From semi- to non-parametric inference in general time scale models
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1 From semi- to non-parametric inference in general time scale models Thierry DUCHESNE Département de mathématiques et de statistique Université Laval Québec, Québec, Canada Research supported by the Natural Sciences and Engineering Research Council of Canada and the Fonds Québécois de la recherche sur la nature et les technologies Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p1
2 Outline 1 Introduction Definitions and notation Ideal time scale 2 Simple models: Collapsible models Definition and interpretation Examples of applications 3 Inference and model checking 4 Concluding remarks Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p2
3 1 Introduction Notation: T : time to death/failure rv y(t): monotone (non-decreasing) covariate (eg, accumulated exposure or usage) up to t, y(0) = 0 θ(t) = y (t): exposure or usage rate at time t θ t = {θ(u), 0 u t}: covariate history up to t Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p3
4 1 Introduction Notation: T : time to death/failure rv y(t): monotone (non-decreasing) covariate (eg, accumulated exposure or usage) up to t, y(0) = 0 θ(t) = y (t): exposure or usage rate at time t θ t = {θ(u), 0 u t}: covariate history up to t Example: For a car, T is its age at failure, θ(t) is its speed at time t, y(t) is its cumulative mileage at time t Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p3
5 1 Introduction mileage y(t) theta(t) t T time Fig 1: Accumulated mileage, y(t), as a function of time, t Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p4
6 1 Introduction Consider regression models for lifetime given covariate history: P[T > t θ t ] = P[T > t {θ(u), 0 u t }] Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p5
7 1 Introduction Consider regression models for lifetime given covariate history: P[T > t θ t ] = P[T > t {θ(u), 0 u t }] Possible purposes: 1 effect of covariate on lifetime (eg, interpretation, hypothesis tests); 2 prediction (eg, warranty or maintenance decisions); 3 joint modeling of (T, Y ), where Y = y(t) Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p5
8 1 Introduction Usually, the effect of covariates on the probability of death is modeled in one of two ways: model the effect on the hazard of failure at time t, for example λ(t θ t ) = lim t 0 P[t T < t + t θ t ]/ t = ψ(λ 0 (t),θ t ); Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p6
9 1 Introduction Usually, the effect of covariates on the probability of death is modeled in one of two ways: model the effect on the hazard of failure at time t, for example λ(t θ t ) = lim t 0 P[t T < t + t θ t ]/ t = ψ(λ 0 (t),θ t ); model the effect by changing the value of time from t to t, for example P[T > t θ t ] = G[t ] = G[φ(t,θ t )], where G[ ] is a (specified) survivor function Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p6
10 1 Introduction Classical regression models (with extensions now available ): Proportional hazards model (Cox, JRSS B 1972): λ(t θ t ) = λ 0 (t)ψ(θ t ;β) eg = λ 0 (t) exp{βy(t)} Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p7
11 1 Introduction Classical regression models (with extensions now available ): Proportional hazards model (Cox, JRSS B 1972): λ(t θ t ) = λ 0 (t)ψ(θ t ;β) eg = λ 0 (t) exp{βy(t)} Accelerated failure time model (eg, Robins and Tsiatis, Biometrika, 1992): [ t ] [ eg t P[T > t θ t ]=G ψ(θ u ;β)du = G 0 0 ] exp{βy(u)}du Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p7
12 1 Introduction Ideal time scale: An ideal time scale (ITS) is any one-to-one transformation of the conditional survivor function of interest that is 0 at t = 0 and is non-decreasing in t More specifically, it is (any increasing transformation of) the φ(, ) functional in P[T > t θ t ] = G[φ(t,θ t )] Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p8
