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.

Patient and physician factors associated with symptomatic undiagnosed asthma or COPD.

BACKGROUND: It remains unclear why some symptomatic individuals with asthma or COPD remain undiagnosed. Here, we compare patient and physician characteristics between symptomatic individuals with obstructive lung disease (OLD) who are undiagnosed and individuals with physician-diagnosed OLD. METHODS: Using random-digit dialling and population-based case finding, we recruited 451 participants with symptomatic undiagnosed OLD and 205 symptomatic control participants with physician-diagnosed OLD. Data on symptoms, quality of life and healthcare utilisation were analysed. We surveyed family physicians of participants in both groups to elucidate differences in physician practices that could contribute to undiagnosed OLD. RESULTS: Participants with undiagnosed OLD had lower mean pre-bronchodilator forced expiratory volume in 1 s percentage predicted compared with those who were diagnosed (75.2% versus 80.8%; OR 0.975, 95% CI 0.963-0.987). They reported greater psychosocial impacts due to symptoms and worse energy and fatigue than those with diagnosed OLD. Undiagnosed OLD was more common in participants whose family physicians were practising for >15 years and in those whose physicians reported that they were likely to prescribe respiratory medications without doing spirometry. Undiagnosed OLD was more common among participants who had never undergone spirometry (OR 10.83, 95% CI 6.18-18.98) or who were never referred to a specialist (OR 5.92, 95% CI 3.58-9.77). Undiagnosed OLD was less common among participants who had required emergency department care (OR 0.44, 95% CI 0.20-0.97). CONCLUSIONS: Individuals with symptomatic undiagnosed OLD have worse pre-bronchodilator lung function and present with greater psychosocial impacts on quality of life compared with their diagnosed counterparts. They were less likely to have received appropriate investigations and specialist referral for their respiratory symptoms.

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.

Disease burden in individuals with symptomatic undiagnosed asthma or COPD.

BACKGROUND: The actual burden of COPD and asthma may be much higher than appreciated, since a large proportion of individuals are not diagnosed. Our study objective was to compare health care utilization, burden of symptoms and quality of life in subjects with self-reported respiratory symptoms who were subsequently found to have undiagnosed airflow obstruction compared to those having no airflow obstruction. METHODS: This cross-sectional case-finding study used data from the Undiagnosed COPD and Asthma Population (UCAP) study. Adult subjects with respiratory symptoms who had no history of diagnosed lung disease were recruited in a two-step case-finding process using random digit-dialling of land lines and cell phones located within a 90-min radius of 16 Canadian study sites. Participants were assessed for COPD, asthma or no airflow obstruction using pre- and post-bronchodilator spirometry based on American Thoracic Society diagnostic criteria. RESULTS: 1660 participants were recruited, of these 1615 had adequate spirometry and 331 (20.5%) subjects met spirometry criteria for undiagnosed asthma or COPD. Subjects with undiagnosed asthma or COPD had increased respiratory symptoms as assessed by the COPD Assessment Test (CAT), and higher St. George's Respiratory Questionnaire (SGRQ) scores indicating worse health-related quality of life, compared to subjects with no airflow obstruction. No between-group differences were found in health care utilization or work or school absenteeism. CONCLUSION: Undiagnosed asthma and COPD are common in Canadian adults experiencing breathing problems and are associated with a greater burden of symptoms and poorer health-related quality of life. These results suggest that patients may benefit from early identification and treatment of undiagnosed asthma and COPD.

Derivation and validation of the UCAP-Q case-finding questionnaire to detect undiagnosed asthma and COPD.

