female with a SBP of 160 mmHg. High: 80 year old black male with a SBP of 180 mmHg

  61. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary (a.0): 80 year old black male, SBP=180 15 semi−parametric (Cox) (a.1) proposed Cumulative incidence (%) 10 5 (b.0) (b.1): 65 year old white female, SBP=160 0 0 1 2 3 4 5 Prospective time (years)

  62. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 0 1 2 3 4 5 6 7 8 9 10 Points Age 60 65 70 75 80 85 90 95 100 1 Male 0 1 Black 0 SBP 155 165 175 185 195 205 215 0 I 1 t 6 0 I.t 6 5 4 3 2 1 0 Total Points 0 2 4 6 8 10 12 14 16 18 20 22 Linear Predictor −6 −5.5 −5 −4.5 −4 −3.5 −3 −2.5 5−year Risk (%) if not treated 3 4 5 6 7 8 9 12 15 18 5−year Risk (%) if treated 2 3 4 5 6 7 8 9 12 15 18

  63. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 1. FEATURES • Smooth-in- t h ( t ) —and CI’s– not new; fitting procedure is. • Keys: 1. representative sampling of the base; 2. offset. • b / c =100 feasible and adequate.

  64. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 1. FEATURES • Smooth-in- t h ( t ) —and CI’s– not new; fitting procedure is. • Keys: 1. representative sampling of the base; 2. offset. • b / c =100 feasible and adequate.

  65. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 1. FEATURES • Smooth-in- t h ( t ) —and CI’s– not new; fitting procedure is. • Keys: 1. representative sampling of the base; 2. offset. • b / c =100 feasible and adequate.

  66. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 1. FEATURES • Smooth-in- t h ( t ) —and CI’s– not new; fitting procedure is. • Keys: 1. representative sampling of the base; 2. offset. • b / c =100 feasible and adequate.

  67. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 2. MODELLING POSSIBILITIES Log-linear modelling for h x ( t ) via logistic regression ... • Standard methods to assess model fit. • Wide range of functional forms for the t -dimension of h x ( t ) . • Effortless handling of censored data. • Flexibility in modeling non-proportionality over t . • Splines for h ( t ) rather than hr ( t ) .

  68. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 2. MODELLING POSSIBILITIES Log-linear modelling for h x ( t ) via logistic regression ... • Standard methods to assess model fit. • Wide range of functional forms for the t -dimension of h x ( t ) . • Effortless handling of censored data. • Flexibility in modeling non-proportionality over t . • Splines for h ( t ) rather than hr ( t ) .

  69. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 2. MODELLING POSSIBILITIES Log-linear modelling for h x ( t ) via logistic regression ... • Standard methods to assess model fit. • Wide range of functional forms for the t -dimension of h x ( t ) . • Effortless handling of censored data. • Flexibility in modeling non-proportionality over t . • Splines for h ( t ) rather than hr ( t ) .

  70. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 2. MODELLING POSSIBILITIES Log-linear modelling for h x ( t ) via logistic regression ... • Standard methods to assess model fit. • Wide range of functional forms for the t -dimension of h x ( t ) . • Effortless handling of censored data. • Flexibility in modeling non-proportionality over t . • Splines for h ( t ) rather than hr ( t ) .

  71. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 2. MODELLING POSSIBILITIES Log-linear modelling for h x ( t ) via logistic regression ... • Standard methods to assess model fit. • Wide range of functional forms for the t -dimension of h x ( t ) . • Effortless handling of censored data. • Flexibility in modeling non-proportionality over t . • Splines for h ( t ) rather than hr ( t ) .

  72. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 2. MODELLING POSSIBILITIES Log-linear modelling for h x ( t ) via logistic regression ... • Standard methods to assess model fit. • Wide range of functional forms for the t -dimension of h x ( t ) . • Effortless handling of censored data. • Flexibility in modeling non-proportionality over t . • Splines for h ( t ) rather than hr ( t ) .

  73. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 2. MODELLING POSSIBILITIES Log-linear modelling for h x ( t ) via logistic regression ... • Standard methods to assess model fit. • Wide range of functional forms for the t -dimension of h x ( t ) . • Effortless handling of censored data. • Flexibility in modeling non-proportionality over t . • Splines for h ( t ) rather than hr ( t ) .

