Threshold regression for survival data with time-varying covariates

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.

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.

A New Sight of Influencing Effects of Major Factors on Cd Transfer from Soil to Wheat (Triticum aestivum L.): Based on Threshold Regression Model

Due to the high toxicity and potential health risk of cadmium (Cd), the influencing effects of major factors (like pH, OM, and clay, etc.) on Cd bioaccumulation and transfer from soil to crop grains are highly concerned. Multiple linear regression models were usually applied in previous literature, but these linear models could not reflect the threshold effects of major factors on Cd transfer under different soil environmental conditions. Soil pH and other factors on Cd transfer in a soil-plant system might pose different or even contrary effects under different soil Cd exposure levels. For this purpose, we try to apply a threshold regression model to analyze the effects of key soil parameters on Cd bioaccumulation and transfer from soil to wheat. The results showed that under different soil pH or Cd levels, several factors, including soil pH, organic matter, exchangeable Cd, clay, P, Zn, and Ca showed obvious threshold effects, and caused different or even contrary impacts on Cd bioaccumulation in wheat grains. Notably, the increase of soil pH inhibited Cd accumulation when pH > 7.98, but had a promotional effect when pH ≤ 7.98. Thus, threshold regression analysis could provide a new insight that can lead to a more integrated understanding of the relevant factors on Cd accumulation and transfer from soil to wheat. In addition, it might give us a new thought on setting regulatory limits on Cd contents in wheat grains, or the inhibitory factors of Cd transfer.

Overview of Equipment Health State Estimation and Remaining Life Prediction Methods

Health state estimation can quantitatively evaluate the current degradation state of equipment, and remaining life prediction can quantitatively predict the remaining service time of equipment. These two technologies can provide a basis for condition-based maintenance and predictive maintenance of equipment, respectively. In recent years, a large amount of research has been implemented in these two technologies. However, there is not any systematic review that covers these two technologies, and their engineering applications, comprehensively. To fill the gap, this paper makes a comparative analysis of existing health state estimation and remaining life prediction methods, and details the characteristics and limitations of various methods. The engineering applications of these two methods are summarized, and their applicable objects are briefly given. Finally, these two methods are summarized, and their feasibility for engineering application is discussed. This work provides guidance for the selection of industrial equipment health assessment and remaining life prediction methods.

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.

Estimating time to event characteristics via longitudinal threshold regression models - an application to cervical dilation progression

In longitudinal studies, it is sometimes of interest to estimate the distribution of the time a longitudinal process takes to traverse from one threshold to another. For example, the distribution of the time it takes a woman's cervical dilation to progress from 3 to 4 cm can aid the decision-making of obstetricians as to whether a stalled labor should be allowed to proceed or stopped in favor of other options. Often researchers treat this type of data structure as interval censored and employ traditional survival analysis methods. However, the traditional interval censoring approaches are inefficient in that they do not use all of the available data. In this paper, we propose utilizing a longitudinal threshold model to estimate the distribution of the elapsed time between two thresholds of the longitudinal process from repeated measurements. We extend this modeling framework to be used with multiple thresholds. A Wiener process under the first hitting time framework is used to represent survival distribution. We demonstrate our model through simulation studies and an analysis of data from the Consortium on Safe Labor study. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.

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.

Parameter inference from hitting times for perturbed Brownian motion

A latent internal process describes the state of some system, e.g. the social tension in a political conflict, the strength of an industrial component or the health status of a person. When this process reaches a predefined threshold, the process terminates and an observable event occurs, e.g. the political conflict finishes, the industrial component breaks down or the person dies. Imagine an intervention, e.g., a political decision, maintenance of a component or a medical treatment, is initiated to the process before the event occurs. How can we evaluate whether the intervention had an effect? To answer this question we describe the effect of the intervention through parameter changes of the law governing the internal process. Then, the time interval between the start of the process and the final event is divided into two subintervals: the time from the start to the instant of intervention, denoted by S, and the time between the intervention and the threshold crossing, denoted by R. The first question studied here is: What is the joint distribution of (S,R)? The theoretical expressions are provided and serve as a basis to answer the main question: Can we estimate the parameters of the model from observations of S and R and compare them statistically? Maximum likelihood estimators are calculated and applied on simulated data under the assumption that the process before and after the intervention is described by the same type of model, i.e. a Brownian motion, but with different parameters. Also covariates and handling of censored observations are incorporated into the statistical model, and the method is illustrated on lung cancer data.

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.

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.

A statistical model to predict one-year risk of death in patients with cystic fibrosis

OBJECTIVES

We constructed a statistical model to assess the risk of death for cystic fibrosis (CF) patients between scheduled annual clinical visits. Our model includes a CF health index that shows the influence of risk factors on CF chronic health and on the severity and frequency of CF exacerbations.

