Incorporation of nested frailties into multi-state models with an application to event-history analysis for multicenter clinical trials

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1 Incorporation of nested frailties into multi-state models with an application to event-history analysis for multicenter clinical trials Federico Rotolo? Catherine Legrand Dipartmento di Scienze Statistiche Institut de Statistique, Biostatistique et Sciences Actuarielles Universita degli Studi di Padova Universite Catholique de Louvain September 14, th PhD day

2 Survival data Survival data Time since an origin event until an event of interest. Ex.: from birth to death, since beginning of therapy until remission, etc. Time T= Incorporation of nested frailties into multi-state models 1/ 21

3 Survival data Survival data Time since an origin event until an event of interest. Ex.: from birth to death, since beginning of therapy until remission, etc. Time T= Right censoring: some values cannot be observed, the only available information being a lower bound. Ex.: migration, change of therapy, loss to follow-up, etc. Time x T> Incorporation of nested frailties into multi-state models 1/ 21

4 Survival data Observable data We have T the event time C the censoring time Incorporation of nested frailties into multi-state models 2/ 21

5 Survival data Observable data We have T the event time C the censoring time We can observe Y = min(t, C) the ev/cens time δ = 1(T C) the censoring indicator Incorporation of nested frailties into multi-state models 2/ 21

6 Survival data Observable data We have T the event time C the censoring time We can observe Y = min(t, C) the ev/cens time δ = 1(T C) the censoring indicator Independent and non-informative type II censoring is assumed. Incorporation of nested frailties into multi-state models 2/ 21

7 Survival data Modeling survival data Because of this peculiarity, instead of modeling the density f (t) of T, the hazard is often considered P[t T < t + t T t] λ(t) = lim = f (t) t 0 t S(t), with S(t) = 1 F (t) = P[T > t] the survival function. Incorporation of nested frailties into multi-state models 3/ 21

8 Survival data Modeling survival data Because of this peculiarity, instead of modeling the density f (t) of T, the hazard is often considered P[t T < t + t T t] λ(t) = lim = f (t) t 0 t S(t), with S(t) = 1 F (t) = P[T > t] the survival function. The most popular regression model for the hazard is arguably the Proportional Hazards (PH) Model by Cox (1972) λ(t; X = x) = λ 0 (t) exp{β x}. Incorporation of nested frailties into multi-state models 3/ 21

9 Motivating problem Motivating problem Multicenter cancer clinical trials Seven EORTC 1 clinical trials to investigate the use of prophylactic treatment following transurethral resection. 1 : European Organisation for Research and Treatment of Cancer (Brussels) - [ Incorporation of nested frailties into multi-state models 4/ 21

10 Motivating problem Motivating problem Research question: Does post-tur treatment improve prognosis of bladder cancer patients? Incorporation of nested frailties into multi-state models 5/ 21

11 Motivating problem Motivating problem Research question: Does post-tur treatment improve prognosis of bladder cancer patients? Clustering Patients, grouped by hospitals, are not independent, neither conditional on the observed covariates between-subjects dependence frailty models Incorporation of nested frailties into multi-state models 5/ 21

12 Motivating problem Motivating problem Research question: Does post-tur treatment improve prognosis of bladder cancer patients? Clustering Patients, grouped by hospitals, are not independent, neither conditional on the observed covariates between-subjects dependence frailty models Rand Multiple endpoints Rec Prog within-subjects dependence multi-state models De Incorporation of nested frailties into multi-state models 5/ 21

13 Motivating problem Motivating problem Research question: Does post-tur treatment improve prognosis of bladder cancer patients? Clustering Patients, grouped by hospitals, are not independent, neither conditional on the observed covariates between-subjects dependence Rand Multiple endpoints Rec Prog within-subjects dependence De frailty models multi-state models Possible integration? Incorporation of nested frailties into multi-state models 5/ 21

14 Background Survival data Motivating problem Background Frailty models Multi-state models Frailty multi-state models Simulation study Case study Conclusion References Outline Incorporation of nested frailties into multi-state models 6/ 21

15 Background Frailty models (Duchateu and Janssen, 2008; Wienke, 2010) Conditional hazard for subject i in cluster h: λ hi (t u h ) = λ 0 (t)u h exp{β x hi }, for h = 1,..., H and i = 1,..., n h. λ 0 (t) is the baseline hazard x hi are the covariates of subject i in cluster h u h > 0 is the frailty term, shared by all subjects in cluster h, unobservable realisation of a random effect U Incorporation of nested frailties into multi-state models 7/ 21

