From semi- to non-parametric inference in general time scale models

Size: px
Start display at page:

Download "From semi- to non-parametric inference in general time scale models"

Transcription

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

On the Collapsibility of Lifetime Regression Models

On the Collapsibility of Lifetime Regression Models On the Collapsibility of Lifetime Regression Models Thierry Duchesne Département de mathématiques et de statistique Pavillon Alexandre-Vachon Université Laval Québec, QC, G1K 7P4, Canada duchesne@mat.ulaval.ca

More information

STAT 331. Accelerated Failure Time Models. Previously, we have focused on multiplicative intensity models, where

STAT 331. Accelerated Failure Time Models. Previously, we have focused on multiplicative intensity models, where STAT 331 Accelerated Failure Time Models Previously, we have focused on multiplicative intensity models, where h t z) = h 0 t) g z). These can also be expressed as H t z) = H 0 t) g z) or S t z) = e Ht

More information

Other Survival Models. (1) Non-PH models. We briefly discussed the non-proportional hazards (non-ph) model

Other Survival Models. (1) Non-PH models. We briefly discussed the non-proportional hazards (non-ph) model Other Survival Models (1) Non-PH models We briefly discussed the non-proportional hazards (non-ph) model λ(t Z) = λ 0 (t) exp{β(t) Z}, where β(t) can be estimated by: piecewise constants (recall how);

More information

PENALIZED LIKELIHOOD PARAMETER ESTIMATION FOR ADDITIVE HAZARD MODELS WITH INTERVAL CENSORED DATA

PENALIZED LIKELIHOOD PARAMETER ESTIMATION FOR ADDITIVE HAZARD MODELS WITH INTERVAL CENSORED DATA PENALIZED LIKELIHOOD PARAMETER ESTIMATION FOR ADDITIVE HAZARD MODELS WITH INTERVAL CENSORED DATA Kasun Rathnayake ; A/Prof Jun Ma Department of Statistics Faculty of Science and Engineering Macquarie University

More information

Survival Analysis Math 434 Fall 2011

Survival Analysis Math 434 Fall 2011 Survival Analysis Math 434 Fall 2011 Part IV: Chap. 8,9.2,9.3,11: Semiparametric Proportional Hazards Regression Jimin Ding Math Dept. www.math.wustl.edu/ jmding/math434/fall09/index.html Basic Model Setup

More information

log T = β T Z + ɛ Zi Z(u; β) } dn i (ue βzi ) = 0,

log T = β T Z + ɛ Zi Z(u; β) } dn i (ue βzi ) = 0, Accelerated failure time model: log T = β T Z + ɛ β estimation: solve where S n ( β) = n i=1 { Zi Z(u; β) } dn i (ue βzi ) = 0, Z(u; β) = j Z j Y j (ue βz j) j Y j (ue βz j) How do we show the asymptotics

More information

Lecture 5 Models and methods for recurrent event data

Lecture 5 Models and methods for recurrent event data Lecture 5 Models and methods for recurrent event data Recurrent and multiple events are commonly encountered in longitudinal studies. In this chapter we consider ordered recurrent and multiple events.

More information

TMA 4275 Lifetime Analysis June 2004 Solution

TMA 4275 Lifetime Analysis June 2004 Solution TMA 4275 Lifetime Analysis June 2004 Solution Problem 1 a) Observation of the outcome is censored, if the time of the outcome is not known exactly and only the last time when it was observed being intact,

More information

Step-Stress Models and Associated Inference

Step-Stress Models and Associated Inference Department of Mathematics & Statistics Indian Institute of Technology Kanpur August 19, 2014 Outline Accelerated Life Test 1 Accelerated Life Test 2 3 4 5 6 7 Outline Accelerated Life Test 1 Accelerated

More information

University of California, Berkeley

University of California, Berkeley University of California, Berkeley U.C. Berkeley Division of Biostatistics Working Paper Series Year 24 Paper 153 A Note on Empirical Likelihood Inference of Residual Life Regression Ying Qing Chen Yichuan

More information

Inference based on the em algorithm for the competing risk model with masked causes of failure

Inference based on the em algorithm for the competing risk model with masked causes of failure Inference based on the em algorithm for the competing risk model with masked causes of failure By RADU V. CRAIU Department of Statistics, University of Toronto, 100 St. George Street, Toronto Ontario,

More information

Chapter 2 Inference on Mean Residual Life-Overview

Chapter 2 Inference on Mean Residual Life-Overview Chapter 2 Inference on Mean Residual Life-Overview Statistical inference based on the remaining lifetimes would be intuitively more appealing than the popular hazard function defined as the risk of immediate