STUDY DESIGN AND SETTING

Our study used Canadian CF registry data for 3,794 CF patients born after 1970. Data up to 2010 were analyzed, yielding 44,390 annual visit records. Our stochastic process model postulates that CF health between annual clinical visits is a superposition of chronic disease progression and an exacerbation shock stream. Death occurs when an exacerbation carries CF health across a critical threshold. The data constitute censored survival data, and hence, threshold regression was used to connect CF death to study covariates. Maximum likelihood estimates were used to determine which clinical covariates were included within the regression functions for both CF chronic health and CF exacerbations.

RESULTS

Lung function, Pseudomonas aeruginosa infection, CF-related diabetes, weight deficiency, pancreatic insufficiency, and the deltaF508 homozygous mutation were significantly associated with CF chronic health status. Lung function, age, gender, age at CF diagnosis, P aeruginosa infection, body mass index <18.5, number of previous hospitalizations for CF exacerbations in the preceding year, and decline in forced expiratory volume in 1 second in the preceding year were significantly associated with CF exacerbations. When combined in one summative model, the regression functions for CF chronic health and CF exacerbation risk provided a simple clinical scoring tool for assessing 1-year risk of death for an individual CF patient. Goodness-of-fit tests of the model showed very encouraging results. We confirmed predictive validity of the model by comparing actual and estimated deaths in repeated hold-out samples from the data set and showed excellent agreement between estimated and actual mortality.

CONCLUSION

Our threshold regression model incorporates a composite CF chronic health status index and an exacerbation risk index to produce an accurate clinical scoring tool for prediction of 1-year survival of CF patients. Our tool can be used by clinicians to decide on optimal timing for lung transplant referral.

Longitudinal multistage model for lung cancer incidence, mortality, and CT detected indolent and aggressive cancers

It is currently not known whether most lung cancers detected by computerized tomography (CT) screening are aggressive and likely to be fatal if left untreated, or if a sizable fraction are indolent and unlikely to cause death during the natural lifetime of the individual. We developed a longitudinal biologically-based model of the relationship between individual smoking histories and the probability for lung cancer incidence, CT screen detection, lung cancer mortality, and other-cause mortality. The longitudinal model relates these different outcomes to an underlying lung cancer disease pathway and an effective other-cause mortality pathway, which are both influenced by the individual smoking history. The longitudinal analysis provides additional information over that available if these outcomes were analyzed separately, including testing if the number of CT detected and histologically-confirmed lung cancers is consistent with the expected number of lung cancers "in the pipeline". We assume indolent nodules undergo Gompertz growth and are detectable by CT, but do not grow large enough to contribute significantly to symptom-based lung cancer incidence or mortality. Likelihood-based model calibration was done jointly to data from 6878 heavy smokers without asbestos exposure in the control (placebo) arm of the Carotene and Retinol Efficacy Trial (CARET); and to 3,642 heavy smokers with comparable smoking histories in the Pittsburgh Lung Screening Study (PLuSS), a single-arm prospective trial of low-dose spiral CT screening for diagnosis of lung cancer. Model calibration was checked using data from two other single-arm prospective CT screening trials, the New York University Lung Cancer Biomarker Center (NYU) (n=1,021), and Moffitt Cancer Center (Moffitt) cohorts (n=677). In the PLuSS cohort, we estimate that at the end of year 2, after the baseline and first annual CT exam, that 33.0 (26.9, 36.9)% of diagnosed lung cancers among females and 7.0 (4.9,11.7)% among males were overdiagnosed due to being indolent cancers. At the end of the PLuSS study, with maximum follow-up of 5.8 years, we estimate that due to early detection by CT and limited follow-up, an additional 2.2 (2.0,2.4)% of all diagnosed cancers among females and 7.1 (6.7,8.0)% among males would not have been diagnosed in the absence of CT screening. We also find a higher apparent cure rate for lung cancer among CARET females than males, consistent with the larger indolent fraction of CT detected and histologically confirmed lung cancers among PLuSS females. This suggests that there are significant gender differences in the aggressiveness of lung cancer. Females may have an inherently higher proportion of indolent lung cancers than males, or aggressive lung cancers may be brought into check by the immune system more frequently among females than males.

Recurrent first hitting times in Wiener diffusion under several observation schemes

Recurrent events are commonly encountered in the natural sciences, engineering, and medicine. The theory of renewal and regenerative processes provides an elegant mathematical foundation for idealized recurrent event processes. In real-world applications, however, the contexts tend to be complicated by a variety of practical intricacies, including observation schemes with different phase and data structures. This paper formulates a recurrent event process as a succession of independent and identically distributed first hitting times for a Wiener sample path as it passes through successive equally-spaced levels. We develop exact mathematical results for statistical inferences based on several observation schemes that include observation initiated at a renewal point, observation of a stationary process over a finite window, and other variants. We also consider inferences drawn from different data structures, including gap times between renewal points (or fragments thereof) and counts of renewal events occurring within an observation window. We explore the precision of estimates using simulated scenarios and develop empirical regression functions for planning the sample size of a recurrent event study. We demonstrate our results using data from a clinical trial for chronic obstructive pulmonary disease in which the recurrent events are successive exacerbations of the condition. The case study demonstrates how covariates can be incorporated into the analysis 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.

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.