A multistate additive relative survival semi-markov model

Transcription

1 46 e Journées de Statistique de la SFdS A multistate additive semi-markov Florence Gillaizeau,, & Etienne Dantan & Magali Giral, & Yohann Foucher, florence.gillaizeau@univ-nantes.fr EA SPHERE - Biostatistique, Pharmacoépidémiologie et Mesures Subjectives en Santé, Université de Nantes, Nantes, France. INSERM UMR1064, Institut Transplantation-Urologie-Néphrologie, Nantes, France. Centre Hospitalier Universitaire de Nantes, Nantes, France. 3 June / 16

2 Context multiple time-to-events data (disease progression, death) multistate s association with excess death associated to the disease analysis litterature Belot et al. [2011] : competing risks + (excess mortality related to colon cancer) Huszti et al. [2012] : Markov NH + (excess mortality related to colon cancer in an illness-death ) Gillaizeau et al. [2014] : relative survival (SMRS) Hakulinen and Tenkanen [1987], Esteve et al. [1990], Perme et al. [2012] 2 / 16

3 Representation Figure 1: The SMRS representation for a multistate including the death related to the disease (X = E) and the death not related to the disease (X = P). Arrows for the transitions to X = E and X = P are represented with dashed lines since the two states can not be distinguished individually. X= 3 X= 1 Disease Diagnostic X= E Death related to the disease ( excess death ) X= 2 X= P Death not related to the disease ( expected death ) 3 / 16

4 Notations T : chronological time from baseline S : duration (or sojourn time) in a state X : finite space of the possible clinical states ɛ : set of possible transitions ij with (i,j) (X,X ), with i transient state with j i X m : state of the patient after the m-th transition occurring at time T m, with T 0 < T 1 <... < T m (T 0 = 0 and X 0 = 1) Z : overall vector of patient characteristics Z ij : subvector of characteristics specifically associated to the transition ij 4 / 16

5 Instantaneous hazard function semi-markovian property transition intensities between two states depend on the duration in the current state instantaneous hazard function specific from state X m = i to the state X m+1 = j after a duration s, given patient characteristics Z ij = z ij : P(s T m+1 T m < s + s,x m+1 = j T m+1 T m > s,x m = i,z ij ) λ ij (s z ij ) = lim s 0 + s (1) with Λ ij (s z ij ) = s 0 λ ij(u z ij )du the corresponding cumulative hazard function. 5 / 16

6 Expected mortality X = E : death related to the disease X = P : death related to other causes A : random variable for patient s age at death a i : patient age observed at entry in state i y : patient s birthyear g : patient s gender Instantaneous hazard function for the mortality not related to the disease after a duration s in the state i : P(s +a i A < s +a i + s,x = P A > s +a i,y,g) λ P (s +a i y,g) = lim s 0 + s (2) calculated from life tables (available by calendar year birthdate gender) 6 / 16

7 Observed mortality Instantaneous hazard function : Cumulative hazard : λ io (s z ie,a i,y,g) = λ ie (s z ie ) + λ P (s +a i y,g) (3) Λ io (s z ie,a i,y,g) = Λ ie (s z ie ) + Λ P (s +a i y,g) Λ P (a i y,g) (4) Λ P (s +a i y,g) Λ P (a i y,g) represents the cumulative hazard of death between age a i and a i +s in the general population. 7 / 16

8 Marginal survival function and subdensity Probability for a patient to sejourn at least a duration s in state i : [ ( S i. (s z,a i,y,g) = exp j:ij ɛ j death equations (1) + (3) + (5) ) ] Λ ij (s z ij ) Λ ie (s z ie ) Λ P (s +a i y,g) + Λ P (a i y,g) density function specific to transition ij, after a duration s : ( ) f ij (s z,a i,y,g) = 1 {j death} λ ij (s z ij ) +1 {j=death} λ io (s z ie,a i,y,g) S i. (s z,a i,y,g) (5) (6) 8 / 16

9 Contribution to the likelihood s ij : duration time in state i before transition to state j δ ij = 1 if the transition ij is observed, δ ij = 0 otherwise Patient in an absorbing state at his/her last time of follow-up {f ij (s ij z,a i,y,g)} δ ij ij ɛ Patient censored in the transient state k (for a duration s k ) at his/her last time of follow-up {f ij (s ij z,a i,y,g)} δ ij S k. (s k z,a k,y,g) ij ɛ λ. (.) : parametric PH s with time-fixed covariates estimations : maximization of the likelihood function + Hessian matrix (Nelder and Mead algorithms) 9 / 16