More information

Quantile Regression for Residual Life and Empirical Likelihood

Quantile Regression for Residual Life and Empirical Likelihood Quantile Regression for Residual Life and Empirical Likelihood Mai Zhou email: mai@ms.uky.edu Department of Statistics, University of Kentucky, Lexington, KY 40506-0027, USA Jong-Hyeon Jeong email: jeong@nsabp.pitt.edu

More information

Frailty Models and Copulas: Similarities and Differences

Frailty Models and Copulas: Similarities and Differences Frailty Models and Copulas: Similarities and Differences KLARA GOETHALS, PAUL JANSSEN & LUC DUCHATEAU Department of Physiology and Biometrics, Ghent University, Belgium; Center for Statistics, Hasselt

More information

Efficient Semiparametric Estimators via Modified Profile Likelihood in Frailty & Accelerated-Failure Models

Efficient Semiparametric Estimators via Modified Profile Likelihood in Frailty & Accelerated-Failure Models NIH Talk, September 03 Efficient Semiparametric Estimators via Modified Profile Likelihood in Frailty & Accelerated-Failure Models Eric Slud, Math Dept, Univ of Maryland Ongoing joint project with Ilia

More information

ST745: Survival Analysis: Cox-PH!

ST745: Survival Analysis: Cox-PH! ST745: Survival Analysis: Cox-PH! Eric B. Laber Department of Statistics, North Carolina State University April 20, 2015 Rien n est plus dangereux qu une idee, quand on n a qu une idee. (Nothing is more

More information

AFT Models and Empirical Likelihood

AFT Models and Empirical Likelihood AFT Models and Empirical Likelihood Mai Zhou Department of Statistics, University of Kentucky Collaborators: Gang Li (UCLA); A. Bathke; M. Kim (Kentucky) Accelerated Failure Time (AFT) models: Y = log(t

More information

Multistate Modeling and Applications

Multistate Modeling and Applications Multistate Modeling and Applications Yang Yang Department of Statistics University of Michigan, Ann Arbor IBM Research Graduate Student Workshop: Statistics for a Smarter Planet Yang Yang (UM, Ann Arbor)

More information

UNIVERSITY OF CALIFORNIA, SAN DIEGO

UNIVERSITY OF CALIFORNIA, SAN DIEGO UNIVERSITY OF CALIFORNIA, SAN DIEGO Estimation of the primary hazard ratio in the presence of a secondary covariate with non-proportional hazards An undergraduate honors thesis submitted to the Department

More information

STAT 6350 Analysis of Lifetime Data. Failure-time Regression Analysis

STAT 6350 Analysis of Lifetime Data. Failure-time Regression Analysis STAT 6350 Analysis of Lifetime Data Failure-time Regression Analysis Explanatory Variables for Failure Times Usually explanatory variables explain/predict why some units fail quickly and some units survive

More information

Tests of independence for censored bivariate failure time data

Tests of independence for censored bivariate failure time data Tests of independence for censored bivariate failure time data Abstract Bivariate failure time data is widely used in survival analysis, for example, in twins study. This article presents a class of χ

More information

Key Words: survival analysis; bathtub hazard; accelerated failure time (AFT) regression; power-law distribution.

Key Words: survival analysis; bathtub hazard; accelerated failure time (AFT) regression; power-law distribution. POWER-LAW ADJUSTED SURVIVAL MODELS William J. Reed Department of Mathematics & Statistics University of Victoria PO Box 3060 STN CSC Victoria, B.C. Canada V8W 3R4 reed@math.uvic.ca Key Words: survival

More information

STAT331. Cox s Proportional Hazards Model

STAT331. Cox s Proportional Hazards Model STAT331 Cox s Proportional Hazards Model In this unit we introduce Cox s proportional hazards (Cox s PH) model, give a heuristic development of the partial likelihood function, and discuss adaptations

More information

Semiparametric Regression

Semiparametric Regression Semiparametric Regression Patrick Breheny October 22 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/23 Introduction Over the past few weeks, we ve introduced a variety of regression models under

More information

MAS3301 / MAS8311 Biostatistics Part II: Survival

MAS3301 / MAS8311 Biostatistics Part II: Survival MAS3301 / MAS8311 Biostatistics Part II: Survival M. Farrow School of Mathematics and Statistics Newcastle University Semester 2, 2009-10 1 13 The Cox proportional hazards model 13.1 Introduction In the

