Nonparametric maximum likelihood estimation for competing risks survival data subject to interval censoring and truncation

We derive the nonparametric maximum likelihood estimate (NPMLE) of the cumulative incidence functions for competing risks survival data subject to interval censoring and truncation. Since the cumulative incidence function NPMLEs give rise to an estimate of the survival distribution which can be undefined over a potentially larger set of regions than the NPMLE of the survival function obtained ignoring failure type, we consider an alternative pseudolikelihood estimator. The methods are then applied to data from a cohort of injecting drug users in Thailand susceptible to infection from HIV-1 subtypes B and E.     

Nonparametric estimation of a Markov 'illness-death' process from interval-censored observations, with application to diabetes survival data

The nonparametric estimation of the cumulative transition intensityfunctions in a threestate time-nonhomogeneous Markov processwith irreversible transitions, an ‘illness-death’model, is considered when times of the intermediate transition,e.g. onset of a disease, are interval-censored. The times of‘death’ are assumed to be known exactly or to beright-censored. In addition the observed process may be left-truncated.Data of this type arise when the process is sampled periodically.For example, when the patients are monitored through periodicexaminations the observations on times of change in their diseasestatus will be interval-censored. Under the sampling schemeconsidered here the Nelson–Aalen estimator (Aalen, 1978)for a cumulative transition intensity is not applicable. Inthe proposed method the maximum likelihood estimators of someof the transition intensities are derived from the estimatorsof the corresponding subdistribution functions. The maximumlikelihood estimators are shown to have a self-consistency property.The self-consistency algorithm is developed for the computationof the estimators. This approach generalises the results fromTurnbull (1976) and Frydman (1992). The methods are illustratedwith diabetes survival data.     

Nonparametric estimation in a Markov "illness-death" process from interval censored observations with missing intermediate transition status

Summary .  In many clinical trials patients are intermittently assessed for the transition to an intermediate state, such as occurrence of a disease-related nonfatal event, and death. Estimation of the distribution of nonfatal event free survival time, that is, the time to the first occurrence of the nonfatal event or death, is the primary focus of the data analysis. The difficulty with this estimation is that the intermittent assessment of patients results in two forms of incompleteness: the times of occurrence of nonfatal events are interval censored and, when a nonfatal event does not occur by the time of the last assessment, a patient's nonfatal event status is not known from the time of the last assessment until the end of follow-up for death. We consider both forms of incompleteness within the framework of an "illness–death" model. We develop nonparametric maximum likelihood (ML) estimation in an "illness–death" model from interval-censored observations with missing status of intermediate transition. We show that the ML estimators are self-consistent and propose an algorithm for obtaining them. This work thus provides new methodology for the analysis of incomplete data that arise from clinical trials. We apply this methodology to the data from a recently reported cancer clinical trial ( Bonner et al., 2006 , New England Journal of Medicine 354, 567–578) and compare our estimation results with those obtained using a Food and Drug Administration recommended convention.     

Discrete-time nonparametric estimation for semi-Markov models of chain-of-events data subject to interval censoring and truncation

Chain-of-events data are longitudinal observations on a succession of events that can only occur in a prescribed order. One goal in an analysis of this type of data is to determine the distribution of times between the successive events. This is difficult when individuals are observed periodically rather than continuously because the event times are then interval censored. Chain-of-events data may also be subject to truncation when individuals can only be observed if a certain event in the chain (e.g., the final event) has occurred. We provide a nonparametric approach to estimate the distributions of times between successive events in discrete time for data such as these under the semi-Markov assumption that the times between events are independent. This method uses a self-consistency algorithm that extends Turnbull's algorithm (1976, Journal of the Royal Statistical Society, Series B 38, 290-295). The quantities required to carry out the algorithm can be calculated recursively for improved computational efficiency. Two examples using data from studies involving HIV disease are used to illustrate our methods.     

Nonparametric estimation of asymptomatic duration from a randomized prospective cancer screening trial

We propose a nonparametric estimation of preclinical duration distribution in cancer based on data from a randomized early detection trial. In cancer screening studies, the preclinical duration of a disease is of great interest for better understanding the natural history of the disease, and for developing optimal screening strategies. To estimate the sojourn time distribution nonparametrically, we first estimate the distribution of the age at onset of preclinical disease nonparametrically using data from the screening arm in a randomized screening trial, and the distribution for the age at onset of clinical disease from the control arm of the randomized screening trial. Finally, by using deconvolution the two estimated distributions lead to a nonparametric estimate of the distribution for the gap time between the onset of preclinical disease and the onset of clinical disease. We illustrate the methodology using data from a randomized breast cancer screening trial.     

