Proportional hazards and threshold regression: their theoretical and practical connections

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.

Development and validation of a simple-to-use nomogram for self-screening the risk of dyslipidemia

This study aimed to help healthy adults achieve self-screening by analyzing the quantitative relationship between body composition index measurements (BMI, waist-to-hip ratio, etc.) and dyslipidemia and establishing a logical risk prediction model for dyslipidemia. We performed a cross-sectional study and collected relevant data from 1115 adults between November 2019 and August 2020. The least absolute shrinkage selection operator (LASSO) regression analysis was performed to select the best predictor variables, and multivariate logistic regression analysis was used to construct the prediction model. In this study, a graphic tool including 10 predictor variables (a "nomogram," see the precise definition in the text) was constructed to predict the risk of dyslipidemia in healthy adults. A calibration diagram, receiver operating characteristic (ROC) curve, and decision curve analysis (DCA) were used to verify the model's utility. Our proposed dyslipidemia nomogram showed good discriminative ability with a C-index of 0.737 (95% confidence interval, 0.70-0.773). In the internal validation, a high C-index value of 0.718 was achieved. DCA showed a dyslipidemia threshold probability of 2-45%, proving the value of the nomogram for clinical application for dyslipidemia. This nomogram may be useful for self-screening the risk of dyslipidemia in healthy adults.

A boosting first-hitting-time model for survival analysis in high-dimensional settings

In this paper we propose a boosting algorithm to extend the applicability of a first hitting time model to high-dimensional frameworks. Based on an underlying stochastic process, first hitting time models do not require the proportional hazards assumption, hardly verifiable in the high-dimensional context, and represent a valid parametric alternative to the Cox model for modelling time-to-event responses. First hitting time models also offer a natural way to integrate low-dimensional clinical and high-dimensional molecular information in a prediction model, that avoids complicated weighting schemes typical of current methods. The performance of our novel boosting algorithm is illustrated in three real data examples.

Accounting for delayed entry into observational studies and clinical trials: length-biased sampling and restricted mean survival time

Individuals in many observational studies and clinical trials for chronic diseases are enrolled well after onset or diagnosis of their disease. Times to events of interest after enrollment are therefore residual or left-truncated event times. Individuals entering the studies have disease that has advanced to varying extents. Moreover, enrollment usually entails probability sampling of the study population. Finally, event times over a short to moderate time horizon are often of interest in these investigations, rather than more speculative and remote happenings that lie beyond the study period. This research report looks at the issue of delayed entry into these kinds of studies and trials. Time to event for an individual is modelled as a first hitting time of an event threshold by a latent disease process, which is taken to be a Wiener process. It is emphasized that recruitment into these studies often involves length-biased sampling. The requisite mathematics for this kind of sampling and delayed entry are presented, including explicit formulas needed for estimation and inference. Restricted mean survival time (RMST) is taken as the clinically relevant outcome measure. Exact parametric formulas for this measure are derived and presented. The results are extended to settings that involve study covariates using threshold regression methods. Methods adapted for clinical trials are presented. An extensive case illustration for a clinical trial setting is then presented to demonstrate the methods, the interpretation of results, and the harvesting of useful insights. The closing discussion covers a number of important issues and concepts.

Group sequential design for randomized trials using "first hitting time" model

Group sequential design (GSD) has become a popular choice in recent clinical trials as it improves trial efficiency by providing options for early termination. The implementation of traditional tests for survival analysis (eg, the log-rank test and the Cox proportional hazard (PH) model) in the GSD setting has been widely discussed. The PH assumption is required for conventional (sequential) design, it is, however, often violated in practice. As an alternative, some generalized tests have been proposed (eg, the Max-Combo test) and their efficacies have been established. In this article, we explore the application of a more flexible, "first hitting time" based threshold regression (TR) model to GSD. TR assumes that subjects' health status is a latent (unobservable) process, and the clinical event of interest occurs when the latent health process hits a pre-specified boundary. The simulation results supported our findings that, in most cases, this comparable new method can successfully control type I error while providing higher early stopping opportunities in the sequential design, even when non-proportional hazard presents.