More information

Efficiency of Profile/Partial Likelihood in the Cox Model

Efficiency of Profile/Partial Likelihood in the Cox Model Efficiency of Profile/Partial Likelihood in the Cox Model Yuichi Hirose School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, New Zealand Summary. This paper shows

More information

Part III. Hypothesis Testing. III.1. Log-rank Test for Right-censored Failure Time Data

Part III. Hypothesis Testing. III.1. Log-rank Test for Right-censored Failure Time Data 1 Part III. Hypothesis Testing III.1. Log-rank Test for Right-censored Failure Time Data Consider a survival study consisting of n independent subjects from p different populations with survival functions

More information

Empirical Processes & Survival Analysis. The Functional Delta Method

Empirical Processes & Survival Analysis. The Functional Delta Method STAT/BMI 741 University of Wisconsin-Madison Empirical Processes & Survival Analysis Lecture 3 The Functional Delta Method Lu Mao lmao@biostat.wisc.edu 3-1 Objectives By the end of this lecture, you will

More information

Exercises. (a) Prove that m(t) =

Exercises. (a) Prove that m(t) = Exercises 1. Lack of memory. Verify that the exponential distribution has the lack of memory property, that is, if T is exponentially distributed with parameter λ > then so is T t given that T > t for

More information

Score tests for dependent censoring with survival data

Score tests for dependent censoring with survival data Score tests for dependent censoring with survival data Mériem Saïd, Nadia Ghazzali & Louis-Paul Rivest (meriem@mat.ulaval.ca, ghazzali@mat.ulaval.ca, lpr@mat.ulaval.ca) Département de mathématiques et

More information

Analysis of Time-to-Event Data: Chapter 4 - Parametric regression models

Analysis of Time-to-Event Data: Chapter 4 - Parametric regression models Analysis of Time-to-Event Data: Chapter 4 - Parametric regression models Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/25 Right censored

More information

Statistical Inference and Methods

Statistical Inference and Methods Department of Mathematics Imperial College London d.stephens@imperial.ac.uk http://stats.ma.ic.ac.uk/ das01/ 31st January 2006 Part VI Session 6: Filtering and Time to Event Data Session 6: Filtering and

More information

Accelerated Failure Time Models: A Review

Accelerated Failure Time Models: A Review International Journal of Performability Engineering, Vol. 10, No. 01, 2014, pp.23-29. RAMS Consultants Printed in India Accelerated Failure Time Models: A Review JEAN-FRANÇOIS DUPUY * IRMAR/INSA of Rennes,

More information

Power and Sample Size Calculations with the Additive Hazards Model

Power and Sample Size Calculations with the Additive Hazards Model Journal of Data Science 10(2012), 143-155 Power and Sample Size Calculations with the Additive Hazards Model Ling Chen, Chengjie Xiong, J. Philip Miller and Feng Gao Washington University School of Medicine

More information

11 Survival Analysis and Empirical Likelihood

11 Survival Analysis and Empirical Likelihood 11 Survival Analysis and Empirical Likelihood The first paper of empirical likelihood is actually about confidence intervals with the Kaplan-Meier estimator (Thomas and Grunkmeier 1979), i.e. deals with

More information

Statistical Inference on Constant Stress Accelerated Life Tests Under Generalized Gamma Lifetime Distributions

Statistical Inference on Constant Stress Accelerated Life Tests Under Generalized Gamma Lifetime Distributions Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS040) p.4828 Statistical Inference on Constant Stress Accelerated Life Tests Under Generalized Gamma Lifetime Distributions

More information

Efficient Estimation of Censored Linear Regression Model

Efficient Estimation of Censored Linear Regression Model 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 25 26 27 28 29 3 3 32 33 34 35 36 37 38 39 4 4 42 43 44 45 46 47 48 Biometrika (2), xx, x, pp. 4 C 28 Biometrika Trust Printed in Great Britain Efficient Estimation

More information

Linear life expectancy regression with censored data

Linear life expectancy regression with censored data Linear life expectancy regression with censored data By Y. Q. CHEN Program in Biostatistics, Division of Public Health Sciences, Fred Hutchinson Cancer Research Center, Seattle, Washington 98109, U.S.A.