10 Performances of 2 s SMRS 5-state SM (causes of death known) X= 3 X= 1 Disease Diagnostic X= E Death related to the disease ( excess death ) X= 2 X= P Death not related to the disease ( expected death ) 10 / 16

11 Distributions s based on kidney transplant recipients data : year of entry into the U([1998 ;2010]) gender g : men B(0.61) explicative variable z B(0.30) age a at baseline truncated N (from 18 to 80 years old) with parameters varying according to g and z The 5 sojourn time distributions W depending on (a, g, z). Scenarios : 3 sample sizes (N=500, N=1000, N=3000 subjects) 3 censoring rates (15%, 30%, 60%) 11 / 16

12 5-state SM Table 1: Estimations of effects with the 5-state SM (100 simulated samples, N=3000 patients, censoring rate=60%) Coefficient Theoretical Mean Absolute RMSE Empiric Asymptotic Coverage value 1 estimate bias SE SE rate (%) β 12 Male β 12 Age β 12 z β 13 Male β 13 Age β 13 z β 1E Male β 1E Age β 1E z β 23 Male β 23 Age β 23 z β 2E Male β 2E Age β 2E z RMSE : Root Mean Square Error, SE : Standard Error 1 Theoretical values for the baseline hazard functions with Weibull distribution (log(scale), log(shape)) : transition 12 (2.5,1.2), transition 13 (5.0,-0.3), transition 1E (1.1,0.2), transition 23 (2.8,-0.4), transition 2E (1.3,-0.1), right-censoring (0.6,-0.2). Theoretical values for effects on expected death : β P Gender=0.4, β P Age= / 16

13 SMRS Table 2: Estimations of effects with the SMRS (100 simulated samples, N=3000 patients, censoring rate=60%) Coefficient Theoretical Mean Absolute RMSE Empiric Asymptotic Coverage value 1 estimate bias SE SE rate (%) β 12 Male β 12 Age β 12 z β 13 Male β 13 Age β 13 z β 1E Male β 1E Age β 1E z β 23 Male β 23 Age β 23 z β 2E Male β 2E Age β 2E z RMSE : Root Mean Square Error, SE : Standard Error 1 Theoretical values for the baseline hazard functions with Weibull distribution (log(scale), log(shape)) : transition 12 (2.5,1.2), transition 13 (5.0,-0.3), transition 1E (1.1,0.2), transition 23 (2.8,-0.4), transition 2E (1.3,-0.1), right-censoring (0.6,-0.2). Theoretical values for effects on expected death : β P Gender=0.4, β P Age= / 16

14 Conclusion Good performances of the SMRS as good as the SM where the causes of death are known similar results for other simulation scenarios Model needs extensions : time-dependent variables non-proportional hazards interval-censored data Package R : esemimarkov ( Application to data from kidney transplant recipients (DIVAT cohort) 14 / 16

15 Application data (goodness-of-fit) Times post transplantation (years) (a) Transition 12 (b) Transition 13 (c) Transition 1O DGF (Aalen Johansen) DGF (SMRS) No DGF (Aalen Johansen) No DGF (SMRS) Times post transplantation (years) (c) Transition 1O DGF (Aalen Johansen) DGF (SMRS) No DGF (Aalen Johansen) No DGF (SMRS) Times post transplantation (years) (d) Transition DGF (Aalen Johansen) DGF (SMRS) No DGF (Aalen Johansen) No DGF (SMRS) Times post transplantation (years) (e) Transition 2O DGF (Aalen Johansen) DGF (SMRS) No DGF (Aalen Johansen) No DGF (SMRS) DGF (Aalen Johansen) DGF (SMRS) No DGF (Aalen Johansen) No DGF (SMRS) DGF (Aalen Johansen) DGF (SMRS) No DGF (Aalen Johansen) No DGF (SMRS) Times post transplantation (years) Times post acute rejection (years) Times post acute rejection (years) (e) Transition 2O DGF (Aalen Johansen) Figure DGF (SMRS) 2: estimates provided by the SMRS (dashed lines) and No DGF (Aalen Johansen) No DGF (SMRS) the Aalen-Johansen nonparametric estimator (solid lines) on data from kidney transplant recipients for the 5 possible transitions and according to the presence of an explicative variable. 15 / 16

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