Nonparametric estimation for cumulative duration of adverse events

Analysis of adverse events (AE) for drug safety assessment presents challenges to statisticians in observational studies as well as in clinical trials since AEs are typically recurrent with varying duration and severity. Routine analyses often concentrate on the number of patients who had at least one occurrence of a specific AE or a group of AEs, or the time to occurrence of the first event. We argue that other information in AE data particularly cumulative duration of events is also important, particularly for benefit-risk assessment. We propose a nonparametric method to estimate the mean cumulative duration (MCD) based on the nonparametric cumulative mean function estimate, together with a robust estimate for the variance of the estimate, as in Lawless and Nadeau (1995). This approach can be easily used to analyze multiple, overlapped and severity weighted AE durations. This method can also be used for estimating the difference between two MCDs. Estimation in the presence of censoring due to informative dropouts and/or a terminal event is also considered. The method can be implemented in standard softwares such as SAS. We illustrate the use of the method with a numerical example. Small sample properties of this approach are examined via simulation.     

Influence of trial duration on the bias of the estimated treatment effect in clinical trials when individual heterogeneity is ignored

In randomized clinical trials where the times to event of two treatment groups are compared under a proportional hazards assumption, it has been established that omitting prognostic factors from the model entails an underestimation of the hazards ratio. Heterogeneity due to unobserved covariates in cancer patient populations is a concern since genomic investigations have revealed molecular and clinical heterogeneity in these populations. In HIV prevention trials, heterogeneity is unavoidable and has been shown to decrease the treatment effect over time. This article assesses the influence of trial duration on the bias of the estimated hazards ratio resulting from omitting covariates from the Cox analysis. The true model is defined by including an unobserved random frailty term in the individual hazard that reflects the omitted covariate. Three frailty distributions are investigated: gamma, log‐normal, and binary, and the asymptotic bias of the hazards ratio estimator is calculated. We show that the attenuation of the treatment effect resulting from unobserved heterogeneity strongly increases with trial duration, especially for continuous frailties that are likely to reflect omitted covariates, as they are often encountered in practice. The possibility of interpreting the long‐term decrease in treatment effects as a bias induced by heterogeneity and trial duration is illustrated by a trial in oncology where adjuvant chemotherapy in stage 1B NSCLC was investigated.     

Double‐Robust Semiparametric Estimator for Differences in Restricted Mean Lifetimes in Observational Studies

Summary Restricted mean lifetime is often of direct interest in epidemiologic studies involving censored survival times. Differences in this quantity can be used as a basis for comparing several groups. For example, transplant surgeons, nephrologists, and of course patients are interested in comparing posttransplant lifetimes among various types of kidney transplants to assist in clinical decision making. As the factor of interest is not randomized, covariate adjustment is needed to account for imbalances in confounding factors. In this report, we use semiparametric theory to develop an estimator for differences in restricted mean lifetimes although accounting for confounding factors. The proposed method involves building working models for the time‐to‐event and coarsening mechanism (i.e., group assignment and censoring). We show that the proposed estimator possesses the double robust property; i.e., when either the time‐to‐event or coarsening process is modeled correctly, the estimator is consistent and asymptotically normal. Simulation studies are conducted to assess its finite‐sample performance and the method is applied to national kidney transplant data.     

Double label estimation of the mean duration of the S-phase

The theoretical values of the double label index and the estimator of the mean duration of the DNA-synthesis phase are obtained for the situation where a steady-state renewal cell population is subjected to two instantaneous labels. The stochastic model used to describe this situation allows for a correlation structure among the durations of the phases of the cell cycle. An explicit form for the double label index is presented for the case where the durations of the phases of the cell cycle are jointly distributed in accordance with a multivariate mixed gamma distribution. The effect of the length sampling bias and the time between label points on the theoretical value of the estimator is demonstrated.     

An estimator for treatment comparisons among survivors in randomized trials

In clinical trials of advanced-stage disease it is often of interest to perform treatment comparisons for endpoints which are defined only for survivors. Examples include time on ventilation in ventilation studies, change in quality of life in health-related quality-of-life studies, and duration of response to therapy in therapeutic trials. Randomized treatment comparisons for these endpoints cannot be performed because the outcomes are only defined in the nonrandomly selected subgroup of survivors. We propose a new evaluation of the survivor average causal effect (SACE) for treatment comparisons of this nature. We provide an estimator of SACE in the presence of no unmeasured confounders, a nontestable assumption, which identifies SACE. We also outline a sensitivity analysis for exploring robustness of conclusions to deviations from this assumption. We apply this method to duration of ventilation in a clinical trial of acute respiratory distress syndrome.     