16 Background Frailty models Dependence λ hi (t u h ) = λ 0 (t)u h exp{β x hi } U h iid fu (u; θ) Marginal independence across clusters: T hi T kj h k Incorporation of nested frailties into multi-state models 8/ 21

17 Background Frailty models Dependence λ hi (t u h ) = λ 0 (t)u h exp{β x hi } U h iid fu (u; θ) Marginal independence across clusters: T hi T kj h k Conditional independence within clusters: T hi T hj U h Incorporation of nested frailties into multi-state models 8/ 21

18 Background Multi-state models (Andersen and Keiding, 2002; Putter, Fiocco, and Geskus, 2007) Transition-specific hazard for transitions of type q of subject i λ qi (t) = λ q0 (t) exp{βq x qi }, for q = 1,..., Q and i = 1,..., n. Rec λ q0(t) is the baseline hazard for transitions of type q x qi are the covariates of subject i, transitions of type q β q are the transition-specific regression coefficients Rand Prog De Incorporation of nested frailties into multi-state models 9/ 21

19 Motivating problem Motivating problem Research question: Does post-tur treatment improve prognosis of bladder cancer patients? Clustering Patients, grouped by hospitals, are not independent, neither conditional on the observed covariates between-subjects dependence Rand Multiple endpoints Rec Prog within-subjects dependence De frailty models multi-state models Possible integration? Incorporation of nested frailties into multi-state models 10/ 21

20 Frailty multi-state models Survival data Motivating problem Background Frailty models Multi-state models Frailty multi-state models Simulation study Case study Conclusion References Outline Incorporation of nested frailties into multi-state models 11/ 21

21 Frailty multi-state models Frailty multi-state models Conditional transition-specific hazard for transitions of type q of subject i in cluster h: λ qhi (t v h, w qh ) = λ q0 (t)v h w qh exp{β q x qhi }, for q = 1,..., Q and i = 1,..., n. λ q0(t) is the baseline hazard for transitions of type q x qhi are the covariates of subject i in cluster h, transitions of type q β q are the transition-specific regression coefficients Incorporation of nested frailties into multi-state models 12/ 21

22 Frailty multi-state models Frailty multi-state models Conditional transition-specific hazard for transitions of type q of subject i in cluster h: λ qhi (t v h, w qh ) = λ q0 (t)v h w qh exp{β q x qhi }, for q = 1,..., Q and i = 1,..., n. λ q0(t) is the baseline hazard for transitions of type q x qhi are the covariates of subject i in cluster h, transitions of type q β q are the transition-specific regression coefficients v h is the frailty term shared by all transitions of all subjects in cluster h w qh is the frailty term shared by all transitions of type q of all subjects in cluster h Incorporation of nested frailties into multi-state models 12/ 21

23 Frailty multi-state models Frailty multi-state models Dependence λ qhi (t v h, w qh ) = λ q0 (t)v h w qh exp{β q x qhi }, with v h and w qh realisations of V h iid fv (v; θ V ), h = 1,..., H, W qh iid fw (w; θ q ), q = 1,..., Q, h = 1,..., H, V h W qh, (h, q). Incorporation of nested frailties into multi-state models 13/ 21

24 Frailty multi-state models Frailty multi-state models Dependence λ qhi (t v h, w qh ) = λ q0 (t)v h w qh exp{β q x qhi }, with v h and w qh realisations of V h iid fv (v; θ V ), h = 1,..., H, W qh iid fw (w; θ q ), q = 1,..., Q, h = 1,..., H, V h W qh, (h, q). { Let U qh = V h W qh, then Cor(U qh, U q h ) = 0 h h, Cor(U qh, U q h ) [0, 1] h = h. Incorporation of nested frailties into multi-state models 13/ 21

25 Frailty multi-state models Frailty multi-state models Dependence λ qhi (t v h, w qh ) = λ q0 (t)v h w qh exp{β q x qhi }, Let U qh = V h W qh, then { Cor(U qh, U q h ) = 0 h h, Cor(U qh, U q h ) [0, 1] h = h. Marginal independence across clusters T qhi T rkj h k Conditional independence within clusters T qhi T rhj V h, W qh, W rh Incorporation of nested frailties into multi-state models 13/ 21

26 Frailty multi-state models Frailty multi-state models Likelihood λ qhi (t v h, w qh ) = λ q0 (t)v h w qh exp{β q x qhi } The conditional likelihood is L C ( λ0 ( ), β; v, w ) = H n h Q h=1 i=1 q=1 { {v h w qh λ q0 (y qhi )e β q x qhi } δqhi [ } exp { v h w qh Λ q0 (y qhi ) Λ q0 (τ qhi ) ]e β q x qhi } I(τqhi < ) with λ 0( ) = (λ 10( ),..., λ Q0 ( )). Incorporation of nested frailties into multi-state models 14/ 21

