Bayesian random-effects threshold regression with application to survival data with nonproportional hazards

Semiparametric predictive inference for failure data using first-hitting-time threshold regression

The progression of disease for an individual can be described mathematically as a stochastic process. The individual experiences a failure event when the disease path first reaches or crosses a critical disease level. This happening defines a failure event and a first hitting time or time-to-event, both of which are important in medical contexts. When the context involves explanatory variables then there is usually an interest in incorporating regression structures into the analysis and the methodology known as threshold regression comes into play. To date, most applications of threshold regression have been based on parametric families of stochastic processes. This paper presents a semiparametric form of threshold regression that requires the stochastic process to have only one key property, namely, stationary independent increments. As this property is frequently encountered in real applications, this model has potential for use in many fields. The mathematical underpinnings of this semiparametric approach for estimation and prediction are described. The basic data element required by the model is a pair of readings representing the observed change in time and the observed change in disease level, arising from either a failure event or survival of the individual to the end of the data record. An extension is presented for applications where the underlying disease process is unobservable but component covariate processes are available to construct a surrogate disease process. Threshold regression, used in combination with a data technique called Markov decomposition, allows the methods to handle longitudinal time-to-event data by uncoupling a longitudinal record into a sequence of single records. Computational aspects of the methods are straightforward. An array of simulation experiments that verify computational feasibility and statistical inference are reported in an online supplement. Case applications based on longitudinal observational data from The Osteoarthritis Initiative (OAI) study are presented to demonstrate the methodology and its practical use.

Estimating duration distribution aided by auxiliary longitudinal measures in presence of missing time origin

Understanding the distribution of an event duration time is essential in many studies. The exact time to the event is often unavailable, and thus so is the full event duration. By linking relevant longitudinal measures to the event duration, we propose to estimate the duration distribution via the first-hitting-time model (e.g. Lee and Whitmore in Stat Sci 21(4):501-513, 2006). The longitudinal measures are assumed to follow a Wiener process with random drift. We apply a variant of the MCEM algorithm to compute likelihood-based estimators of the parameters in the longitudinal process model. This allows us to adapt the well-known empirical distribution function to estimate the duration distribution in the presence of missing time origin. Estimators with smooth realizations can then be obtained by conventional smoothing techniques. We establish the consistency and weak convergence of the proposed distribution estimator and present its variance estimation. We use a collection of wildland fire records from Alberta, Canada to motivate and illustrate the proposed approach. The finite-sample performance of the proposed estimator is examined by simulation. Viewing the available data as interval-censored times, we show that the proposed estimator can be more efficient than the well-established Turnbull estimator, an alternative that is often applied in such situations.

Semi-parametric survival analysis via Dirichlet process mixtures of the First Hitting Time model

Time-to-event data often violate the proportional hazards assumption inherent in the popular Cox regression model. Such violations are especially common in the sphere of biological and medical data where latent heterogeneity due to unmeasured covariates or time varying effects are common. A variety of parametric survival models have been proposed in the literature which make more appropriate assumptions on the hazard function, at least for certain applications. One such model is derived from the First Hitting Time (FHT) paradigm which assumes that a subject's event time is determined by a latent stochastic process reaching a threshold value. Several random effects specifications of the FHT model have also been proposed which allow for better modeling of data with unmeasured covariates. While often appropriate, these methods often display limited flexibility due to their inability to model a wide range of heterogeneities. To address this issue, we propose a Bayesian model which loosens assumptions on the mixing distribution inherent in the random effects FHT models currently in use. We demonstrate via simulation study that the proposed model greatly improves both survival and parameter estimation in the presence of latent heterogeneity. We also apply the proposed methodology to data from a toxicology/carcinogenicity study which exhibits nonproportional hazards and contrast the results with both the Cox model and two popular FHT models.

Ornstein-Uhlenbeck threshold regression for time-to-event data with and without a cure fraction

In this paper we propose a threshold regression (TR) model for time to event data related to subject health using a latent Ornstein-Uhlenbeck (OU) process that fails once it hits a boundary value for the first time. Baseline covariates are incorporated into the analysis using a log-link function for the initial state of the health process. The model provides clinically meaningful covariate effects and does not require the proportional hazards assumption of the commonly used Cox model. Unlike TR models based on the Wiener process, the OU model allows increments in the health process to depend on previous values and drifts toward a state of equilibrium or homeostasis, which are present in many biological applications. We also extend our model to incorporate a cure rate for applications with improper survival functions, such as time to tumor recurrence in a cancer clinical trial. Our models are applied to overall and relapse-free survival data of melanoma patients undergoing definitive surgery.

Evidence synthesis through a degradation model applied to myocardial infarction

We propose an evidence synthesis approach through a degradation model to estimate causal influences of physiological factors on myocardial infarction (MI) and coronary heart disease (CHD). For instance several studies give incidences of MI and CHD for different age strata, other studies give relative or absolute risks for strata of main risk factors of MI or CHD. Evidence synthesis of several studies allows incorporating these disparate pieces of information into a single model. For doing this we need to develop a sufficiently general dynamical model; we also need to estimate the distribution of explanatory factors in the population. We develop a degradation model for both MI and CHD using a Brownian motion with drift, and the drift is modeled as a function of indicators of obesity, lipid profile, inflammation and blood pressure. Conditionally on these factors the times to MI or CHD have inverse Gaussian ([Formula: see text]) distributions. The results we want to fit are generally not conditional on all the factors and thus we need marginal distributions of the time of occurrence of MI and CHD; this leads us to manipulate the inverse Gaussian normal distribution ([Formula: see text]) (an [Formula: see text] whose drift parameter has a normal distribution). Another possible model arises if a factor modifies the threshold. This led us to define an extension of [Formula: see text] obtained when both drift and threshold parameters have normal distributions. We applied the model to results published in five important studies of MI and CHD and their risk factors. The fit of the model using the evidence synthesis approach was satisfactory and the effects of the four risk factors were highly significant.

Proportional hazards and threshold regression: their theoretical and practical connections

Proportional hazards (PH) regression is a standard methodology for analyzing survival and time-to-event data. The proportional hazards assumption of PH regression, however, is not always appropriate. In addition, PH regression focuses mainly on hazard ratios and thus does not offer many insights into underlying determinants of survival. These limitations have led statistical researchers to explore alternative methodologies. Threshold regression (TR) is one of these alternative methodologies (see Lee and Whitmore, Stat Sci 21:501-513, 2006, for a review). The connection between PH regression and TR has been examined in previous published work but the investigations have been limited in scope. In this article, we study the connections between these two regression methodologies in greater depth and show that PH regression is, for most purposes, a special case of TR. We show two methods of construction by which TR models can yield PH functions for survival times, one based on altering the TR time scale and the other based on varying the TR boundary. We discuss how to estimate the TR time scale and boundary, with or without the PH assumption. A case demonstration is used to highlight the greater understanding of scientific foundations that TR can offer in comparison to PH regression. Finally, we discuss the potential benefits of positioning PH regression within the first-hitting-time context of TR regression.