More information

Machine Learning. Module 3-4: Regression and Survival Analysis Day 2, Asst. Prof. Dr. Santitham Prom-on

Machine Learning. Module 3-4: Regression and Survival Analysis Day 2, Asst. Prof. Dr. Santitham Prom-on Machine Learning Module 3-4: Regression and Survival Analysis Day 2, 9.00 16.00 Asst. Prof. Dr. Santitham Prom-on Department of Computer Engineering, Faculty of Engineering King Mongkut s University of

More information

Improving Efficiency of Inferences in Randomized Clinical Trials Using Auxiliary Covariates

Improving Efficiency of Inferences in Randomized Clinical Trials Using Auxiliary Covariates Improving Efficiency of Inferences in Randomized Clinical Trials Using Auxiliary Covariates Anastasios (Butch) Tsiatis Department of Statistics North Carolina State University http://www.stat.ncsu.edu/

More information

Causal exposure effect on a time-to-event response using an IV.

Causal exposure effect on a time-to-event response using an IV. Faculty of Health Sciences Causal exposure effect on a time-to-event response using an IV. Torben Martinussen 1 Stijn Vansteelandt 2 Eric Tchetgen 3 1 Department of Biostatistics University of Copenhagen

More information

Estimators of the regression parameters of the zeta distribution

Estimators of the regression parameters of the zeta distribution Insurance: Mathematics and Economics 30 2002) 439 450 Estimators of the regression parameters of the zeta distribution Louis G. Doray, Michel Arsenault Département de Mathématiques et de Statistique, Université

More information

Survival Analysis. Stat 526. April 13, 2018

Survival Analysis. Stat 526. April 13, 2018 Survival Analysis Stat 526 April 13, 2018 1 Functions of Survival Time Let T be the survival time for a subject Then P [T < 0] = 0 and T is a continuous random variable The Survival function is defined

More information

8. Parametric models in survival analysis General accelerated failure time models for parametric regression

8. Parametric models in survival analysis General accelerated failure time models for parametric regression 8. Parametric models in survival analysis 8.1. General accelerated failure time models for parametric regression The accelerated failure time model Let T be the time to event and x be a vector of covariates.

More information

Optimal Treatment Regimes for Survival Endpoints from a Classification Perspective. Anastasios (Butch) Tsiatis and Xiaofei Bai

Optimal Treatment Regimes for Survival Endpoints from a Classification Perspective. Anastasios (Butch) Tsiatis and Xiaofei Bai Optimal Treatment Regimes for Survival Endpoints from a Classification Perspective Anastasios (Butch) Tsiatis and Xiaofei Bai Department of Statistics North Carolina State University 1/35 Optimal Treatment

More information

Modeling and Analysis of Recurrent Event Data

Modeling and Analysis of Recurrent Event Data Edsel A. Peña (pena@stat.sc.edu) Department of Statistics University of South Carolina Columbia, SC 29208 New Jersey Institute of Technology Conference May 20, 2012 Historical Perspective: Random Censorship

More information

Competing risks data analysis under the accelerated failure time model with missing cause of failure

Competing risks data analysis under the accelerated failure time model with missing cause of failure Ann Inst Stat Math 2016 68:855 876 DOI 10.1007/s10463-015-0516-y Competing risks data analysis under the accelerated failure time model with missing cause of failure Ming Zheng Renxin Lin Wen Yu Received:

More information

Size and Shape of Confidence Regions from Extended Empirical Likelihood Tests

Size and Shape of Confidence Regions from Extended Empirical Likelihood Tests Biometrika (2014),,, pp. 1 13 C 2014 Biometrika Trust Printed in Great Britain Size and Shape of Confidence Regions from Extended Empirical Likelihood Tests BY M. ZHOU Department of Statistics, University

More information

Introduction to Reliability Theory (part 2)

Introduction to Reliability Theory (part 2) Introduction to Reliability Theory (part 2) Frank Coolen UTOPIAE Training School II, Durham University 3 July 2018 (UTOPIAE) Introduction to Reliability Theory 1 / 21 Outline Statistical issues Software

More information

You know I m not goin diss you on the internet Cause my mama taught me better than that I m a survivor (What?) I m not goin give up (What?

You know I m not goin diss you on the internet Cause my mama taught me better than that I m a survivor (What?) I m not goin give up (What? You know I m not goin diss you on the internet Cause my mama taught me better than that I m a survivor (What?) I m not goin give up (What?) I m not goin stop (What?) I m goin work harder (What?) Sir David

More information

EMPIRICAL LIKELIHOOD ANALYSIS FOR THE HETEROSCEDASTIC ACCELERATED FAILURE TIME MODEL

EMPIRICAL LIKELIHOOD ANALYSIS FOR THE HETEROSCEDASTIC ACCELERATED FAILURE TIME MODEL Statistica Sinica 22 (2012), 295-316 doi:http://dx.doi.org/10.5705/ss.2010.190 EMPIRICAL LIKELIHOOD ANALYSIS FOR THE HETEROSCEDASTIC ACCELERATED FAILURE TIME MODEL Mai Zhou 1, Mi-Ok Kim 2, and Arne C.