13 1 Introduction Ideal time scale: An ideal time scale (ITS) is any one-to-one transformation of the conditional survivor function of interest that is 0 at t = 0 and is non-decreasing in t More specifically, it is (any increasing transformation of) the φ(, ) functional in P[T > t θ t ] = G[φ(t,θ t )] Also known as/related to intrinsic time scale, load invariant scale, virtual age, generalized residual, Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p8
14 1 Introduction Ideal time scale: An ideal time scale (ITS) is any one-to-one transformation of the conditional survivor function of interest that is 0 at t = 0 and is non-decreasing in t More specifically, it is (any increasing transformation of) the φ(, ) functional in P[T > t θ t ] = G[φ(t,θ t )] Also known as/related to intrinsic time scale, load invariant scale, virtual age, generalized residual, If t θ (x) = φ(x, θ x ) is age in ITS, then equal age in ITS means equal quantile of lifetime distribution Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p8
15 1 Introduction Examples of ideal time scales Cox model: t θ (x) = x 0 λ 0(u) exp{βy(u)} du Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p9
16 1 Introduction Examples of ideal time scales Cox model: t θ (x) = x 0 λ 0(u) exp{βy(u)} du Aalen s additive hazards model: t θ (x) = x 0 {β 0(u) + β 1 (u)y(u)} du, which simplifies to B 0 (x) + B 1 (x)y(x) θ x 0 B 1(u) du when θ(u) = θ u Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p9
17 1 Introduction Examples of ideal time scales Cox model: t θ (x) = x 0 λ 0(u) exp{βy(u)} du Aalen s additive hazards model: t θ (x) = x 0 {β 0(u) + β 1 (u)y(u)} du, which simplifies to B 0 (x) + B 1 (x)y(x) θ x 0 B 1(u) du when θ(u) = θ u AFT model: t θ (x) = x exp{βy(u)} du, which simplifies 0 to [exp{βy(x)} 1]/(βθ) when θ(u) = θ u Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p9
18 1 Introduction Examples of ideal time scales Cox model: t θ (x) = x 0 λ 0(u) exp{βy(u)} du Aalen s additive hazards model: t θ (x) = x 0 {β 0(u) + β 1 (u)y(u)} du, which simplifies to B 0 (x) + B 1 (x)y(x) θ x 0 B 1(u) du when θ(u) = θ u AFT model: t θ (x) = x exp{βy(u)} du, which simplifies 0 to [exp{βy(x)} 1]/(βθ) when θ(u) = θ u Linear transformation model (Cheng, Wei, Ying, Biometrika 1995): H(T) = βy(t) + ε P[T > x θ x ] = G{H(x) βy(x)} t θ (x) = H(x) βy(x) Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p9
19 2 Collapsible models Term collapsibility introduced by Oakes (LIDA 1995) Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p10
20 2 Collapsible models Term collapsibility introduced by Oakes (LIDA 1995) Definition: A model is collapsible when P[T > t θ t ] = G[φ{t, y(t)}] Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p10
21 2 Collapsible models Term collapsibility introduced by Oakes (LIDA 1995) Definition: A model is collapsible when P[T > t θ t ] = G[φ{t, y(t)}] In a collapsible model, the ITS only depends on t and the value y(t) = t 0 θ(u)du, not on the whole path θ t = {θ(u), 0 u t} Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p10
22 2 Collapsible models Let C α = {(x,y) : G[φ(x,y)] = 1 α φ(x,y) = t α } be the α th age curve corresponding to ITS φ Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p11
23 2 Collapsible models Let C α = {(x,y) : G[φ(x,y)] = 1 α φ(x,y) = t α } be the α th age curve corresponding to ITS φ y(x) usage path 1 usage path 2 t=1 t=2 t=3 t=4 usage path 3 usage path 4 Fig 2: Age curves, P[T > x θ x ] = G[(1 β)x + βy(x)] x Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p11
24 2 Collapsible models Why use these models? Nice interpretability Eg, y(t) = number of cigarettes smoked by age t (in minutes) Model P[T > t θ t ] = G[t + 11y(t)] means each cigarette smoked makes you 11 minutes older! Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p12
