Sieve estimation in a Markov illness-death process under dual censoring

Biostatistics (2016), 17, 2, pp. 350–363 doi:10.1093/biostatistics/kxv042 Advance Access publication on November 22, 2015 AUDREY BORUVKA∗ , RICHARD J. COOK Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada N2L 3G1 SUMMARY Semiparametric methods are well established for the analysis of a progressive Markov illness-death process observed up to a noninformative right censoring time. However, often the intermediate and terminal events are censored in different ways, leading to a dual censoring scheme. In such settings, unbiased estimation of the cumulative transition intensity functions cannot be achieved without some degree of smoothing. To overcome this problem, we develop a sieve maximum likelihood approach for inference on the hazard ratio. A simulation study shows that the sieve estimator offers improved finite-sample performance over common imputation-based alternatives and is robust to some forms of dependent censoring. The proposed method is illustrated using data from cancer trials. Keywords: Cox model; Interval censoring; Method of sieves; Profile likelihood; Progression-free survival. 1. INTRODUCTION Vital status for individuals in a clinical trial is often readily available. Detection of non-fatal events requires closer surveillance, which can prove difficult and costly to maintain over time. As a result survival times are subject to right censoring, but the occurrence of intermediate events may be right-censored earlier or interval-censored between assessments. In general, we refer to this scenario as dual censoring. Various forms of dual censoring arise in trials involving tumor progression. Guidelines call for the analysis of socalled time to progression (TTP), coinciding with detection of progression, or progression-free survival (PFS), given by the earliest of TTP and death (FDA, 2007). TTP is typically right-censored at death or the preceding (negative) assessment, which induces dependent censoring. PFS is thus deemed preferable to TTP (FDA, 2007, p. 8), but this outcome is subject to systematic imputation. Multistate models have been suggested as a more natural framework for assessing treatment effects on progression and death. A chain of events model (Figure 1, left), for example, is useful for settings in which progression always precedes death (Frydman, 1995b). Semicompeting risks (Figure 1, middle) have been proposed for the case where death may precede progression (Hu and Tsodikov, 2014). Xu and others (2010) observe that semicompeting risks essentially amount to the progressive illness-death ∗ To whom correspondence should be addressed. c The Author 2015. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: .  Sieve estimation in a Markov illness-death process under dual censoring Markov illness-death process under dual censoring 351 model (Fix and Neyman, 1951; Figure 1, right), which is fully specified by the state-transition intensity functions. Among the three state-transition structures, methods to deal with specific instances of dual censoring are most developed for the illness-death model. Frydman (1995a) considers the nonparametric maximum likelihood estimator (NPMLE) from interval-censored progression times with known progression status. This is generalized by Frydman and Szarek (2009) to account for unknown status, which often arises when the last assessment is negative and long precedes right-censoring or death. Bebchuk and Betensky (2001) combine local likelihood and multiple imputation to estimate transition intensities under progression times right-censored before death. Joly and others (2002) propose spline-based penalized likelihood for the (Cox, 1972) proportional hazards model for an interval-censored variant of this observation scheme. Jackson (2011) considers a piecewise exponential analog by way of time-dependent covariates. These works recognize that progression and death are observed in different ways, but the broader problem of dual censoring has not yet been considered. Methods for time-to-event endpoints that leave any dependence on time unspecified are generally preferred in practice. However, non- and semi-parametric maximum likelihood estimators require the locations of support for the distribution of each transition time, and these are ambiguous whenever the progression status is unknown. To address these issues, we develop a sieve estimator for a multistate extension of the Cox model and compare its numerical performance with routine analysis of imputation-based PFS under a variety of censoring scenarios. 2. DUAL CENSORING OF THE PROGRESSIVE ILLNESS-DEATH PROCESS Let Nh j be a one-jump counting process representing the transition from state h to state j (h = j) in the progressive illness-death model and Th j be the corresponding transition time. So T01 is the time to progression, T02 is the time to progression-free death, and T12 is the time of death following progression. Over the observation period [0, τ ], τ < ∞, suppose that the survival time T02 ∧ T12 is observed up to a right censoring time D, 0 < D  τ , but progression status 1(T01  t) is not necessarily known for all t ∈ (0, V ], V = T02 ∧ T12 ∧ D. For example, progression may be right-censored at some random time preceding D. Alternatively, progression status could assessed periodically, leading to interval censoring. Whatever the form of this inspection process, we presume that it yields a potential censoring interval (L , R] for the progression time T01 . We say “potential” because we may not know with certainty that T01 ∈ (L , R]. Put Δ2 = 1(T02 ∧ T12  D) to denote whether or not the survival time is observed. Let Δ0 = 1 whenever progression status is known to be negative at V and Δ0 = 0 otherwise. Similarly, let Δ1 indicate that progression status is known to be positive at V . So Δ1 = 1 denotes that, based on the available data, we are certain T01 ∈ (L , R] for some L < R  V . Otherwise either Δ0 = 1, indicating that T01 > V , or progression status is unknown at V . If the status is unknown, then Δ0 = Δ1 = 0 and we cannot rule out the possibility that either T01 ∈ (L , R] or T01 > R. Fig. 1. Multistate alternatives to TTP and PFS: chain of events (left), semicompeting risks (middle), and progressive illness-death (right) models. 352 A. BORUVKA AND R. J. COOK 2.1 Example: Bone lesions and their complications Dual right-censored data are encountered in cancer trials evaluating the effect of bisphosphonates on bone metastases and their complications, known as skeletal-related events (SREs). The time of an SRE is often self-evident, but can otherwise be measured accurately through frequent clinic visits, so SREs are typically considered subject only to right censoring. Growth of new or existing bone lesions is assessed by radiographic surveys, which are carried out less frequently. This results in interval-censored lesion pro (...truncated)