Ensemble methods for survival function estimation with time-varying covariates.

Survival data with time-varying covariates are common in practice. If relevant, they can improve on the estimation of a survival function. However, the traditional survival forests-conditional inference forest, relative risk forest and random survival forest-have accommodated only time-invariant covariates. We generalize the conditional inference and relative risk forests to allow time-varying covariates. We also propose a general framework for estimation of a survival function in the presence of time-varying covariates. We compare their performance with that of the Cox model and transformation forest, adapted here to accommodate time-varying covariates, through a comprehensive simulation study in which the Kaplan-Meier estimate serves as a benchmark, and performance is compared using the integrated L2 difference between the true and estimated survival functions. In general, the performance of the two proposed forests substantially improves over the Kaplan-Meier estimate. Taking into account all other factors, under the proportional hazard setting, the best method is always one of the two proposed forests, while under the non-proportional hazard setting, it is the adapted transformation forest. K-fold cross-validation is used as an effective tool to choose between the methods in practice.

An ensemble method for interval-censored time-to-event data.

Interval-censored data analysis is important in biomedical statistics for any type of time-to-event response where the time of response is not known exactly, but rather only known to occur between two assessment times. Many clinical trials and longitudinal studies generate interval-censored data; one common example occurs in medical studies that entail periodic follow-up. In this article, we propose a survival forest method for interval-censored data based on the conditional inference framework. We describe how this framework can be adapted to the situation of interval-censored data. We show that the tuning parameters have a non-negligible effect on the survival forest performance and guidance is provided on how to tune the parameters in a data-dependent way to improve the overall performance of the method. Using Monte Carlo simulations, we find that the proposed survival forest is at least as effective as a survival tree method when the underlying model has a tree structure, performs similarly to an interval-censored Cox proportional hazards model fit when the true relationship is linear, and outperforms the survival tree method and Cox model when the true relationship is nonlinear. We illustrate the application of the method on a tooth emergence data set.

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 {0,1,2} 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 the cumulative intensities in an interval censored competing risks model.

The nonparametric maximum likelihood estimation (NPMLE) of the distribution function from the interval censored (IC) data has been extensively studied in the extant literature. The NPMLE was also developed for the subdistribution functions in an IC competing risks model and in an illness-death model under various interval-censoring scenarios. But the important problem of estimation of the cumulative intensities (CIs) in the interval-censored models has not been considered previously. We develop the NPMLE of the CI in a simple alive/dead model and of the CIs in a competing risks model. Assuming that data are generated by a discrete and finite mixed case interval censoring mechanism we provide a discussion and the simulation study of the asymptotic properties of the NPMLEs of the CIs. In particular we show that they are asymptotically unbiased; in contrast the ad hoc estimators presented in extant literature are substantially biased. We illustrate our methods with the data from a prospective cohort study on the longevity of dental veneers.

Estimation of overall survival in an 'illness-death' model with application to the vertical transmission of HIV-1.

We derive a nonparametric maximum likelihood estimate of the overall survival distribution in an illness-death model from interval censored observations with unknown status of the nonfatal event. This expanded model is applied to the re-analysis of data from a randomized trial where infants, born to women infected with HIV-1 that were randomly assigned to breastfeeding or counseling for formula feeding, were followed for 24 months for HIV-1 positivity, HIV-1-free survival, and overall survival. HIV-1 positivity, assessed by postpartum venous blood tests, is the interval censored nonfatal event, and HIV-1 positivity status is unknown for a subset of infants due to periodic assessment. The analysis demonstrates that estimation of the overall and the pre- and post-nonfatal event survival distributions with the proposed methods provide novel insights into how overall survival is influenced by the occurrence of the nonfatal event. More generally, it suggests the usefulness of this expanded illness-death model when evaluating composite endpoints as potential surrogates for overall survival in a given disease setting.

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

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 Medicine354, 567-578) and compare our estimation results with those obtained using a Food and Drug Administration recommended convention.