25 2 Collapsible models Why use these models? Nice interpretability Eg, y(t) = number of cigarettes smoked by age t (in minutes) Model P[T > t θ t ] = G[t + 11y(t)] means each cigarette smoked makes you 11 minutes older! Used with success to model the life of miners exposed to asbestos dust (Oakes, LIDA 1995), aircraft reliability (K & G, LIDA 1997) and the fatigue life of steel specimens (K & G, LIDA 1997 and D & L, LIDA 2000) Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p12
26 2 Collapsible models They also lead to simplified prediction problems: Conditioned based maintenance with decision in real time : no need to store the whole usage history (only y(t) if decision based on P[T > t] or y(t) and θ(t) if decision based on λ(t θ t ) ) Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p13
27 2 Collapsible models They also lead to simplified prediction problems: Conditioned based maintenance with decision in real time : no need to store the whole usage history (only y(t) if decision based on P[T > t] or y(t) and θ(t) if decision based on λ(t θ t ) ) Taylor (2005): Y (t ij ) = Z i (t ij ) + ε ij, Z i (t) = Xi β + a i + b i t and λ(t) = λ 0 (t)e γzi(t), but need P[T i > t i + s T i > t i,x i, Ω i ] Simpler if P[T i > x X i, Ω i ] = G[φ{x,Z i (x)}]? Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p13
28 3 Inference and diagnostics Data: n independent observations (t i,δ i,θ i t i ), i = 1,,n Consider inference about η and β in P[T i > t θ i t ] = G[φ{t,y i (t);β};η] Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p14
29 3 Inference and diagnostics Data: n independent observations (t i,δ i,θ i t i ), i = 1,,n Consider inference about η and β in P[T i > t θ i t ] = G[φ{t,y i (t);β};η] Parametric inference (dim(η) <, dim(β) < ): Maximum likelihood (Oakes, LIDA 1995; D & L, LIDA 2000): L(η,β) = n G[φ{t,y i (t);β};η] i=1 ( [ ]) δi d h G (t i ) φ{t i,y i (t i )}, dt i where h G (t) = d/dt ln G(t) Careful, Corr(ˆη, ˆβ) possibly high! Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p14
30 3 Inference and diagnostics Semiparametric inference (dim(η) =, dim(β) < ): K & G (LIDA 1997) only consider the case φ{t,y(t);β} = t + βy(t) and suggest to use ˆβ = arg min β ĈV 2 [φ{t i,y i (t i );β}], where ĈV 2 is the square of the sample coefficient of variation Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p15
31 3 Inference and diagnostics Semiparametric inference (dim(η) =, dim(β) < ): K & G (LIDA 1997) only consider the case φ{t,y(t);β} = t + βy(t) and suggest to use ˆβ = arg min β ĈV 2 [φ{t i,y i (t i );β}], where ĈV 2 is the square of the sample coefficient of variation Pro s: Closed form expression, consistent, intuitively sensible Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p15
32 3 Inference and diagnostics Semiparametric inference (dim(η) =, dim(β) < ): K & G (LIDA 1997) only consider the case φ{t,y(t);β} = t + βy(t) and suggest to use ˆβ = arg min β ĈV 2 [φ{t i,y i (t i );β}], where ĈV 2 is the square of the sample coefficient of variation Pro s: Closed form expression, consistent, intuitively sensible Con s: Difficulties arise with censored data, not invariant to non linear transformations, loses efficiency if G not Weibull Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p15
33 3 Inference and diagnostics Semiparametric inference (dim(η) =, dim(β) < ): D & L (LIDA 2002) propose an adaptation of R & T (Biometrika 1992) and L & Y (JSPI 1995) Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p16
34 3 Inference and diagnostics Semiparametric inference (dim(η) =, dim(β) < ): D & L (LIDA 2002) propose an adaptation of R & T (Biometrika 1992) and L & Y (JSPI 1995) Define Ñi(t;β) = I[φ{t i,y i (t i );β} t,δ i = 1], Ỹ i (t;β) = I[φ{t i,y i (t i );β} t] and dˆλ(t;β) = n j=1 Ỹj(t;β)dÑj(t;β)/ n l=1 Ỹl(t;β) Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p16