More information

Joint Modeling of Longitudinal Item Response Data and Survival

Joint Modeling of Longitudinal Item Response Data and Survival Joint Modeling of Longitudinal Item Response Data and Survival Jean-Paul Fox University of Twente Department of Research Methodology, Measurement and Data Analysis Faculty of Behavioural Sciences Enschede,

More information

Statistics for Engineers Lecture 4 Reliability and Lifetime Distributions

Statistics for Engineers Lecture 4 Reliability and Lifetime Distributions Statistics for Engineers Lecture 4 Reliability and Lifetime Distributions Chong Ma Department of Statistics University of South Carolina chongm@email.sc.edu February 15, 2017 Chong Ma (Statistics, USC)

More information

Chapter 17. Failure-Time Regression Analysis. William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University

Chapter 17. Failure-Time Regression Analysis. William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University Chapter 17 Failure-Time Regression Analysis William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University Copyright 1998-2008 W. Q. Meeker and L. A. Escobar. Based on the authors

More information

Introduction to repairable systems STK4400 Spring 2011

Introduction to repairable systems STK4400 Spring 2011 Introduction to repairable systems STK4400 Spring 2011 Bo Lindqvist http://www.math.ntnu.no/ bo/ bo@math.ntnu.no Bo Lindqvist Introduction to repairable systems Definition of repairable system Ascher and

More information

Issues on quantile autoregression

Issues on quantile autoregression Issues on quantile autoregression Jianqing Fan and Yingying Fan We congratulate Koenker and Xiao on their interesting and important contribution to the quantile autoregression (QAR). The paper provides

More information

Cox s proportional hazards model and Cox s partial likelihood

Cox s proportional hazards model and Cox s partial likelihood Cox s proportional hazards model and Cox s partial likelihood Rasmus Waagepetersen October 12, 2018 1 / 27 Non-parametric vs. parametric Suppose we want to estimate unknown function, e.g. survival function.

More information

A Resampling Method on Pivotal Estimating Functions

A Resampling Method on Pivotal Estimating Functions A Resampling Method on Pivotal Estimating Functions Kun Nie Biostat 277,Winter 2004 March 17, 2004 Outline Introduction A General Resampling Method Examples - Quantile Regression -Rank Regression -Simulation

More information

Some Theories about Backfitting Algorithm for Varying Coefficient Partially Linear Model

Some Theories about Backfitting Algorithm for Varying Coefficient Partially Linear Model Some Theories about Backfitting Algorithm for Varying Coefficient Partially Linear Model 1. Introduction Varying-coefficient partially linear model (Zhang, Lee, and Song, 2002; Xia, Zhang, and Tong, 2004;

More information

STAT 6350 Analysis of Lifetime Data. Probability Plotting

STAT 6350 Analysis of Lifetime Data. Probability Plotting STAT 6350 Analysis of Lifetime Data Probability Plotting Purpose of Probability Plots Probability plots are an important tool for analyzing data and have been particular popular in the analysis of life

More information

Density estimation Nonparametric conditional mean estimation Semiparametric conditional mean estimation. Nonparametrics. Gabriel Montes-Rojas

Density estimation Nonparametric conditional mean estimation Semiparametric conditional mean estimation. Nonparametrics. Gabriel Montes-Rojas 0 0 5 Motivation: Regression discontinuity (Angrist&Pischke) Outcome.5 1 1.5 A. Linear E[Y 0i X i] 0.2.4.6.8 1 X Outcome.5 1 1.5 B. Nonlinear E[Y 0i X i] i 0.2.4.6.8 1 X utcome.5 1 1.5 C. Nonlinearity

More information

MAP Examples. Sargur Srihari

MAP Examples. Sargur Srihari MAP Examples Sargur srihari@cedar.buffalo.edu 1 Potts Model CRF for OCR Topics Image segmentation based on energy minimization 2 Examples of MAP Many interesting examples of MAP inference are instances

More information

Survival Regression Models

Survival Regression Models Survival Regression Models David M. Rocke May 18, 2017 David M. Rocke Survival Regression Models May 18, 2017 1 / 32 Background on the Proportional Hazards Model The exponential distribution has constant