Nonparametric estimation in an “illness‐death” model when all transition times are interval censored

We develop nonparametric maximum likelihood estimation for the parameters of an irreversible Markov chain on states from the observations with interval censored times of 0 → 1, 0 → 2 and 1 → 2 transitions. The distinguishing aspect of the data is that, in addition to all transition times being interval censored, the times of two events (0 → 1 and 1 → 2 transitions) can be censored into the same interval. This development was motivated by a common data structure in oral health research, here specifically illustrated by the data from a prospective cohort study on the longevity of dental veneers. Using the self‐consistency algorithm we obtain the maximum likelihood estimators of the cumulative incidences of the times to events 1 and 2 and of the intensity of the 1 → 2 transition. This work generalizes previous results on the estimation in an “illness‐death” model from interval censored observations.     

Nonparametric estimation of a delay distribution based on left-censored and right-truncated data

This paper proposes a nonparametric method for estimating a delay distribution based on left-censored and right-truncated data. A variance-covariance estimator is provided. The method is applied to the Australian AIDS data in which some data are left censored and some data are not left censored. This situation arises with AIDS case-reporting data in Australia because reporting delays were recorded only from November 1990 rather than from the beginning of the epidemic there. It is shown that inclusion of the left-censored data, as opposed to analyzing only the uncensored data, improves the precision of the estimate.     

Nonparametric estimation of the survival function for ordered multivariate failure time data: A comparative study

In longitudinal studies of disease, patients may experience several events through a follow‐up period. In these studies, the sequentially ordered events are often of interest and lead to problems that have received much attention recently. Issues of interest include the estimation of bivariate survival, marginal distributions, and the conditional distribution of gap times. In this work, we consider the estimation of the survival function conditional to a previous event. Different nonparametric approaches will be considered for estimating these quantities, all based on the Kaplan–Meier estimator of the survival function. We explore the finite sample behavior of the estimators through simulations. The different methods proposed in this article are applied to a dataset from a German Breast Cancer Study. The methods are used to obtain predictors for the conditional survival probabilities as well as to study the influence of recurrence in overall survival.     

Joint modeling of longitudinal and survival data via a common frailty

We develop a joint model for the analysis of longitudinal and survival data in the presence of data clustering. We use a mixed effects model for the repeated measures that incorporates both subject- and cluster-level random effects, with subjects nested within clusters. A Cox frailty model is used for the survival model in order to accommodate the clustering. We then link the two responses via the common cluster-level random effects, or frailties. This model allows us to simultaneously evaluate the effect of covariates on the two types of responses, while accounting for both the relationship between the responses and data clustering. The model was motivated by a study of end-stage renal disease patients undergoing hemodialysis, where we wished to evaluate the effect of iron treatment on both the patients' hemoglobin levels and survival times, with the patients clustered by enrollment site.     

Nonparametric estimation of the causal effect of a stochastic threshold-based intervention

Identifying a biomarker or treatment-dose threshold that marks a specified level of risk is an important problem, especially in clinical trials. In view of this goal, we consider a covariate-adjusted threshold-based interventional estimand, which happens to equal the binary treatment–specific mean estimand from the causal inference literature obtained by dichotomizing the continuous biomarker or treatment as above or below a threshold. The unadjusted version of this estimand was considered in Donovan et al.. Expanding upon Stitelman et al., we show that this estimand, under conditions, identifies the expected outcome of a stochastic intervention that sets the treatment dose of all participants above the threshold. We propose a novel nonparametric efficient estimator for the covariate-adjusted threshold-response function for the case of informative outcome missingness, which utilizes machine learning and targeted minimum-loss estimation (TMLE). We prove the estimator is efficient and characterize its asymptotic distribution and robustness properties. Construction of simultaneous 95% confidence bands for the threshold-specific estimand across a set of thresholds is discussed. In the Supporting Information, we discuss how to adjust our estimator when the biomarker is missing at random, as occurs in clinical trials with biased sampling designs, using inverse probability weighting. Efficiency and bias reduction of the proposed estimator are assessed in simulations. The methods are employed to estimate neutralizing antibody thresholds for virologically confirmed dengue risk in the CYD14 and CYD15 dengue vaccine trials.     

Inference for the effect of treatment on survival probability in randomized trials with noncompliance and administrative censoring

In many clinical studies with a survival outcome, administrative censoring occurs when follow-up ends at a prespecified date and many subjects are still alive. An additional complication in some trials is that there is noncompliance with the assigned treatment. For this setting, we study the estimation of the causal effect of treatment on survival probability up to a given time point among those subjects who would comply with the assignment to both treatment and control. We first discuss the standard instrumental variable (IV) method for survival outcomes and parametric maximum likelihood methods, and then develop an efficient plug-in nonparametric empirical maximum likelihood estimation (PNEMLE) approach. The PNEMLE method does not make any assumptions on outcome distributions, and makes use of the mixture structure in the data to gain efficiency over the standard IV method. Theoretical results of the PNEMLE are derived and the method is illustrated by an analysis of data from a breast cancer screening trial. From our limited mortality analysis with administrative censoring times 10 years into the follow-up, we find a significant benefit of screening is present after 4 years (at the 5% level) and this persists at 10 years follow-up.