27 Frailty multi-state models Frailty multi-state models Semiparametric approach The conditional loglikelihood l C ( λ0 ( ), β; v, w ) is profiled w.r.t. the baseline (conditional) partial loglikelihood { [ l P (β;v,w)= H h=1 d h log v h + Q q=1 log w qh + { } nh i=1 (δ )]} qhi βq x qhi log kj Rq(hi) v kw kj exp βq x qkj, with R q(hi) the risk set for transition q at t qhi. Incorporation of nested frailties into multi-state models 15/ 21

28 Frailty multi-state models Frailty multi-state models Semiparametric approach The conditional loglikelihood l C ( λ0 ( ), β; v, w ) is profiled w.r.t. the baseline (conditional) partial loglikelihood { [ l P (β;v,w)= H h=1 d h log v h + Q q=1 log w qh + { } nh i=1 (δ )]} qhi βq x qhi log kj Rq(hi) v kw kj exp βq x qkj. The full or penalised partial loglikelihood l PP (β, θ) = l P (β; v, w) + log f V,W (v, w; θ) can be maximised via the EM-PL method proposed by Horny (2009) for multilevel frailty models. Incorporation of nested frailties into multi-state models 15/ 21

29 Frailty multi-state models Frailty multi-state models Semiparametric approach l PP (β, θ) = l P (β; v, w) + log f V,W (v, w; θ) EMPPL estimation algorithm Expectation ṽ h = max vh l PP ( ˆβ, θ) with offsets log w w qh = max wqh l PP ( ˆβ, θ) with offsets log ṽ Maximisation ˆβ = max β l P (β; ṽ, w) ˆθ = max θ log f V,W (ṽ, w; θ) Incorporation of nested frailties into multi-state models 16/ 21

30 Frailty multi-state models Simulation study Multi-state structure LR NED De DM Study settings Duration : 12 6 months Recruitment period: 12 4 months Incorporation of nested frailties into multi-state models 17/ 21

31 Frailty multi-state models Simulation study Multi-state structure LR Treatment effect transitions β Treat NED LR log(0.5) NED DM log(0.9) {NED, LR, DM} De 0 NED De Frailties V h iid Gam(θ V = 0.25) W qh iid Gam(θ q = 0.75), q = 1,... 5 DM Study settings Duration : 12 6 months Recruitment period: 12 4 months Hospitals number size Scenario #1: Scenario #2: Scenario #3: Incorporation of nested frailties into multi-state models 17/ 21

32 Frailty multi-state models Simulation study Multi-state structure LR Treatment effect transitions β Treat NED LR log(0.5) NED DM log(0.9) {NED, LR, DM} De 0 NED De Frailties V h iid Gam(θ V = 0.25) W qh iid Gam(θ q = 0.75), q = 1,... 5 DM Study settings Duration : 12 6 months Recruitment period: 12 4 months Hospitals number size Scenario #1: Scenario #2: Scenario #3: Replications: 200 datasets Incorporation of nested frailties into multi-state models 17/ 21

33 Frailty multi-state models Simulation study Scenario #1 Scenario #2 Scenario #3 βtreat1 True values I No frailty I Shared frailty I Nested frailties βtreat1 True values I No frailty I Shared frailty I Nested frailties βtreat1 True values I No frailty I Shared frailty I Nested frailties βtreat2 βtreat2 βtreat2 βtreat3 βtreat3 βtreat3 βtreat4 βtreat4 βtreat4 βtreat5 βtreat5 βtreat hospitals of size hospitals of size hospitals of size 125 Incorporation of nested frailties into multi-state models 18/ 21

34 Frailty multi-state models Scenario #3 lty lties β Treat1 I I I True values No frailty Shared frailty Nested frailties β Treat2 β Treat3 β Treat4 β Treat hospitals of size 125 Incorporation of nested frailties into multi-state models 18/ 21

35 Frailty multi-state models Frailty multi-state models Bladder cancer dataset 2523 patients from 63 European centers with stage Ta-T1 cancer 1116 Rec (54.2%) received further intravesical treatment, 1155 (45.8%) did not Rand De 829 Tot: 2523 Prog 91 Incorporation of nested frailties into multi-state models 19/ 21