BACKGROUND: Many people with asthma and COPD remain undiagnosed. We developed and validated a new case-finding questionnaire to identify symptomatic adults with undiagnosed obstructive lung disease. METHODS: Adults in the community with no prior history of physician-diagnosed lung disease who self-reported respiratory symptoms were contacted via random-digit dialling. Pre- and post-bronchodilator spirometry was used to confirm asthma or COPD. Predictive questions were selected using multinomial logistic regression with backward elimination. Questionnaire performance was assessed using sensitivity, predictive values and area under the receiver operating characteristic curve (AUC). The questionnaire was assessed for test-retest reliability, acceptability and readability. External validation was prospectively conducted in an independent sample and predictive performance re-evaluated. RESULTS: A 13-item Undiagnosed COPD and Asthma Population Questionnaire (UCAP-Q) case-finding questionnaire to predict undiagnosed asthma or COPD was developed. The most appropriate risk cut-off was determined to be 6% for either disease. Applied to the derivation sample (n=1615), the questionnaire yielded a sensitivity of 92% for asthma and 97% for COPD; specificity of 17%; and an AUC of 0.69 (95% CI 0.64-0.74) for asthma and 0.82 (95% CI 0.78-0.86) for COPD. Prospective validation using an independent sample (n=471) showed sensitivities of 93% and 92% for asthma and COPD, respectively; specificity of 19%; with AUCs of 0.70 (95% CI 0.62-0.79) for asthma and 0.81 (95% CI 0.74-0.87) for COPD. AUCs for UCAP-Q were higher compared to AUCs for currently recommended case-finding questionnaires for asthma or COPD. CONCLUSIONS: The UCAP-Q demonstrated high sensitivities and AUCs for identifying undiagnosed asthma or COPD. A web-based calculator allows for easy calculation of risk probabilities for each disease.

Placebo Effects in Clinical Trials Evaluating Patients with Uncontrolled Persistent Asthma.

Rationale: Patients with uncontrolled, persistent asthma can show substantial health improvements when administered placebo.Objectives: We analyzed five randomized, placebo-controlled clinical trials that assessed subjects with uncontrolled, persistent asthma to determine the magnitudes of placebo effects across different clinical outcomes.Methods: Placebo effects for objective asthma-related outcomes, healthcare utilization outcomes, and patient-reported outcomes were estimated, with adjustments for regression to the mean.Results: Statistically significant improvements in all clinical outcomes were seen in patients randomized to placebo across all trials. Placebo effects were largest for healthcare utilization outcomes, including exacerbations (median reduction, 0.44 events/yr; 31% risk reduction; range, 19-56%), emergency department visits (median reduction, 0.19 events/yr; 50% risk reduction; range, 36-82%), and hospitalizations for asthma (median reduction, 0.26 events/yr; 66% risk reduction; range, 61-74%). Patient-reported outcomes exhibited intermediate placebo effects. Median improvements in the Asthma Control Questionnaire and St. George's Respiratory Questionnaire scores in placebo-treated patients were 0.53 units (25% improvement; range, 18-30%) and 8.3 units (19.5% improvement; range 19-20%), respectively. Forced expiratory volume in 1 second exhibited the smallest relative placebo effects (median increase, 77 ml; 4.2% improvement; range, 3.4-4.9%). Subgroup analyses did not reveal patient subgroups that were more susceptible to placebo effects. Pre- and postrandomization counts for asthma exacerbations showed patterns consistent with the expected negative binomial distribution except for significant departures in prerandomization exacerbations for two trials.Conclusions: Patients with uncontrolled asthma derived consistent benefit from randomization to placebo. Observed placebo effects may represent beneficial effects of both sham therapy and a structured asthma regimen dictated by the study protocol. In the case of healthcare utilization outcomes, recall errors in self-reported healthcare events may have introduced biases that inflated placebo effect estimates.

Influence of country-level differences on COPD prevalence.