More information

Estimation and Inference of Quantile Regression. for Survival Data under Biased Sampling

Estimation and Inference of Quantile Regression. for Survival Data under Biased Sampling Estimation and Inference of Quantile Regression for Survival Data under Biased Sampling Supplementary Materials: Proofs of the Main Results S1 Verification of the weight function v i (t) for the lengthbiased

More information

ST495: Survival Analysis: Maximum likelihood

ST495: Survival Analysis: Maximum likelihood ST495: Survival Analysis: Maximum likelihood Eric B. Laber Department of Statistics, North Carolina State University February 11, 2014 Everything is deception: seeking the minimum of illusion, keeping

More information

Statistics 262: Intermediate Biostatistics Non-parametric Survival Analysis

Statistics 262: Intermediate Biostatistics Non-parametric Survival Analysis Statistics 262: Intermediate Biostatistics Non-parametric Survival Analysis Jonathan Taylor & Kristin Cobb Statistics 262: Intermediate Biostatistics p.1/?? Overview of today s class Kaplan-Meier Curve

More information

Accelerated Failure Time Models

Accelerated Failure Time Models Accelerated Failure Time Models Patrick Breheny October 12 Patrick Breheny University of Iowa Survival Data Analysis (BIOS 7210) 1 / 29 The AFT model framework Last time, we introduced the Weibull distribution

More information

Survival Distributions, Hazard Functions, Cumulative Hazards

Survival Distributions, Hazard Functions, Cumulative Hazards BIO 244: Unit 1 Survival Distributions, Hazard Functions, Cumulative Hazards 1.1 Definitions: The goals of this unit are to introduce notation, discuss ways of probabilistically describing the distribution

More information

Continuous case Discrete case General case. Hazard functions. Patrick Breheny. August 27. Patrick Breheny Survival Data Analysis (BIOS 7210) 1/21

Continuous case Discrete case General case. Hazard functions. Patrick Breheny. August 27. Patrick Breheny Survival Data Analysis (BIOS 7210) 1/21 Hazard functions Patrick Breheny August 27 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/21 Introduction Continuous case Let T be a nonnegative random variable representing the time to an event

More information

Approximation of Survival Function by Taylor Series for General Partly Interval Censored Data

Approximation of Survival Function by Taylor Series for General Partly Interval Censored Data Malaysian Journal of Mathematical Sciences 11(3): 33 315 (217) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal Approximation of Survival Function by Taylor

More information

ST745: Survival Analysis: Nonparametric methods

ST745: Survival Analysis: Nonparametric methods ST745: Survival Analysis: Nonparametric methods Eric B. Laber Department of Statistics, North Carolina State University February 5, 2015 The KM estimator is used ubiquitously in medical studies to estimate

More information

Investigation of goodness-of-fit test statistic distributions by random censored samples

Investigation of goodness-of-fit test statistic distributions by random censored samples d samples Investigation of goodness-of-fit test statistic distributions by random censored samples Novosibirsk State Technical University November 22, 2010 d samples Outline 1 Nonparametric goodness-of-fit

More information

e 4β e 4β + e β ˆβ =0.765

e 4β e 4β + e β ˆβ =0.765 SIMPLE EXAMPLE COX-REGRESSION i Y i x i δ i 1 5 12 0 2 10 10 1 3 40 3 0 4 80 5 0 5 120 3 1 6 400 4 1 7 600 1 0 Model: z(t x) =z 0 (t) exp{βx} Partial likelihood: L(β) = e 10β e 10β + e 3β + e 5β + e 3β

More information

Survival Analysis for Case-Cohort Studies

Survival Analysis for Case-Cohort Studies Survival Analysis for ase-ohort Studies Petr Klášterecký Dept. of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, harles University, Prague, zech Republic e-mail: petr.klasterecky@matfyz.cz

More information

A Sampling of IMPACT Research:

A Sampling of IMPACT Research: A Sampling of IMPACT Research: Methods for Analysis with Dropout and Identifying Optimal Treatment Regimes Marie Davidian Department of Statistics North Carolina State University http://www.stat.ncsu.edu/

More information

Goodness-of-Fit Tests With Right-Censored Data by Edsel A. Pe~na Department of Statistics University of South Carolina Colloquium Talk August 31, 2 Research supported by an NIH Grant 1 1. Practical Problem

More information

Accounting for extreme-value dependence in multivariate data

Accounting for extreme-value dependence in multivariate data Accounting for extreme-value dependence in multivariate data 38th ASTIN Colloquium Manchester, July 15, 2008 Outline 1. Dependence modeling through copulas 2. Rank-based inference 3. Extreme-value dependence

More information

3003 Cure. F. P. Treasure

3003 Cure. F. P. Treasure 3003 Cure F. P. reasure November 8, 2000 Peter reasure / November 8, 2000/ Cure / 3003 1 Cure A Simple Cure Model he Concept of Cure A cure model is a survival model where a fraction of the population

More information

Multivariate Survival Data With Censoring.