36 Frailty multi-state models Frailty multi-state models from Rand to Rec from Rand to Prog from Rand to De MS [ ] MS [ ] MS [ ] ST [ ] ST [ ] ST [ ] SF1 [ ] SF1 [ ] SF1 [ ] SF2 [ ] SF2 [ ] SF2 [ ] NF [ ] NF [ ] NF [ ] HR HR HR from Rec to Prog from Rec to De from Prog to De MS [ ] MS [ ] MS [ ] ST [ ] ST [ ] ST [ ] SF1 [ ] SF1 [ ] SF1 [ ] SF2 [ ] SF2 [ ] SF2 [ ] NF [ ] NF [ ] NF [ ] HR HR HR Incorporation of nested frailties into multi-state models 20/ 21

37 Conclusion Conclusion Possible integration? Yes! Incorporation of nested frailties into multi-state models 21/ 21

38 Conclusion Conclusion Possible integration? Yes! in many ways... Cor(U qh, U q'h ) 1 0 shared frailties positively dependent frailties independent frailties λ qhi (t v h ) = λ q0 (t)v h exp{β q x qhi } λ qhi (t v h, w qh ) = λ q0 (t)v h w qh exp{β q x qhi } λ qhi (t w qh ) = λ q0 (t) w qh exp{β q x qhi } Incorporation of nested frailties into multi-state models 21/ 21

39 Conclusion Conclusion Possible integration? Yes! in many ways... Cor(U qh, U q'h ) 1 0 shared frailties positively dependent frailties independent frailties λ qhi (t v h ) = λ q0 (t)v h exp{β q x qhi } λ qhi (t v h, w qh ) = λ q0 (t)v h w qh exp{β q x qhi } λ qhi (t w qh ) = λ q0 (t) w qh exp{β q x qhi }! joint modelling of different transitions in the event history! evidence of more consistent estimators, accounting for both within clusters and within transitions dependence! distinct measures of within clusters and within transitions dependence Incorporation of nested frailties into multi-state models 21/ 21

40 PhD Student at University of Padova and Visiting PhD Student at Univeriste catholique de Louvain C. Legrand Professor of Biostatistics at Univeriste catholique de Louvain in collaboration with prof. M. Chiogna, UniPd

41 References References I Ana P. Amorim, Jacobo de Uña-Alvarez, and Luís Meira-Machado. Presmoothing the transition probabilities in the illness death model. Statistics and Probability Letters, 81: , doi: /j.spl P. K. Andersen and N. Keiding. Multi-state models for event history analysis. Statistical Methods In Medical Research, 11:91 115, doi: / SM276ra. Ralf Bender, Thomas Augustin, and Maria Blettner. Generating survival times to simulate cox proportional hazards models. Statistics in Medicine, 24(11): , doi: /sim D. R. Cox. Regression models and life-tables. Journal of the Royal Statistical Society. Series B (Methodological), 34(2): , Jacobo de Uña-Álvarez and Luis F. Meira-Machado. A simple estimator of the bivariate distribution function for censored gap times. Statistics & Probability Letters, 78(15): , doi: /j.spl Luc Duchateu and Paul Janssen. The Frailty Model. Springer New York, New York, NY, ISBN doi: / Incorporation of nested frailties into multi-state models 23/ 21

42 References References II Filippo Grillo Ruggieri, M. P. Pace, F. Bunkeila, F. Cartei, B. M. Panizza, L. Fabbietti, G. Moroni, S. Cammelli, P. Api, C. Giorgetti, and E. Barbieri. Subcutaneous amifostine in head and neck cancer radiotherapy. I supplementi di Tumori, 4(1), G. Horny. Inference in mixed proportional hazard models with k random effects. Statistical Papers, 50(3): , doi: /s y. H. Putter, M. Fiocco, and R. B. Geskus. Tutorial in biostatistics: competing risks and multi-state models. Statistics in Medicine, 26(11): , doi: /sim Federico Rotolo, Catherine Legrand, and Ingrid Van Keilegom. A simulation procedure based on copulas to generate clustered multi-state survival data. Computer Methods and Programs in Biomedicine, in press. doi: /j.cmpb I. Van Keilegom, Jacobo de Uña-Alvarez, and Luís Meira-Machado. Nonparametric location-scale models for censored successive survival times. Journal of Statistical Planning and Inference, 141(3): , doi: /j.jspi A. Wienke. Frailty Models in Survival Analysis. Chapman & Hall/CRC biostatistics series. Taylor and Francis, doi: / Incorporation of nested frailties into multi-state models 24/ 21

43 Selection effect Selection effect Shared frailty model. Conditional hazard of time variable T hi U h for patient i in hospital h: λ hi (t) = λ 0 (t)u h exp{β x hi }. Selection effect: the higher the frailty term, the higher the risk and the earlier the event will occur (on average) ignoring clustering yields underestimation of the risk at late times! Incorporation of nested frailties into multi-state models 25/ 21