PURPOSE: Studies suggest that COPD prevalence may vary between countries. We conducted an ecological study of data from COPD prevalence articles to assess the influence of differences in country-level risk factors on COPD prevalence. PATIENTS AND METHODS: Our study covered English language articles published during 2003-2014. Qualified articles used spirometry to assess COPD prevalence and used representative samples from national or subnational populations. Stepwise binomial regression was used to analyze associations between study- and country-level factors and COPD prevalence. RESULTS: Eighty articles provided 1,583 measures of COPD prevalence for subjects in different sex, age, and smoking categories for 112 districts in 41 countries. Adjusted prevalence rates for COPD were significantly lower for Australia/New Zealand and the Mediterranean and significantly higher for Latin America, compared to North America, Southeast Asia, and Northern Europe. Country-level socioeconomic development variables had an uneven and mixed association with COPD prevalence. High elevation above sea level was shown to be a protective factor for COPD. Study-level variables for the established risk factors of sex, age, and smoking explained 64% of variability in COPD prevalence. Country-level risk factors raised the explanatory power to 72%. Approximately 28% of worldwide variability in COPD prevalence remained unexplained. CONCLUSION: Our study suggests that COPD prevalence varies across world regions, even after adjustment for established risk factors. Major country-level risk factors contributing to the worldwide epidemic of COPD remain to be investigated.

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.

A contemporary survival analysis of individuals with cystic fibrosis: a cohort study.

Previously established predictors of survival may no longer apply in the current era of cystic fibrosis (CF) care. Our objective was to identify risk factors associated with survival in a contemporary CF population. We used the Canadian CF Registry, a population-based cohort, to calculate median age of survival and summarise patient characteristics from 1990 to 2012. Clinical, demographic and geographical factors, and survival were estimated for a contemporary cohort (2000-2012) using Cox proportional hazards models. There were 5787 individuals in the registry between 1990 and 2012. Median survival age increased from 31.9 years (95% CI 28.3-35.2 years) in 1990 to 49.7 years (95% CI 46.1-52.2 years) in the most current 5-year window ending in 2012. Median forced expiratory volume in 1 s improved (p=0.04) and fewer subjects were malnourished (p<0.001) over time. Malnourished patients (hazard ratio (HR) 2.1, 95% CI 1.6-2.8), those with multiple exacerbations (HR 4.5, 95% CI 3.2-6.4) and women with CF-related diabetes (HR 1.8, 95% CI 1.2-2.7) were at increased risk of death. Life expectancy in Canadians with CF is increasing. Modifiable risk factors such as malnutrition and pulmonary exacerbations are associated with an increased risk of death. The sex gap in CF survival may be explained by an increased hazard for death in women with CF-related diabetes.

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.

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.

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.

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

In epidemiological and clinical studies, time-to-event data often violate the assumptions of Cox regression due to the presence of time-dependent covariate effects and unmeasured risk factors. An alternative approach, which does not require proportional hazards, is to use a first hitting time model which treats a subject's health status as a latent stochastic process that fails when it reaches a threshold value. Although more flexible than Cox regression, existing methods do not account for unmeasured covariates in both the initial state and the rate of the process. To address this issue, we propose a Bayesian methodology that models an individual's health status as a Wiener process with subject-specific initial state and drift. Posterior inference proceeds via a Markov chain Monte Carlo methodology with data augmentation steps to sample the final health status of censored observations. We apply our method to data from melanoma patients with nonproportional hazards and find interesting differences from a similar model without random effects. In a simulation study, we show that failure to account for unmeasured covariates can lead to inaccurate estimates of survival probabilities.

A case-control study relating railroad worker mortality to diesel exhaust exposure using a threshold regression model.

A case-control study of lung cancer mortality in U.S. railroad workers in jobs with and without diesel exhaust exposure is reanalyzed using a new threshold regression methodology. The study included 1256 workers who died of lung cancer and 2385 controls who died primarily of circulatory system diseases. Diesel exhaust exposure was assessed using railroad job history from the US Railroad Retirement Board and an industrial hygiene survey. Smoking habits were available from next-of-kin and potential asbestos exposure was assessed by job history review. The new analysis reassesses lung cancer mortality and examines circulatory system disease mortality. Jobs with regular exposure to diesel exhaust had a survival pattern characterized by an initial delay in mortality, followed by a rapid deterioration of health prior to death. The pattern is seen in subjects dying of lung cancer, circulatory system diseases, and other causes. The unique pattern is illustrated using a new type of Kaplan-Meier survival plot in which the time scale represents a measure of disease progression rather than calendar time. The disease progression scale accounts for a healthy-worker effect when describing the effects of cumulative exposures on mortality.