Multivariate Threshold Regression Models with Cure Rates: Identification and Estimation in the Presence of the Esscher Property

The first hitting time of a boundary or threshold by the sample path of a stochastic process is the central concept of threshold regression models for survival data analysis. Regression functions for the process and threshold parameters in these models are multivariate combinations of explanatory variates. The stochastic process under investigation may be a univariate stochastic process or a multivariate stochastic process. The stochastic processes of interest to us in this report are those that possess stationary independent increments (i.e., Lévy processes) as well as the Esscher property. The Esscher transform is a transformation of probability density functions that has applications in actuarial science, financial engineering, and other fields. Lévy processes with this property are often encountered in practical applications. Frequently, these applications also involve a ‘cure rate’ fraction because some individuals are susceptible to failure and others not. Cure rates may arise endogenously from the model alone or exogenously from mixing of distinct statistical populations in the data set. We show, using both theoretical analysis and case demonstrations, that model estimates derived from typical survival data may not be able to distinguish between individuals in the cure rate fraction who are not susceptible to failure and those who may be susceptible to failure but escape the fate by chance. The ambiguity is aggravated by right censoring of survival times and by minor misspecifications of the model. Slightly incorrect specifications for regression functions or for the stochastic process can lead to problems with model identification and estimation. In this situation, additional guidance for estimating the fraction of non-susceptibles must come from subject matter expertise or from data types other than survival times, censored or otherwise. The identifiability issue is confronted directly in threshold regression but is also present when applying other kinds of models commonly used for survival data analysis. Other methods, however, usually do not provide a framework for recognizing or dealing with the issue and so the issue is often unintentionally ignored. The theoretical foundations of this work are set out, which presents new and somewhat surprising results for the first hitting time distributions of Lévy processes that have the Esscher property.

Model-assisted estimators for time-to-event data from complex surveys

We develop model-assisted estimators for complex survey data for the proportion of a population that experienced some event by a specified time t. Theory for the new estimators uses time-to-event models as the underlying framework but have both good model-based and design-based properties. The estimators are compared in a simulation to traditional survey estimation methods and are also applied to a study of nurses' health. The new estimators take advantage of covariates predictive of the event and reduce standard errors compared to conventional alternatives.

Variable Selection in Threshold Regression Model with Applications to HIV Drug Adherence Data

The threshold regression model is an effective alternative to the Cox proportional hazards regression model when the proportional hazards assumption is not met. This paper considers variable selection for threshold regression. This model has separate regression functions for the initial health status and the speed of degradation in health. This flexibility is an important advantage when considering relevant risk factors for a complex time-to-event model where one needs to decide which variables should be included in the regression function for the initial health status, in the function for the speed of degradation in health, or in both functions. In this paper, we extend the broken adaptive ridge (BAR) method, originally designed for variable selection for one regression function, to simultaneous variable selection for both regression functions needed in the threshold regression model. We establish variable selection consistency of the proposed method and asymptotic normality of the estimator of non-zero regression coefficients. Simulation results show that our method outperformed threshold regression without variable selection and variable selection based on the Akaike information criterion. We apply the proposed method to data from an HIV drug adherence study in which electronic monitoring of drug intake is used to identify risk factors for non- adherence.

Estimating dose-response for time to remission with instrumental variable adjustment: the obscuring effects of drug titration in Genome Based Therapeutic Drugs for Depression Trial (GENDEP): clinical trial data

Background

Threshold regression, in which time to remission is modelled as a stochastic drift towards a boundary, is an alternative to the proportional hazards survival model and has a clear conceptual mechanism for examining the effects of drug dose. However, for both threshold regression and proportional hazard models, when dose titration occurs during treatment, the estimated causal effect of dose can be biased by confounding. An instrumental variable analysis can be used to minimise such bias.

Method

Weekly antidepressant dose was measured in 380 men and women with major depression treated with escitalopram or nortriptyline for 12 weeks as part of the Genome Based Therapeutic Drugs for Depression (GENDEP) study. The averaged dose relative to maximum prescribing dose was calculated from the 12 trial weeks and tested for association with time to depression remission. We combined the instrumental variable approach, utilising randomised treatment as an instrument, with threshold regression and proportional hazard survival models.

Results

The threshold model was constructed with two linear predictors. In the naïve models, averaged daily dose was not associated with reduced time to remission. By contrast, the instrumental variable analyses showed a clear and significant relationship between increased dose and faster time to remission, threshold regression (velocity estimate: 0.878, 95% confidence interval [CI]: 0.152–1.603) and proportional hazards (log hazards ratio: 3.012, 95% CI: 0.086–5.938).

Conclusions

We demonstrate, using the GENDEP trial, the benefits of these analyses to estimate causal parameters rather than those that estimate associations. The results for the trial dataset show the link between antidepressant dose and time to depression remission. The threshold regression model more clearly distinguishes the factors associated with initial severity from those influencing treatment effect. Additionally, applying the instrumental variable estimator provides a more plausible causal estimate of drug dose on treatment effect. This validity of these results is subject to meeting the assumptions of instrumental variable analyses.