44 Selection effect Selection effect Shared frailty model. Conditional hazard of time variable T hi U h for patient i in hospital h: λ hi (t) = λ 0 (t)u h exp{β x hi }. λ(t) hazard U=1 marginal hazard t Incorporation of nested frailties into multi-state models 25/ 21

45 Frailty distribution Frailty distribution λ hi (t u h ) = λ 0 (t)u h exp{β x hi } U h iid fu (u; θ) Incorporation of nested frailties into multi-state models 26/ 21

46 Frailty distribution Frailty distribution λ hi (t u h ) = λ 0 (t)u h exp{β x hi } U h iid fu (u; θ) Many choices are available for the frailty distribution gamma: very very very popular due to its analytical tractability positive stable: analytical integration is possible inverse Gaussian: analytical integration is possible lognormal: the most consistent with the GLMM theory power-variance-function: it extends gamma, inverse Gaussian and positive stable, but it is very difficult to use Incorporation of nested frailties into multi-state models 26/ 21

47 Frailty distribution Frailty distribution λ hi (t u h ) = λ 0 (t)u h exp{β x hi } U h iid fu (u; θ) Distribution Laplace Transform L(s) gamma (1 + θs) 1/θ positive stable exp ( s θ) inverse Gaussian exp [{ 1 (1 + 2θs) 1/2}/ θ ] { } ν compound Poisson and P[U = 0] = exp [ { θ(ν 1) ( ν power variance function exp θs ) }] 1 ν θ(1 ν) ν lognormal ( 1 quadratic hazard 1 + 2σ 2 s exp (1 ) σ2 )s 1 + 2σ 2 s log Student t Lévy exp{ σφ(s)} Incorporation of nested frailties into multi-state models 26/ 21

48 Frailty distribution Frailty distribution λ hi (t u h ) = λ 0 (t)u h exp{β x hi } U h iid fu (u; θ) PvF* µ = 1 IG* ν = 1/2 ν 1 Gam* µ = 1 ν 1 cp* µ = 1 ν fixed θ = νµ (1/ν) 1 /(1 ν) 1/ν µ PS* ν (0, 1) LN* µ = 0 θ > 0 Incorporation of nested frailties into multi-state models 26/ 21

49 Simulations Cox s PH models: established practice (Bender, Augustin, and Blettner, 2005) Incorporation of nested frailties into multi-state models 27/ 21

50 Simulations Cox s PH models: established practice (Bender, Augustin, and Blettner, 2005) frailty models: 1. frailties from f U ( ) 2. survival data from λ(t u) (e. g.,?, Sec. 5) Incorporation of nested frailties into multi-state models 27/ 21

51 Simulations Cox s PH models: established practice (Bender, Augustin, and Blettner, 2005) frailty models: 1. frailties from f U ( ) 2. survival data from λ(t u) (e. g.,?, Sec. 5) multi-state models: most of the papers show only real examples simulations are done only for simple and very particular models (e. g., de Uña-Álvarez and Meira-Machado, 2008; Amorim et al., 2011; Van Keilegom et al., 2011) no general method exists Incorporation of nested frailties into multi-state models 27/ 21

52 Simulations Cox s PH models: established practice (Bender, Augustin, and Blettner, 2005) frailty models: 1. frailties from f U ( ) 2. survival data from λ(t u) (e. g.,?, Sec. 5) multi-state models: most of the papers show only real examples simulations are done only for simple and very particular models (e. g., de Uña-Álvarez and Meira-Machado, 2008; Amorim et al., 2011; Van Keilegom et al., 2011) no general method exist sed Incorporation of nested frailties into multi-state models 27/ 21

53 Rotolo, Legrand, and Van Keilegom (in press) A simulation method should be able to generate LR the dependence of times of competing events NED De DM Incorporation of nested frailties into multi-state models 28/ 21

54 Rotolo, Legrand, and Van Keilegom (in press) A simulation method should be able to generate LR the dependence of times of competing events the dependence of times of subsequent events NED De DM Incorporation of nested frailties into multi-state models 28/ 21

55 Rotolo, Legrand, and Van Keilegom (in press) A simulation method should be able to generate LR LR LR LR the dependence of times of competing events NED De NED LR DM DM NED De De NED De NED LR DM DM NED De De the dependence of times of subsequent events NED LR DM DM De NED LR DM De NED LR DM DM De NED LR DM De the dependence between clustered observations LR LR LR LR NED De NED De NED De NED De DM LR DM DM LR DM NED De NED De LR DM LR LR DM LR NED De NED De NED De NED De DM DM DM DM Incorporation of nested frailties into multi-state models 28/ 21

