Analysis of Multivariate Survival Data under Semiparametric Copula Models

Modelling multivariate survival data is complicated by the complex association structure among responses. To balance model flexibility and interpretability, we propose a semiparametric copula to modulate data, with marginal distributions of response components described linear transformation models. conduct inference about parameters, develop two-stage maximum likelihood method three-stage pseudo-likelihood estimation procedure. We investigate impact misspecification on covariate effects identify scenario in which consistent parameters retained even when misspecified. The proposed methods are justified both theoretically empirically. An application real dataset provided demonstrate utility method. Modéliser les données de survie multivariées s'avère être un processus complexe en raison la d'association sophistiquée entre différentes réponses. Pour gérer efficacement ces tout maintenant une interprétation claire et accessible, auteurs du présent article proposent modèle copule semiparamétrique, dans lequel marginales des composantes réponse sont représentées par modèles linéaire semiparamétriques. réaliser inférences ou estimer paramètres proposé, ils élaborent méthode vraisemblance deux étapes procédure d'estimation pseudovraisemblance trois étapes. Ensuite, examinent l'impact d'une spécification inadéquate sur l'estimation effets covariables identifient le scénario qui assure convergence marginaux, même cas mauvaise copule. conclure, valident approches proposées tant plan théorique qu'empirique présentent basée ensemble réelles pour illustrer l'efficacité proposée. Correlated commonly arise from clinical trials, epidemiological studies, cancer research. Examples include studies times family members cancer, where primary endpoint (e.g., death) secondary time tumour progression) gathered for study subjects. Different models have been developed describe various types dependence correlated covariates. Common modelling strategies models, frailty When employed, focus characterizing covariates, outcomes typically left unspecified. Inference procedures usually carried out using estimating function theory. Some early work this was discussed Wei, Lin & Weissfeld (1989) Cai Prentice (1995), Cox proportional hazards used. In contrast accommodate through random effects, called frailties. Conditional frailties as well assumed be independent. This strategy basically employs capture times, distribution assumed. often performed working function, integrates joint frailties, given scheme can found Clayton (1978), together Cuzick (1985), Anderson Louis Oakes Jeong (1998), Fan Li (2002). Unlike no or little interest, offer different perspective characterize (Joe, 1997; Nelsen, 2006). allows us separate survivor functions characterization associations, particularly useful flexible facilitating structures transparent manner Shih Louis, 1995; Glidden, 2000; Andersen, 2005; Li, Lin, 2008). While copulas conceptually simple that they separately delineate most existing parametric structure. Though treatment convenient implement, it has two notable limitations. First, induces higher risk than nonparametric approaches. Second, inflexible accommodating structures. Although Chen, Tsyrennikov, 2006; Chen Yu, 2012), developments largely rely empirical explorations. Rigorous theoretical justifications still need explored. article, (Dabrowska Doksum, 1988) components. preserves transparency interpretability framework, yet employ times. comes at price complications developing establishing properties. For procedures. former procedure more efficient latter but harder implement. Further, rigorously establish properties resulting estimators. remainder organized follows. Basic notation set-up Section 2. procedures, asymptotic estimators, presented Sections 3 4, respectively. Model 5. 6, simulation conducted assess variety scenarios, an methods. Concluding remarks final section. Consider sample n subjects who experience m events. i ∈ { 1 , … } j let T denote th event subject C associated censoring time. Let δ = I ( ≤ ) right indicator ˜ min observed · function. Suppose X vector covariates corresponding . write ⊤ V Assume conditionally independent Expression (2) includes used instance, taking g u log specifying F cumulative generalized extreme value shape parameter 0, scale 1, location i.e., − exp leads model. Taking standard logistic / + yields odds 1988; Cheng, Wei Ying, Jin 2002). fact, since completely unspecified, flexibly accommodates broad class viewed regression ease exposition, consider common all ϵ Extensions component-dependent transformations treated same will notionally involved. However, involves unknown impossible obtain estimator θ directly maximizing ℓ ⋯ respect resolve issue, handle separately. first stage, fix estimate nonparametrically; second use replaced its stage. specific, present form (3) ≥ 2 ω β r × entries being 0 - Denote | any real-valued -variate G ∂ does not appear if Now explicitly implementing Equations (6)–(7) iteratively. choose initial value, say ^ ϕ Then iteration repeat following stages: Stage 1: Nonparametric Estimation Transformation Function t k K Equation (8) shows monotonicity preserved estimates : By monotonically interpolating denoted Profile Pseudo-score Remark 1.The equation (7) advantageous because computational simplicity. It overcomes difficulty score equation, very difficult solve under constraint. convenience achieved incurring efficiency loss relative likelihood-based estimation. On other hand, based differs Ying (2002), considered unbiased martingale property counting process. Furthermore, our univariate setting examined (2002) contains alone. 2.Our related Yu (2012), development here differs. (2012) focused only Clayton–Oakes linear-transformation modelling, general. ours well. stage martingales, via Our separates finite-dimensional implementation simple. now obtained Pr > Θ space stand