  74. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 3. CLINICAL POSSIBILITIES / DESIDERATA • PDAs (personal digital assistants) → online information. • Profile-specific risk estimates for various interventions. • Already, online calculators: risk of MI, Breast/Lung Cancer; probability of extra-organ spread of cancer. • RCT reports should contain: suitably designed risk function, fitted parameters of h x ( t ) , and risk function. • (Offline:) risk scores → risks via nomogram/table.

  75. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 3. CLINICAL POSSIBILITIES / DESIDERATA • PDAs (personal digital assistants) → online information. • Profile-specific risk estimates for various interventions. • Already, online calculators: risk of MI, Breast/Lung Cancer; probability of extra-organ spread of cancer. • RCT reports should contain: suitably designed risk function, fitted parameters of h x ( t ) , and risk function. • (Offline:) risk scores → risks via nomogram/table.

  76. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 3. CLINICAL POSSIBILITIES / DESIDERATA • PDAs (personal digital assistants) → online information. • Profile-specific risk estimates for various interventions. • Already, online calculators: risk of MI, Breast/Lung Cancer; probability of extra-organ spread of cancer. • RCT reports should contain: suitably designed risk function, fitted parameters of h x ( t ) , and risk function. • (Offline:) risk scores → risks via nomogram/table.

  77. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 3. CLINICAL POSSIBILITIES / DESIDERATA • PDAs (personal digital assistants) → online information. • Profile-specific risk estimates for various interventions. • Already, online calculators: risk of MI, Breast/Lung Cancer; probability of extra-organ spread of cancer. • RCT reports should contain: suitably designed risk function, fitted parameters of h x ( t ) , and risk function. • (Offline:) risk scores → risks via nomogram/table.

  78. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 3. CLINICAL POSSIBILITIES / DESIDERATA • PDAs (personal digital assistants) → online information. • Profile-specific risk estimates for various interventions. • Already, online calculators: risk of MI, Breast/Lung Cancer; probability of extra-organ spread of cancer. • RCT reports should contain: suitably designed risk function, fitted parameters of h x ( t ) , and risk function. • (Offline:) risk scores → risks via nomogram/table.

  79. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 3. CLINICAL POSSIBILITIES / DESIDERATA • PDAs (personal digital assistants) → online information. • Profile-specific risk estimates for various interventions. • Already, online calculators: risk of MI, Breast/Lung Cancer; probability of extra-organ spread of cancer. • RCT reports should contain: suitably designed risk function, fitted parameters of h x ( t ) , and risk function. • (Offline:) risk scores → risks via nomogram/table.

  80. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 4. SUMMARY • Profile-specific risk (CI) functions are important. • Two paths to CI, via... • Steps-in-time S 0 ( t ) • Smooth-in-time ID x ( t ) . • New simple estimation method for broad class of smooth-in-time ID / hazard functions. • Biostatistics & Epidemiology methods: a little more unified?

  81. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 4. SUMMARY • Profile-specific risk (CI) functions are important. • Two paths to CI, via... • Steps-in-time S 0 ( t ) • Smooth-in-time ID x ( t ) . • New simple estimation method for broad class of smooth-in-time ID / hazard functions. • Biostatistics & Epidemiology methods: a little more unified?

  82. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 4. SUMMARY • Profile-specific risk (CI) functions are important. • Two paths to CI, via... • Steps-in-time S 0 ( t ) • Smooth-in-time ID x ( t ) . • New simple estimation method for broad class of smooth-in-time ID / hazard functions. • Biostatistics & Epidemiology methods: a little more unified?

  83. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 4. SUMMARY • Profile-specific risk (CI) functions are important. • Two paths to CI, via... • Steps-in-time S 0 ( t ) • Smooth-in-time ID x ( t ) . • New simple estimation method for broad class of smooth-in-time ID / hazard functions. • Biostatistics & Epidemiology methods: a little more unified?

  84. Introduction Smooth-in-time hazard functions How we fit fully-parametric hazard model Illustration Comments/Summary 4. SUMMARY • Profile-specific risk (CI) functions are important. • Two paths to CI, via... • Steps-in-time S 0 ( t ) • Smooth-in-time ID x ( t ) . • New simple estimation method for broad class of smooth-in-time ID / hazard functions. • Biostatistics & Epidemiology methods: a little more unified?

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