56 Rotolo, Legrand, and Van Keilegom (in press) A simulation method should be able to generate LR the dependence of times of competing events the dependence of times of subsequent events NED x De the dependence between clustered observations the censoring due to competing events occurrence x DM Incorporation of nested frailties into multi-state models 28/ 21

57 Rotolo, Legrand, and Van Keilegom (in press) A simulation method should be able to generate NED x LR x De the dependence of times of competing events the dependence of times of subsequent events the dependence between clustered observations the censoring due to competing events occurrence x the censoring due to end of the study or loss to follow up DM Incorporation of nested frailties into multi-state models 28/ 21

58 Rotolo, Legrand, and Van Keilegom (in press) A simulation method should be able to generate T 1 LR T 4 the dependence of times of competing events the dependence of times of subsequent events NED T 3 De the dependence between clustered observations the censoring due to competing events occurrence T 2 T 5 the censoring due to end of the study or loss to follow up DM the event-specific covariates effect Incorporation of nested frailties into multi-state models 28/ 21

59 Rotolo, Legrand, and Van Keilegom (in press) LR T 1 Marginal survival functions freely chosen S 1(t), S 2(t) and S 3(t) NED T 3 De T 2 DM Incorporation of nested frailties into multi-state models 29/ 21

60 Rotolo, Legrand, and Van Keilegom (in press) LR T 1 Marginal survival functions freely chosen S 1(t), S 2(t) and S 3(t) Joint survival function by Clayton Copula NED T 3 De S 123(t) = ( 3 i=1 S i(t i ) θ 2 ) 1/θ, θ > 0 T 2 DM Incorporation of nested frailties into multi-state models 29/ 21

61 Rotolo, Legrand, and Van Keilegom (in press) LR T 1 Marginal survival functions freely chosen S 1(t), S 2(t) and S 3(t) Joint survival function by Clayton Copula NED T 3 De S 123(t) = ( 3 i=1 S i(t i ) θ 2 ) 1/θ, θ > 0 Conditional survivals from the joint [ ) ] θ 1/θ 1 S 2 1 (t 2 t 1) = 1 + S1(t 1) θ ( S1 (t 1 ) S 2 (t 2 ) T 2 DM Incorporation of nested frailties into multi-state models 29/ 21

62 Rotolo, Legrand, and Van Keilegom (in press) NED T 1 T 2 LR T 3 De Marginal survival functions freely chosen S 1(t), S 2(t) and S 3(t) Joint survival function by Clayton Copula S 123(t) = ( 3 i=1 S i(t i ) θ 2 ) 1/θ, θ > 0 Conditional survivals from the joint [ ( ) ] θ 1/θ 1 S 2 1 (t 2 t 1) = 1 + S1 (t 1 ) S 2 (t 2 ) S1(t 1) θ ( S 3 12 (t 3 t 1, t 2) = 1 + S 3 (t 3 ) θ 1 S 1 (t 1 ) θ +S 2 (t 2 ) θ 1 ) 1/θ 2 DM Incorporation of nested frailties into multi-state models 29/ 21

63 Rotolo, Legrand, and Van Keilegom (in press) NED T 1 LR De Marginal survival functions freely chosen S 1(t), S 2(t) and S 3(t) Joint survival function by Clayton Copula S 123(t) = ( 3 i=1 S i(t i ) θ 2 ) 1/θ, θ > 0 Conditional survivals from the joint [ ( ) ] θ 1/θ 1 S 2 1 (t 2 t 1) = 1 + S1 (t 1 ) S 2 (t 2 ) S1(t 1) θ ( S 3 12 (t 3 t 1, t 2) = 1 + S 3 (t 3 ) θ 1 S 1 (t 1 ) θ +S 2 (t 2 ) θ 1 ) 1/θ 2 DM Incorporation of nested frailties into multi-state models 29/ 21

64 Rotolo, Legrand, and Van Keilegom (in press) LR T 1 Marginal survival functions freely chosen S 1(t), S 2(t) and S 3(t) Joint survival function by Clayton Copula NED T 2 T 3 De S 123(t) = ( 3 i=1 S i(t i ) θ 2 ) 1/θ, θ > 0 Conditional survivals from the joint [ ( ) ] θ 1/θ 1 S 2 1 (t 2 t 1) = 1 + S1 (t 1 ) S 2 (t 2 ) S1(t 1) θ ( S 3 12 (t 3 t 1, t 2) = 1 + S 3 (t 3 ) θ 1 S 1 (t 1 ) θ +S 2 (t 2 ) θ 1 ) 1/θ 2 DM Incorporation of nested frailties into multi-state models 29/ 21