35 3 Inference and diagnostics Semiparametric inference (dim(η) =, dim(β) < ): D & L (LIDA 2002) propose an adaptation of R & T (Biometrika 1992) and L & Y (JSPI 1995) Define Ñi(t;β) = I[φ{t i,y i (t i );β} t,δ i = 1], Ỹ i (t;β) = I[φ{t i,y i (t i );β} t] and dˆλ(t;β) = n j=1 Ỹj(t;β)dÑj(t;β)/ n l=1 Ỹl(t;β) Consider the estimating function U(β) = n i=1 τ 0 Ỹ i (t;β)w(t,θ i,β){dñi(t;β) dˆλ(t;β)} Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p16
36 3 Inference and diagnostics Semiparametric inference (dim(η) =, dim(β) < ): Optimal weight function: W opt (t,θ,β ) = / β ln t θ {t 1 θ (t,β );β} Simplifies β=β to {θ(t) 1}/{1 β + β θ(t)} in linear scale case Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p17
37 3 Inference and diagnostics Semiparametric inference (dim(η) =, dim(β) < ): Optimal weight function: W opt (t,θ,β ) = / β ln t θ {t 1 θ (t,β );β} Simplifies β=β to {θ(t) 1}/{1 β + β θ(t)} in linear scale case Let ˆβ = arg min β U(β) U(β) Then n(ˆβ β) is asymptotically normal with mean 0 and variance given by D & L (2002) Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p17
38 3 Inference and diagnostics Semiparametric inference (dim(η) =, dim(β) < ): Optimal weight function: W opt (t,θ,β ) = / β ln t θ {t 1 θ (t,β );β} Simplifies β=β to {θ(t) 1}/{1 β + β θ(t)} in linear scale case Let ˆβ = arg min β U(β) U(β) Then n(ˆβ β) is asymptotically normal with mean 0 and variance given by D & L (2002) In simulations, this method did much better than minimun CV Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p17
39 3 Inference and diagnostics Semiparametric inference (dim(η) =, dim(β) < ): K & G (LIDA 1997) consider the fatigue life of steel specimens, with y(t) = θt How do scales (1 β)t + βy(t) and traditional AFT tθ η t β y(t) 1 β compare? Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p18
40 3 Inference and diagnostics Semiparametric inference (dim(η) =, dim(β) < ): K & G (LIDA 1997) consider the fatigue life of steel specimens, with y(t) = θt How do scales (1 β)t + βy(t) and traditional AFT tθ η t β y(t) 1 β compare? t(eta) Linear scale path t(eta) Multiplicative scale path Failure times in estimated linear scale, (1 ˆβ)t+ ˆβy(t) (LHS), and estimated multiplicative scale, tˆβy(t) 1 ˆβ (RHS) vs θ Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p18
41 3 Inference and diagnostics Nonparametric inference (dim(η) =, dim(β) = ): Can we estimate G and φ in P[T > t θ] = G[φ{t,y(t)}] completely nonparametrically? Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p19
42 3 Inference and diagnostics Nonparametric inference (dim(η) =, dim(β) = ): Can we estimate G and φ in P[T > t θ] = G[φ{t,y(t)}] completely nonparametrically? Obviously, G and φ are only identifiable up to monotone transformations Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p19
43 3 Inference and diagnostics Nonparametric inference (dim(η) =, dim(β) = ): Can we estimate G and φ in P[T > t θ] = G[φ{t,y(t)}] completely nonparametrically? Obviously, G and φ are only identifiable up to monotone transformations But we can estimate the age curves C α = {(x,y) : G[φ{x,y}] = α} Once the age curves are known, we fix G to whatever we want and we can identify φ Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p19
44 3 Inference and diagnostics Nonparametric inference (dim(η) =, dim(β) = ): Why estimate age curves nonparametrically? Model specification: different scales φ{x, y} produce age curves of different shapes Eg, the linear scale (1 β)x + βy has age curves that are parallel straight lines with slope (1 β)/β, while multiplicative scale x β y 1 β has age curves given by y = k α x β/(1 β) Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p20