Multivariate Survival Data With Censoring. 1 Multivariate Survival Data With Censoring. Shulamith Gross and Catherine Huber-Carol Baruch College of the City University of New York, Dept of Statistics and CIS, Box 11-220, 1 Baruch way, 10010 NY.

More information

REGRESSION ANALYSIS FOR TIME-TO-EVENT DATA THE PROPORTIONAL HAZARDS (COX) MODEL ST520

REGRESSION ANALYSIS FOR TIME-TO-EVENT DATA THE PROPORTIONAL HAZARDS (COX) MODEL ST520 REGRESSION ANALYSIS FOR TIME-TO-EVENT DATA THE PROPORTIONAL HAZARDS (COX) MODEL ST520 Department of Statistics North Carolina State University Presented by: Butch Tsiatis, Department of Statistics, NCSU

More information

ST5212: Survival Analysis

ST5212: Survival Analysis ST51: Survival Analysis 8/9: Semester II Tutorial 1. A model for lifetimes, with a bathtub-shaped hazard rate, is the exponential power distribution with survival fumction S(x) =exp{1 exp[(λx) α ]}. (a)

More information

MAS3301 / MAS8311 Biostatistics Part II: Survival

MAS3301 / MAS8311 Biostatistics Part II: Survival MAS330 / MAS83 Biostatistics Part II: Survival M. Farrow School of Mathematics and Statistics Newcastle University Semester 2, 2009-0 8 Parametric models 8. Introduction In the last few sections (the KM

More information

Nonparametric Model Construction

Nonparametric Model Construction Nonparametric Model Construction Chapters 4 and 12 Stat 477 - Loss Models Chapters 4 and 12 (Stat 477) Nonparametric Model Construction Brian Hartman - BYU 1 / 28 Types of data Types of data For non-life

More information

Modelling geoadditive survival data

Modelling geoadditive survival data Modelling geoadditive survival data Thomas Kneib & Ludwig Fahrmeir Department of Statistics, Ludwig-Maximilians-University Munich 1. Leukemia survival data 2. Structured hazard regression 3. Mixed model

More information

Problem Set 3: Bootstrap, Quantile Regression and MCMC Methods. MIT , Fall Due: Wednesday, 07 November 2007, 5:00 PM

Problem Set 3: Bootstrap, Quantile Regression and MCMC Methods. MIT , Fall Due: Wednesday, 07 November 2007, 5:00 PM Problem Set 3: Bootstrap, Quantile Regression and MCMC Methods MIT 14.385, Fall 2007 Due: Wednesday, 07 November 2007, 5:00 PM 1 Applied Problems Instructions: The page indications given below give you

More information

On nonlinear weighted least squares fitting of the three-parameter inverse Weibull distribution

On nonlinear weighted least squares fitting of the three-parameter inverse Weibull distribution MATHEMATICAL COMMUNICATIONS 13 Math. Commun., Vol. 15, No. 1, pp. 13-24 (2010) On nonlinear weighted least squares fitting of the three-parameter inverse Weibull distribution Dragan Juić 1 and Darija Marović

More information

Analysis of Gamma and Weibull Lifetime Data under a General Censoring Scheme and in the presence of Covariates

Analysis of Gamma and Weibull Lifetime Data under a General Censoring Scheme and in the presence of Covariates Communications in Statistics - Theory and Methods ISSN: 0361-0926 (Print) 1532-415X (Online) Journal homepage: http://www.tandfonline.com/loi/lsta20 Analysis of Gamma and Weibull Lifetime Data under a

More information

Empirical Likelihood in Survival Analysis

Empirical Likelihood in Survival Analysis Empirical Likelihood in Survival Analysis Gang Li 1, Runze Li 2, and Mai Zhou 3 1 Department of Biostatistics, University of California, Los Angeles, CA 90095 vli@ucla.edu 2 Department of Statistics, The