A partially parametric estimator of survival in the presence of randomly censored data

Many biological or medical experiments have as their goal to estimate the survival function of a specified population of subjects when the time to the specified event may be censored due to loss to follow-up, the occurrence of another event that precludes the occurrence of the event of interest, or the study being terminated before the event of interest occurs. This paper suggests an improvement of the Kaplan-Meier product-limit estimator when the censoring mechanism is random. The proposed estimator treats the uncensored observations nonparametrically and uses a parametric model only for the censored observations. One version of this proposed estimator always has a smaller bias and mean squared error than the product-limit estimator. An example estimating the survival function of patients enrolled in the Ohio State University Bone Marrow Transplant Program is presented.     

Corrections for measurement error due to delayed onset of illness for case-crossover designs

Epidemiologic studies of the short-term effects of ambient particulate matter (PM) on the risk of acute cardiovascular or cerebrovascular events often use data from administrative databases in which only the date of hospitalization is known. A common study design for analyzing such data is the case-crossover design, in which exposure at a time when a patient experiences an event is compared to exposure at times when the patient did not experience an event within a case-control paradigm. However, the time of true event onset may precede hospitalization by hours or days, which can yield attenuated effect estimates. In this article, we consider a marginal likelihood estimator, a regression calibration estimator, and a conditional score estimator, as well as parametric bootstrap versions of each, to correct for this bias. All considered approaches require validation data on the distribution of the delay times. We compare the performance of the approaches in realistic scenarios via simulation, and apply the methods to analyze data from a Boston-area study of the association between ambient air pollution and acute stroke onset. Based on both simulation and the case study, we conclude that a two-stage regression calibration estimator with a parametric bootstrap bias correction is an effective method for correcting bias in health effect estimates arising from delayed onset in a case-crossover study.     

Inverse Probability of Censoring Weighted Estimates of Kendall's τ for Gap Time Analyses

Summary In life history studies, interest often lies in the analysis of the interevent, or gap times and the association between event times. Gap time analyses are challenging however, even when the length of follow‐up is determined independently of the event process, because associations between gap times induce dependent censoring for second and subsequent gap times. This article discusses nonparametric estimation of the association between consecutive gap times based on Kendall's τ in the presence of this type of dependent censoring. A nonparametric estimator that uses inverse probability of censoring weights is provided. Estimates of conditional gap time distributions can be obtained following specification of a particular copula function. Simulation studies show the estimator performs well and compares favorably with an alternative estimator. Generalizations to a piecewise constant Clayton copula are given. Several simulation studies and illustrations with real data sets are also provided.     

Exploratory analysis of longitudinal trials with staggered intervention times

Longitudinal trials involving surgical interventions commonly have subject-specific intervention times, due to constraints on the availability of surgeons and operating theatres. Moreover, the intervention often effects a discontinuous change in the mean response. We propose a nonparametric estimator for the mean response profile of longitudinal data with staggered intervention times and a discontinuity at the times of intervention, as an exploratory tool to assist the formulation of a suitable parametric model. We use an adaptation of the standard generalized additive model algorithm for estimation, with smoothing constants chosen by a cross-validation criterion. We illustrate the method using longitudinal data from a trial to assess the effect of lung resection surgery in the treatment of emphysema patients.     

Cox model with interval‐censored covariate in cohort studies

In cohort studies the outcome is often time to a particular event, and subjects are followed at regular intervals. Periodic visits may also monitor a secondary irreversible event influencing the event of primary interest, and a significant proportion of subjects develop the secondary event over the period of follow‐up. The status of the secondary event serves as a time‐varying covariate, but is recorded only at the times of the scheduled visits, generating incomplete time‐varying covariates. While information on a typical time‐varying covariate is missing for entire follow‐up period except the visiting times, the status of the secondary event are unavailable only between visits where the status has changed, thus interval‐censored. One may view interval‐censored covariate of the secondary event status as missing time‐varying covariates, yet missingness is partial since partial information is provided throughout the follow‐up period. Current practice of using the latest observed status produces biased estimators, and the existing missing covariate techniques cannot accommodate the special feature of missingness due to interval censoring. To handle interval‐censored covariates in the Cox proportional hazards model, we propose an available‐data estimator, a doubly robust‐type estimator as well as the maximum likelihood estimator via EM algorithm and present their asymptotic properties. We also present practical approaches that are valid. We demonstrate the proposed methods using our motivating example from the Northern Manhattan Study.     