45 3 Inference and diagnostics Nonparametric inference (dim(η) =, dim(β) = ): Why estimate age curves nonparametrically? Model specification: different scales φ{x, y} produce age curves of different shapes Eg, the linear scale (1 β)x + βy has age curves that are parallel straight lines with slope (1 β)/β, while multiplicative scale x β y 1 β has age curves given by y = k α x β/(1 β) Model checking: overlay model-based age curves with nonparametric age curves on same graph Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p20
46 3 Inference and diagnostics Nonparametric inference (dim(η) =, dim(β) = ): When y i (t) = θ i t, Duchesne (2000) proposes an ad hoc method: 1 Partition values of θ in a few groups; 2 Estimate α th quantile of T in each group; 3 Plot the pairs (t i,y i (t i ) = θ i t i ), and draw the lines separating the groups of step 1 on the plot, as well as the mean value of θ in each group; 4 Along each of the mean θ, find the corresponding α th quantile of T Draw a segment between each of these points Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p21
47 3 Inference and diagnostics Nonparametric inference (dim(η) =, dim(β) = ): x y(x) T~Weibull(5,9), T=X+(Theta)X atan(theta)~u(0,pi/2), sample size=400 Estimated quintiles True quintiles Region delimiters Mean paths x y(x) ~Weibull(5,9), T=X^2+[(Theta)X atan(theta)~u(0,pi/2), sample size=400 Estimated quintiles True quintiles Region delimiters Mean paths Nearest neighbor estimation of age curves from data simulated from models with scales t + y(t) (LHS) and t 2 + y(t) 2 (RHS) Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p22
48 3 Inference and diagnostics Nonparametric inference (dim(η) =, dim(β) = ): When y i (t) = f(t;θ i ), then this is completely equivalent to quantile regression With censored data, much on semiparametric quantile regression, but not much on smooth nonparametric quantile regression Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p23
49 3 Inference and diagnostics Nonparametric inference (dim(η) =, dim(β) = ): When y i (t) = f(t;θ i ), then this is completely equivalent to quantile regression With censored data, much on semiparametric quantile regression, but not much on smooth nonparametric quantile regression Let ξ α (θ) = arg min a E[ρ α (T a) θ], with ρ α (u) = u{α I(u < 0)} Then ξ α (θ) is the α th quantile of the conditional distribution of T θ Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p23
50 3 Inference and diagnostics Nonparametric inference (dim(η) =, dim(β) = ): To estimate ξ α (θ) from censored data, A & D (2005) propose to modify the method of Fan et al (SJS 1994): ˆξ α (θ) = â, where (â,ˆb) = arg min a,b ρ α {T i a b(θ i θ)}k h {(θ i θ)/h} i by replacing K h {(θ i θ)/h} with K h {(θ i θ)/h}w i,n Eg, S & W (Ann Stat 1993) propose W i,n = δ [i:n] /(n i + 1) i 1 {(n j)/(n j + 1)} δ [j:n] j=1 Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p24
51 4 Concluding remarks Age curve estimation when y i (t) f(t;θ i ) much more challenging Can estimated age curves be used for more general model checking? Can formal goodness-of-fit tests be based on age in ITS? Can collapsible model simplify some prediction problems? Test of the collapsibility assumption? Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p25
52 References Thank you for your attention! Abdous & Duchesne (2005) Work in progress Cheng et al (1995) Biometrika 82, Cox (1972) JRSS B 34, Duchesne (2000) Recent advances in reliability theory, Duchesne & Lawless (2000) Lifetime Data Analysis 6, Duchesne & Lawless (2002) Lifetime Data Analysis 8, Kordonsky & Gertsbakh (1997) Lifetime Data Analysis 2, Lin & Ying (1995) J Statist Planning & Infe 44, Oakes (1995) Lifetime Data Analysis 1, 7 18 Robins & Tsiatis (1992) Biometrika 79, Stute & Wang (1993) Ann Stat 21, Taylor (2005) Talk given at Workshop on Incomplete Longitudinal Data, Fields Institute Thierry Duchesne Slides available at Workshop on Survival Analysis, Montréal, November p26
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