Trial registration

EudraCT, 2004–001723-38; ISRCTN, 03693000. Registered on 27 September 2007.

Dynamic Prediction in Clinical Survival Analysis Using Temporal Convolutional Networks

Accurate prediction of disease trajectories is critical for early identification and timely treatment of patients at risk. Conventional methods in survival analysis are often constrained by strong parametric assumptions and limited in their ability to learn from high-dimensional data. This paper develops a novel convolutional approach that addresses the drawbacks of both traditional statistical approaches as well as recent neural network models for survival. We present Match-Net: a missingness-aware temporal convolutional hitting-time network, designed to capture temporal dependencies and heterogeneous interactions in covariate trajectories and patterns of missingness. To the best of our knowledge, this is the first investigation of temporal convolutions in the context of dynamic prediction for personalized risk prognosis. Using real-world data from the Alzheimer's disease neuroimaging initiative, we demonstrate state-of-the-art performance without making any assumptions regarding underlying longitudinal or time-to-event processes-attesting to the model's potential utility in clinical decision support.

Using threshold regression to analyze survival data from complex surveys: With application to mortality linked NHANES III Phase II genetic data

The Cox proportional hazards (PH) model is a common statistical technique used for analyzing time-to-event data. The assumption of PH, however, is not always appropriate in real applications. In cases where the assumption is not tenable, threshold regression (TR) and other survival methods, which do not require the PH assumption, are available and widely used. These alternative methods generally assume that the study data constitute simple random samples. In particular, TR has not been studied in the setting of complex surveys that involve (1) differential selection probabilities of study subjects and (2) intracluster correlations induced by multistage cluster sampling. In this paper, we extend TR procedures to account for complex sampling designs. The pseudo-maximum likelihood estimation technique is applied to estimate the TR model parameters. Computationally efficient Taylor linearization variance estimators that consider both the intracluster correlation and the differential selection probabilities are developed. The proposed methods are evaluated by using simulation experiments with various complex designs and illustrated empirically by using mortality-linked Third National Health and Nutrition Examination Survey Phase II genetic data.

A threshold regression model to predict return to work after traumatic limb injury

OBJECTIVE

The study aims to examine the severity of initial impairment and recovery rate of return-to-work (RTW) predictors among workers with traumatic limb injury.

METHODS

This 2-year prospective cohort study recruited 1124 workers with traumatic limb injury during the first 2 weeks of hospital admission. Baseline data were obtained by questionnaire and chart review. Patient follow-up occurred at 1, 3, 6, 12, 18, and 24 months post injury. The primary outcome was the time of first RTW. The impact of potential predictors on initial impairment and rate of recovery towards RTW was estimated by threshold regression (TR).

RESULTS

A total of 846 (75.27%) participants returned to work during the follow-up period. Our model revealed that the initial impairment level in elderly workers and lower limb injuries were 33% and 35% greater than their counterparts, respectively. Workers with >12 years of education, part-time job, and moderate and higher self-efficacy were less impaired at initial injury compared with their counterparts. In terms of the rate of recovery leading to RTW, workers with older age, part-time jobs, lower limbs, or combined injuries had a significantly slower recovery rate, while workers with 9-12 years of education and >12 years of education had a significantly faster recovery rate.

CONCLUSIONS

Our study provides researchers and clinicians with evidence to understand the baseline impairment and rate of recovery towards RTW by explaining the predictors of RTW among workers with traumatic limb injuries.

A model for time to fracture with a shock stream superimposed on progressive degradation: the Study of Osteoporotic Fractures

Osteoporotic hip fractures in the elderly are associated with a high mortality in the first year following fracture and a high incidence of disability among survivors. We study first and second fractures of elderly women using data from the Study of Osteoporotic Fractures. We present a new conceptual framework, stochastic model, and statistical methodology for time to fracture. Our approach gives additional insights into the patterns for first and second fractures and the concomitant risk factors. Our modeling perspective involves a novel time-to-event methodology called threshold regression, which is based on the plausible idea that many events occur when an underlying process describing the health or condition of a person or system encounters a critical boundary or threshold for the first time. In the parlance of stochastic processes, this time to event is a first hitting time of the threshold. The underlying process in our model is a composite of a chronic degradation process for skeletal health combined with a random stream of shocks from external traumas, which taken together trigger fracture events.