More information

Goodness-of-fit test for the Cox Proportional Hazard Model

Goodness-of-fit test for the Cox Proportional Hazard Model Goodness-of-fit test for the Cox Proportional Hazard Model Rui Cui rcui@eco.uc3m.es Department of Economics, UC3M Abstract In this paper, we develop new goodness-of-fit tests for the Cox proportional hazard

More information

Asymptotics for posterior hazards

Asymptotics for posterior hazards Asymptotics for posterior hazards Pierpaolo De Blasi University of Turin 10th August 2007, BNR Workshop, Isaac Newton Intitute, Cambridge, UK Joint work with Giovanni Peccati (Université Paris VI) and

More information

FULL LIKELIHOOD INFERENCES IN THE COX MODEL

FULL LIKELIHOOD INFERENCES IN THE COX MODEL October 20, 2007 FULL LIKELIHOOD INFERENCES IN THE COX MODEL BY JIAN-JIAN REN 1 AND MAI ZHOU 2 University of Central Florida and University of Kentucky Abstract We use the empirical likelihood approach

More information

Goodness-Of-Fit for Cox s Regression Model. Extensions of Cox s Regression Model. Survival Analysis Fall 2004, Copenhagen

Goodness-Of-Fit for Cox s Regression Model. Extensions of Cox s Regression Model. Survival Analysis Fall 2004, Copenhagen Outline Cox s proportional hazards model. Goodness-of-fit tools More flexible models R-package timereg Forthcoming book, Martinussen and Scheike. 2/38 University of Copenhagen http://www.biostat.ku.dk

More information

Product-limit estimators of the survival function with left or right censored data

Product-limit estimators of the survival function with left or right censored data Product-limit estimators of the survival function with left or right censored data 1 CREST-ENSAI Campus de Ker-Lann Rue Blaise Pascal - BP 37203 35172 Bruz cedex, France (e-mail: patilea@ensai.fr) 2 Institut

More information

Statistical Analysis of Competing Risks With Missing Causes of Failure

Statistical Analysis of Competing Risks With Missing Causes of Failure Proceedings 59th ISI World Statistics Congress, 25-3 August 213, Hong Kong (Session STS9) p.1223 Statistical Analysis of Competing Risks With Missing Causes of Failure Isha Dewan 1,3 and Uttara V. Naik-Nimbalkar

More information

Does Cox analysis of a randomized survival study yield a causal treatment effect?

Does Cox analysis of a randomized survival study yield a causal treatment effect? Published in final edited form as: Lifetime Data Analysis (2015), 21(4): 579 593 DOI: 10.1007/s10985-015-9335-y Does Cox analysis of a randomized survival study yield a causal treatment effect? Odd O.

More information

Survival Analysis. Lu Tian and Richard Olshen Stanford University

Survival Analysis. Lu Tian and Richard Olshen Stanford University 1 Survival Analysis Lu Tian and Richard Olshen Stanford University 2 Survival Time/ Failure Time/Event Time We will introduce various statistical methods for analyzing survival outcomes What is the survival

More information

Cox s proportional hazards/regression model - model assessment

Cox s proportional hazards/regression model - model assessment Cox s proportional hazards/regression model - model assessment Rasmus Waagepetersen September 27, 2017 Topics: Plots based on estimated cumulative hazards Cox-Snell residuals: overall check of fit Martingale

More information

ANALYSIS OF COMPETING RISKS DATA WITH MISSING CAUSE OF FAILURE UNDER ADDITIVE HAZARDS MODEL

ANALYSIS OF COMPETING RISKS DATA WITH MISSING CAUSE OF FAILURE UNDER ADDITIVE HAZARDS MODEL Statistica Sinica 18(28, 219-234 ANALYSIS OF COMPETING RISKS DATA WITH MISSING CAUSE OF FAILURE UNDER ADDITIVE HAZARDS MODEL Wenbin Lu and Yu Liang North Carolina State University and SAS Institute Inc.

More information

Discussion of the paper Inference for Semiparametric Models: Some Questions and an Answer by Bickel and Kwon

Discussion of the paper Inference for Semiparametric Models: Some Questions and an Answer by Bickel and Kwon Discussion of the paper Inference for Semiparametric Models: Some Questions and an Answer by Bickel and Kwon Jianqing Fan Department of Statistics Chinese University of Hong Kong AND Department of Statistics

More information

Analysis of Time-to-Event Data: Chapter 6 - Regression diagnostics

Analysis of Time-to-Event Data: Chapter 6 - Regression diagnostics Analysis of Time-to-Event Data: Chapter 6 - Regression diagnostics Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/25 Residuals for the

More information