Analysis of clustered interval‐censored data using a class of semiparametric partly linear frailty transformation models

Abstract

A flexible class of semiparametric partly linear frailty transformation models is considered for analyzing clustered interval‐censored data, which arise naturally in complex diseases and dental research. This class of models features two nonparametric components, resulting in a nonparametric baseline survival function and a potential nonlinear effect of a continuous covariate. The dependence among failure times within a cluster is induced by a shared, unobserved frailty term. A sieve maximum likelihood estimation method based on piecewise linear functions is proposed. The proposed estimators of the regression, dependence, and transformation parameters are shown to be strongly consistent and asymptotically normal, whereas the estimators of the two nonparametric functions are strongly consistent with optimal rates of convergence. An extensive simulation study is conducted to study the finite‐sample performance of the proposed estimators. We provide an application to a dental study for illustration.

Partly linear single-index cure models with a nonparametric incidence link function

“…In model ( 2), we assume a linear effect of W on the log-hazard, but the effect of Z is potentially nonlinear. An appealing feature of the PLSIM is that the class of models includes both the partially linear survival model (with Z being univariate) 32,33 and the single-index survival models (in the absence of W ) 24,25 as special cases. Based on (1) and ( 2), the population survival function is…”

“…To fit SIM/PLSIM, a variety of estimation methods have been proposed, which include but are not limited to the kernel smoothing methods 9,21,24 and spline approximations; 25,27 piecewise linear functions 32,33 are special cases of splines. A particular choice of splines is the Bernstein polynomial (BP), 34 which has been adopted to approximate the baseline cumulative hazard functions in the bivariate transformation survival models, 35 to approximate nonlinear covariate effects in the additive Cox model for interval-censored data, 36 and to approximate the distribution function in the semiparametric transformation non-mixture cure models.…”

Semiparametric copula method for semi-competing risks data subject to interval censoring and left truncation: Application to disability in elderly

“…The sieve approach is one of the most well-developed methods for estimating infinitedimensional parameters. [31][32][33] Specifically, we use Bernstein polynomials to build a sieve space…”

“…The sieve approach is one of the most well-developed methods for estimating infinite-dimensional parameters. 31–33 Specifically, we use Bernstein polynomials to build a sieve space Θ n = falsefalse{ θ n = false( β T , α , κ , Λ 1 n , Λ 2 n ) T ∈ B ⊗ scriptM n 1 ⊗ scriptM n 2 falsefalse}. Here, scriptM n j is the space defined by Bernstein polynomials:where B .1em j , k false( t , m .1em j n , c , u false) represents the Bernstein basis polynomial defined as:with degree m .1em j n = o false( n ν j false) for some ν j ∈ false( 0 , 1 false).…”

Marginal analysis of current status data with informative cluster size using a class of semiparametric transformation cure models

“…Sun and Ding 19 considered the semi‐parametric transformation models for bivariate general interval‐censored survival data based on the two‐parameter Achimedean copula model. Lee et al 20 studied a class of semi‐parametric partly linear frailty transformation models for clustered interval‐censored data. All the aforementioned papers adopted the sieve maximum likelihood estimation (sieve‐MLE) approach where the estimators are shown to be consistent, asymptotically normal and efficient, but their methods do not accommodate data with a cure fraction.…”