Analysis of the effect of occupational exposure to asbestos based on threshold regression modeling of case-control data

We analyze the effect of occupational exposure to asbestos on the occurrence of lung cancer based on a recent French case-control (CC) study. We build a large collection of threshold regression models, data-adaptively select a better model by CC-weighted likelihood-based cross-validation and then fit this better model by CC-weighted maximum likelihood. The CC-weighting allows us to draw valid inferences from CC data without relying on a logistic regression. This is possible because the joint distribution of the indicator of being a case and matching variable is available beforehand owing to two studies independent from our data set. The implications of the fitted model in terms of years of life free of lung cancer lost due to the exposure to asbestos are discussed.

Constructing normalcy and discrepancy indexes for birth weight and gestational age using a threshold regression mixture model

Birth weight and gestational age are important measures of a newborn's intrinsic health, serving both as outcome measures and explanatory variables in health studies. The measures are highly correlated but occasionally inconsistent. We anticipate that health researchers and other scientists would be helped by summary indexes of birth weight and gestational age that give more precise indications of whether the birth outcome is healthy or not. We propose a pair of indexes that we refer to as the birth normalcy index or BNI and birth discrepancy index or BDI. Both indexes are simple functions of birth weight and gestational age and in logarithmic form are orthogonal by construction. The BNI gauges whether the birth weight and gestational age combination are in a normal range. The BDI gauges whether birth weight and gestational age are consistent. We present a three-component mixture model for BNI, with the components representing premature, at-risk, and healthy births. The BNI distribution is derived from a stochastic model of fetal development proposed by Whitmore and Su (2007, Lifetime Data Analysis 13, 161-190) and takes the form of a mixture of inverse Gaussian distributions. We present a noncentral t-distribution as a model for BDI. BNI and BDI are also well suited for making comparisons of birth outcomes in different reference populations. A simple z-score and t-score are proposed for such comparisons. The BNI and BDI distributions can be estimated for births in any reference population of interest using threshold regression.

A threshold regression model for recurrent exacerbations in chronic obstructive pulmonary disease

OBJECTIVE

Respiratory exacerbations are a major source of morbidity in patients with chronic obstructive pulmonary disease (COPD). In this article, we model COPD health status as a formal stochastic process. A successful model will provide a suitable statistical structure for analysis of the effects of medical interventions on a patient's health status, and, possibly, offer new insights into the underlying disease process.

STUDY DESIGN AND SETTING

Our approach uses a regression methodology for time-to-event data called threshold regression (TR). We test the methodology on COPD data from a randomized clinical trial. Two TR models are studied: one based on a Poisson process and the other, a Wiener diffusion process.

RESULTS

Both models provide reasonably accurate fits to the clinical trial data. The insights offered by the fitted models are interpreted. Analysis of the clinical trial data set using these TR models revealed that patients who experienced multiple exacerbations showed a progressive acceleration in rate of exacerbations, and successive shortening of stable intervals between exacerbations.

CONCLUSION

TR techniques allow for realistic modeling of the COPD health state. A hybrid Poisson/Wiener diffusion TR model that incorporates the causal determinants of disease operating in each patient may be preferable.

Threshold regression for survival data with time-varying covariates

Time-to-event data with time-varying covariates pose an interesting challenge for statistical modeling and inference, especially where the data require a regression structure but are not consistent with the proportional hazard assumption. Threshold regression (TR) is a relatively new methodology based on the concept that degradation or deterioration of a subject's health follows a stochastic process and failure occurs when the process first reaches a failure state or threshold (a first-hitting-time). Survival data with time-varying covariates consist of sequential observations on the level of degradation and/or on covariates of the subject, prior to the occurrence of the failure event. Encounters with this type of data structure abound in practical settings for survival analysis and there is a pressing need for simple regression methods to handle the longitudinal aspect of the data. Using a Markov property to decompose a longitudinal record into a series of single records is one strategy for dealing with this type of data. This study looks at the theoretical conditions for which this Markov approach is valid. The approach is called threshold regression with Markov decomposition or Markov TR for short. A number of important special cases, such as data with unevenly spaced time points and competing risks as stopping modes, are discussed. We show that a proportional hazards regression model with time-varying covariates is consistent with the Markov TR model. The Markov TR procedure is illustrated by a case application to a study of lung cancer risk. The procedure is also shown to be consistent with the use of an alternative time scale. Finally, we present the connection of the procedure to the concept of a collapsible survival model.