Estimating survival from Gray’s Outline flexible model I. Introduction II. Semi–parametric survival models III. Gray’s model introduction Zdenek Valenta IV. Survival estimates based on semi-parametric models V. Estimating survival from Gray’s model (with examples) VI. Impact of misspecifying the survival model – simulation study results Department of Medical Informatics Institute of Computer Science VII. Discussion Academy of Sciences of the Czech Republic • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit I. Introduction I. Introduction • Let F ( Y ) denote a cumulative distribution function of the ran- dom variable Y , i.e. F ( Y ) = P ( Y ≤ y ) . We assume that Y • Let Y be a random variable capturing time to occurrence of a is absolutely continuous with density f ( y ) . The expression (1) certain event of interest. The hazard function h ( y ) is at time may be then written as: y formally defined as follows: P ( y ≤ Y < y + ∆ y | Y ≥ y ) P ( y ≤ Y < y + ∆ y | Y ≥ y ) h ( y ) = lim h ( y ) = lim (1) , ∆ y ∆ y → 0 ∆ y ∆ y → 0 P ( y ≤ Y < y + ∆ y ) = lim where P ( . ) denotes conditional probability, that an event of P ( Y ≥ y )∆ y ∆ y → 0 (2) interest would occur immediately after time y , given it did not 1 f ( y ) d = dyF ( y ) = prior to this time. P ( Y ≥ y ) P ( Y ≥ y ) = f ( y ) S ( y ) , • It follows from (1) that the hazard function may only take non– negative values. where S ( y ) denotes the value of the survival function at time y . • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
I. Introduction II. Semi–parametric survival models • We define a cumulative hazard function H ( t ) at time t as: • Multiplicative models: � t – Cox PH model: H ( t ) = h ( y ) dy (3) 0 h ( y | Z ) = h 0 ( y ) · exp ( β ′ Z ) (6) It follows from (2) that S ( . ) and H ( . ) capture equivalent infor- – Gray’s flexible model: mation: h ( y | Z ) = h 0 ( y ) · exp { β ( y ) ′ Z } H ( t ) = − ln ( S ( t )) (4) (7) • Furthermore, it follows from (4) that we can determine the • Additive models: value of the survival function S ( t ) at time t whenever we are – Aalen’s linear model: able to evaluate the cumulative hazard function H ( t ) : h ( y | Z ) = h 0 ( y ) + β ( y ) ′ Z (8) S ( t ) = exp {− H ( t ) } (5) • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit III. Gray’s model introduction II. Semi–parametric survival models Note: Aalen’s linear model (8) may be embedded in the class of • Let us recall the definition of Gray’s flexible model : (7): multiplicative models : h ( y | Z ) = h 0 ( y ) · exp { β ( y ) ′ Z } • Aalen’s model : • Gray’s model uses penalized B-splines for modelling time- h ( y | Z ) = h 0 ( y ) + β ( y ) ′ Z varying effects β ( y ) . B-splines allow for flexible modelling of the covariate effects β ( y ) and the hazard function over time. h 0 ( y ) + β ( y ) ′ Z � � exp ( h ( y | Z )) = exp (9) • In the context of Gray’s model using piecewise-constant time-varying regression coefficients the β ( y ) remain con- β ( y ) ′ Z h 1 ( y | Z ) = h 1 � � 0 ( y ) · exp stant for y ∈ [ τ j − 1 , τ j ) . We can thus write β ( y ) = β j = ( β j 1 , β j 2 , . . . , β jp ) , where p denotes the number of model co- The class of multiplicative models represented by the Cox PH and variates and j = 1 , . . . , M + 1 indexes the intervals on time Gray’s flexible model includes the whole class of models pro- axis. Here τ j denote the knots that allow for a change of the posed by Aalen . regression coefficients β j . • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Piecewise-constant vs. quadratic Piecewise-constant vs. cubic penalized splines