65 Rotolo, Legrand, and Van Keilegom (in press) NED T 1 T 2 LR T 3 De Marginal survival functions freely chosen S 1(t), S 2(t) and S 3(t) Joint survival function by Clayton Copula S 123(t) = ( 3 i=1 S i(t i ) θ 2 ) 1/θ, θ > 0 Conditional survivals from the joint [ ( ) ] θ 1/θ 1 S 2 1 (t 2 t 1) = 1 + S1 (t 1 ) S 2 (t 2 ) S1(t 1) θ ( S 3 12 (t 3 t 1, t 2) = 1 + S 3 (t 3 ) θ 1 S 1 (t 1 ) θ +S 2 (t 2 ) θ 1 ) 1/θ 2 DM Incorporation of nested frailties into multi-state models 29/ 21

66 Rotolo, Legrand, and Van Keilegom (in press) NED LR Second and following transitions: T 1 T 4 [ ) ] θ 1/θ 1 S 4 1 (t 4 t 1) = 1 + S1(t 1) θ De S 5 2 (t 5 t 2) = [ 1 + ( S1 (t 1 ) S 4 (t 4 ) ( ) ] θ 1/θ 1 S2 (t 2 ) S 5 (t 5 ) S2(t 2) θ and the same algorithm is used to simulate second transition times, conditionally on first transition ones. DM Incorporation of nested frailties into multi-state models 30/ 21

67 Rotolo, Legrand, and Van Keilegom (in press) NED LR De Second and following transitions: [ ) ] θ 1/θ 1 S 4 1 (t 4 t 1) = 1 + S1(t 1) θ S 5 2 (t 5 t 2) = [ 1 + ( S1 (t 1 ) S 4 (t 4 ) ( ) ] θ 1/θ 1 S2 (t 2 ) S 5 (t 5 ) S2(t 2) θ and the same algorithm is used to simulate second transition times, conditionally on first transition ones. T 2 T 5 DM Incorporation of nested frailties into multi-state models 30/ 21

68 Research question: Does post-tur treatment improve prognosis of bladder cancer patients? Clustering Patients, grouped by hospitals, are not independent, neither conditional on the observed covariates between-subjects dependence frailty models Rand Multiple endpoints Rec Prog within-subjects dependence multi-state models De Incorporation of nested frailties into multi-state models 31/ 21

69 Simulation of clustering... Clustering can be generated and added in a PH way λ qh (t U h = u h ) = u h λ q0 (t), with λ q0 (t) the baseline hazard for transition q. Incorporation of nested frailties into multi-state models 32/ 21

70 Simulation of clustering... Clustering can be generated and added in a PH way λ qh (t U h = u h ) = u h λ q0 (t), with λ q0 (t) the baseline hazard for transition q. Then, we can i simulate U h from f U ( ) ii simulate times conditionally on frailties from S qh (t U h = u h ) = exp { u h Λ q0 (t)} = [S q0 (t)] u h Incorporation of nested frailties into multi-state models 32/ 21

71 Simulation of clustering... Clustering can be generated and added in a PH way λ qh (t U h = u h ) = u h λ q0 (t), with λ q0 (t) the baseline hazard for transition q. Then, we can i simulate U h from f U ( ) ii simulate times conditionally on frailties from S qh (t U h = u h ) = exp { u h Λ q0 (t)} = [S q0 (t)] u h... and of covariates i simulate U h from f U ( ) and (transition-specific) covariates x qhi from their distribution ii simulate times conditionally on frailties and covariates from S qh (t U h = u h ) = [S q0 (t)] u he β q x qhi Incorporation of nested frailties into multi-state models 32/ 21

72 Choice of simulation parameters Typically, when performing simulations, one wants to reproduce some precise situations. Here, simulation parameters should be chosen in order to obtain particular target values for p q probabilities of competing events and censoring Incorporation of nested frailties into multi-state models 33/ 21

73 Choice of simulation parameters Typically, when performing simulations, one wants to reproduce some precise situations. Here, simulation parameters should be chosen in order to obtain particular target values for p q probabilities of competing events and censoring m q median of uncensored times of each transition Incorporation of nested frailties into multi-state models 33/ 21

74 Choice of simulation parameters Typically, when performing simulations, one wants to reproduce some precise situations. Here, simulation parameters should be chosen in order to obtain particular target values for p q probabilities of competing events and censoring m q median of uncensored times of each transition but it is not possible to analytically express these quantities as functions of the parameters. Incorporation of nested frailties into multi-state models 33/ 21