A threshold regression mixture model for assessing treatment efficacy in a multiple myeloma clinical trial.

A first-hitting-time (FHT) survival model postulates a health status process for a patient that gradually declines until the patient dies when the level first reaches a critical threshold. Threshold regression (TR) is a new regression methodology that incorporates the effects of covariates on the threshold and process parameters of this FHT model. In this study, we use TR to analyze data from a randomized clinical trial of treatment for multiple myeloma. The trial compares VELCADE and high-dose dexamethasone, the former a new therapy and the latter an established therapy for this disease. Patients are switched between the two drugs based on patient response. The novel contribution of this work is the modeling of this clinical trial design using a mixture of TR models. Specifically, we propose a mixture FHT model to fit the survival distribution. The model includes a composite time scale that differentiates the rate of disease progression before and after switching. The analysis shows significant benefit from initial treatment by VELCADE. A comparison is made with a Cox proportional hazards regression analysis of the same data.

Modeling low birth weights using threshold regression: results for U.S. birth data.

Babies born live under 2,500 g or with a gestational age under 37 weeks are often inadequately developed and have elevated risks of infant mortality, congenital malformations, mental retardation, and other physical and neurological impairments. In this paper, we model birth weight as a first hitting time (FHT) of a birthing boundary in a Wiener process representing fetal development. We associate the parameters of the process and boundary with covariates describing maternal characteristics and the birthing environment using a relatively new regression methodology called threshold regression. Two FHT models for birth weight are developed. One is a mixture model and the other a competing risks model. These models are tested in a case demonstration using a 4%-systematic sample of the more than four million live births in the United States in 2002. An extensive data set for these births was provided by the National Center for Health Statistics. The focus of this paper is on the conceptual framework, models and methodology. A full empirical study is deferred to a later occasion.

Nonparametric methods for microarray data based on exchangeability and borrowed power.

This article proposes nonparametric inference procedures for analyzing microarray gene expression data that are reliable, robust, and simple to implement. They are conceptually transparent and require no special-purpose software. The analysis begins by normalizing gene expression data in a unique way. The resulting adjusted observations consist of gene-treatment interaction terms (representing differential expression) and error terms. The error terms are considered to be exchangeable, which is the only substantial assumption. Thus, under a family null hypothesis of no differential expression, the adjusted observations are exchangeable and all permutations of the observations are equally probable. The investigator may use the adjusted observations directly in a distribution-free test method or use their ranks in a rank-based method, where the ranking is taken over the whole data set. For the latter, the essential steps are as follows: (1) Calculate a Wilcoxon rank-sum difference or a corresponding Kruskal-Wallis rank statistic for each gene. (2) Randomly permute the observations and repeat the previous step. (3) Independently repeat the random permutation a suitable number of times. Under the exchangeability assumption, the permutation statistics are independent random draws from a null cumulative distribution function (c.d.f) approximated by the empirical c.d.f Reference to the empirical c.d.f tells if the test statistic for a gene is outlying and, hence, shows differential expression. This feature is judged by using an appropriate rejection region or computing a p-value for each test statistic, taking into account multiple testing. The distribution-free analog of the rank-based approach is also available and has parallel steps which are described in the article. The proposed nonparametric analysis tends to give good results with no additional refinement, although a few refinements are presented that may interest some investigators. The implementation is illustrated with a case application involving differential gene expression in wild-type and knockout mice of an E. coli lipopolysaccharide (LPS) endotoxin treatment, relative to a baseline untreated condition.