penalized splines Intervention Age Diabetes Mellitus Intervention Age Diabetes Mellitus 2 2 3 3 0.2 1 1 0.2 Log Hazard Ratio 0 Log Hazard Ratio Log Hazard Ratio Log Hazard Ratio Log Hazard Ratio Log Hazard Ratio 0 2 2 0.1 0.1 −1 −1 1 1 −2 −2 0.0 0.0 0 0 −3 −3 −0.1 −0.1 −4 −1 −4 −1 500 1000 2000 500 1000 2000 500 1000 2000 500 1000 2000 500 1000 2000 500 1000 2000 Days of Follow−up Days of Follow−up Days of Follow−up Days of Follow−up Days of Follow−up Days of Follow−up Intervention Age Diabetes Mellitus Intervention Age Diabetes Mellitus 8 0.4 0.4 2 4 5 6 0.3 3 0.2 Log Hazard Ratio Log Hazard Ratio Log Hazard Ratio Log Hazard Ratio 0 Log Hazard Ratio Log Hazard Ratio 0.2 4 2 0 0.0 −2 0.1 2 1 −0.2 0.0 −5 0 −4 0 −0.4 −0.1 −1 −2 −6 −0.2 500 1000 2000 500 1000 2000 500 1000 2000 500 1000 2000 500 1000 2000 500 1000 2000 Days of Follow−up Days of Follow−up Days of Follow−up Days of Follow−up Days of Follow−up Days of Follow−up • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit IV. Survival estimates based on semi-parametric models IV. Survival estimates based on semi-parametric models • Aalen’s linear model: h ( y | Z ) = h 0 ( y ) + β ( y ) ′ Z • Cox PH model: (11) � t β ( y ) ′ ∼ ∼ � � h ( y | Z ) = Z, h 0 ( y ) · exp ( β ′ Z ) dy S ( t | Z ) = exp − 0 ∼ ∼ while β ( y ) = ( h 0 ( y ) , β ( y )) a Z = (1 , Z ) . = exp {− H 0 ( t ) · exp ( β ′ Z ) } = [ S 0 ( t )] exp( β ′ Z ) , (10) • Survival function estimates based on Aalen’s model use cumu- � t ∼ ∼ ∼ where S 0 ( t ) represents baseline survival function estimate lative regression coefficients B ( t ) , where B i ( t ) = β i ( y ) dy . 0 at time t . Estimating survival based on Aalen’s model may then proceed as follows: � � B ( t ) ′ ∼ ∼ S ( t | Z ) = exp − Z (12) • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
V. Estimating survival from Gray’s model V. Estimating survival from Gray’s model • Survival function estimate based on Gray’s model a us- ing piecewise–constant penalized splines may be obtained as follows: • Derivation of confidence limits for the survival function esti- mate based on Gray’s model uses the Delta method . � � M +1 � β ′ � � S ( t | Z ) = exp − H 0 j ( t ) · exp j Z , (13) • Recall the Taylor formulae for a function f ( X ) of a random j =1 variable X with expectation µ : where Z denotes p -dimensional vector of patient’s characteris- n tics, and f ( k ) ( µ ) ( X − µ ) k + R n � f ( X ) = (15) � k ! H 0 j ( t ) = I ( u ≤ t ) dH 0 ( u ) (14) k =0 [ τ j − 1 ,τ j ) • Delta method : represents a contribution to the cumulative baseline Var ( f ( X )) ≈ Var [ f ( µ ) + f ′ ( µ )( X − µ )] hazard function H 0 ( t ) on the interval [ τ j − 1 , τ j ) , j = (16) 2 · Var ( X ) 1 , . . . , M + 1 . = [ f ′ ( µ )] a Valenta Z et al, Statistics in Medicine 2002. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit “coxspline” package in R V. Estimating survival from Gray’s model • If X is a random vector the Delta method G ( X ) takes the form: Var ( G ( X )) ≈ ▽ G ′ ( µ ) · Var ( X ) · ▽ G ( µ ) , (17) where ▽ G ( µ ) is a column vector of first partial derivatives of G . • Confidence limits estimates a were derived for a log– and log(- log)–transformed survival function S ( t ) and are reported simultaneously in R. a Valenta Z et al, Statistics in Medicine 2002. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
“cox.spline” R-routine “cox.spline” R-routine (cont.) • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit R-function “gsurv.R” R-function “gsurv.R” (cont.) • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
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