75 Choice of simulation parameters To find appropriate parameters for given target values {p q, m q }, we minimize the criterion function { Q [ ] p 2 [ ] } q m 2 q Υ(Π) = log + log 0, ˆp q (Π) ˆm q (Π) q=1 with Π the simulation parameters and {ˆp i (Π), ˆm i (Π)} the observed values in a simulated dataset with parameters Π. Incorporation of nested frailties into multi-state models 34/ 21

76 Choice of simulation parameters To find appropriate parameters for given target values {p q, m q }, we minimize the criterion function { Q [ ] p 2 [ ] } q m 2 q Υ(Π) = log + log 0, ˆp q (Π) ˆm q (Π) q=1 with Π the simulation parameters and {ˆp i (Π), ˆm i (Π)} the observed values in a simulated dataset with parameters Π. Note that [ log a b ] 2 = [ log b a ] 2 Incorporation of nested frailties into multi-state models 34/ 21

77 An example A dataset of size 44 is available from a multi-center study on head and neck cancer (Grillo Ruggieri et al., 2005). Target values {p q} and {m q} 8 LR 15 7 NED De 4 4 Tot: 44 0 DM Incorporation of nested frailties into multi-state models 35/ 21

78 An example A dataset of size 44 is available from a multi-center study on head and neck cancer (Grillo Ruggieri et al., 2005). Target values {p q} and {m q} 15 8 LR 7 Frailty term 40 Hospitals random sizes U Gam(1, 0.5) NED De 4 4 Tot: 44 0 DM Incorporation of nested frailties into multi-state models 35/ 21

79 An example A dataset of size 44 is available from a multi-center study on head and neck cancer (Grillo Ruggieri et al., 2005). Target values {p q} and {m q} 15 8 LR 7 Frailty term 40 Hospitals random sizes U Gam(1, 0.5) NED 22 Tot: DM 4 14 De Covariates Age N (60, 7) with log(0.8)/10 q = 1 β q,age = log(0.9)/10 q = 2 log(1.2)/10 q = 3, 4, 5 Treat Bin(0.5) with log(1/3) q = 1 β q,treat = 0 q = 2 log(1.2) q = 3, 4, 5 Incorporation of nested frailties into multi-state models 35/ 21

80 Results First transitions. The algorithm is run with datasets of size 10 4, maxit = 10 and th = 0.1. NED {LR,DM,De} λ 1 λ 2 λ 3 λ C ρ 1 ρ 2 ρ Incorporation of nested frailties into multi-state models 36/ 21

81 Results First transitions. The algorithm is run with datasets of size 10 4, maxit = 10 and th = 0.1. NED {LR,DM,De} λ 1 λ 2 λ 3 λ C ρ 1 ρ 2 ρ p q NED {LR,DM,De} LR DM De C LR DM De Target Simulated m q Υ 123 (Π 123 ) = 0.24 Incorporation of nested frailties into multi-state models 36/ 21

82 Results Second transitions. Conditionally on first transitions data, the algorithm is run for second transitions from LR and DM with maxit = 6 and th = LR De λ 4 λ C4 ρ DM De λ 5 λ C5 ρ Incorporation of nested frailties into multi-state models 37/ 21

83 Results Second transitions. Conditionally on first transitions data, the algorithm is run for second transitions from LR and DM with maxit = 6 and th = LR De λ 4 λ C4 ρ DM De λ 5 λ C5 ρ LR De DM De p q m q p q m q De C De De C De Target Simulated Υ 4 (Π 4 ) = Υ 5 (Π 5 ) = Back Incorporation of nested frailties into multi-state models 37/ 21

84 Positive correlation of nested frailties Positive correlation of nested frailties Cov(U qh, U kj ) = E(U qh U kj ) E(U qh )E(U kj ) = E(V h V j W qh W kj ) E(V h )E(V j )E(W qh )E(W kj ). If h = j, then Cov(U qh, U kh ) = E(Vh 2 )E(W qh)e(w kh ) E(V h ) 2 E(W qh )E(W kh ) = V(V h ) [E(W qh )E(W kh )] > 0. Back Incorporation of nested frailties into multi-state models 38/ 21

85 Positive correlation of nested frailties Simulation study Scenario #1 Scenario #2 Scenario #3 True values I Shared frailty I Nested frailties True values I Shared frailty I Nested frailties True values I Shared frailty I Nested frailties θv θv θv θw θw θw hospitals of size hospitals of size hospitals of size 125 Incorporation of nested frailties into multi-state models 39/ 21