Utilizing Propensity Scores to Estimate Causal Treatment Effects with Censored Time-Lagged Data

Observational studies frequently are conducted to compare long-term effects of treatments. Without randomization, patients receiving one treatment are not guaranteed to be prognostically comparable to those receiving another treatment. Furthermore, the response of interest may be right-censored because of incomplete follow-up. Statistical methods that do not account for censoring and confounding may lead to biased estimates. This article presents a method for estimating treatment effects in nonrandomized studies with right-censored responses. We review the assumptions required to estimate average causal effects and derive an estimator for comparing two treatments by applying inverse weights to the complete cases. The weights are determined according to the estimated probability of receiving treatment conditional on covariates and the estimated treatment-specific censoring distribution. By utilizing martingale representations, the estimator is shown to be asymptotically normal and an estimator for the asymptotic variance is derived. Simulation results are presented to evaluate the properties of the estimator. These methods are applied to an observational data set of acute coronary syndrome patients from Duke University Medical Center to estimate the effect of a treatment strategy on the mean 5-year medical cost.     

Semiparametric analysis of survival data with left truncation and dependent right censoring

Studies of chronic life-threatening diseases often involve both mortality and morbidity. In observational studies, the data may also be subject to administrative left truncation and right censoring. Because mortality and morbidity may be correlated and mortality may censor morbidity, the Lynden-Bell estimator for left-truncated and right-censored data may be biased for estimating the marginal survival function of the non-terminal event. We propose a semiparametric estimator for this survival function based on a joint model for the two time-to-event variables, which utilizes the gamma frailty specification in the region of the observable data. First, we develop a novel estimator for the gamma frailty parameter under left truncation. Using this estimator, we then derive a closed-form estimator for the marginal distribution of the non-terminal event. The large sample properties of the estimators are established via asymptotic theory. The methodology performs well with moderate sample sizes, both in simulations and in an analysis of data from a diabetes registry.     

Estimation of separable direct and indirect effects in continuous time

Many research questions involve time-to-event outcomes that can be prevented from occurring due to competing events. In these settings, we must be careful about the causal interpretation of classical statistical estimands. In particular, estimands on the hazard scale, such as ratios of cause-specific or subdistribution hazards, are fundamentally hard to interpret causally. Estimands on the risk scale, such as contrasts of cumulative incidence functions, do have a clear causal interpretation, but they only capture the total effect of the treatment on the event of interest; that is, effects both through and outside of the competing event. To disentangle causal treatment effects on the event of interest and competing events, the separable direct and indirect effects were recently introduced. Here we provide new results on the estimation of direct and indirect separable effects in continuous time. In particular, we derive the nonparametric influence function in continuous time and use it to construct an estimator that has certain robustness properties. We also propose a simple estimator based on semiparametric models for the two cause-specific hazard functions. We describe the asymptotic properties of these estimators and present results from simulation studies, suggesting that the estimators behave satisfactorily in finite samples. Finally, we reanalyze the prostate cancer trial from Stensrud et al. (2020).     

Graphical exploration of covariate effects on survival data through nonparametric quantile curves

Kaplan-Meier curves provide an effective means of presenting the distributional pattern in a sample of survival data. However, in order to assess the effect of a covariate, a standard scatterplot is often difficult to interpret because of the presence of censored observations. Several authors have proposed a running median as an effective way of indicating the effect of a covariate. This article proposes a form of kernel estimation, employing double smoothing, that can be applied in a simple and efficient manner to construct an estimator of a percentile of the survival distribution as a function of one or two covariates. Permutations and bootstrap samples can be used to construct reference bands that help identify whether particular features of the estimates indicate real features of the underlying curve or whether this may be due simply to random variation. The techniques are illustrated on data from a study of kidney transplant patients.     

Maximum likelihood estimation in the additive hazards model

The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As the non-parametric model, additive hazards offer a very flexible way of modeling time-varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper, we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non-negative for all covariate values in their domain. We show that the maximum likelihood estimator may be obtained by separately maximizing the log-likelihood contribution of each event time point, and we show that the maximizing problem is equivalent to fitting a series of Poisson regression models with an identity link under non-negativity constraints. We derive an analytic solution to the maximum likelihood estimator. We contrast the maximum likelihood estimator with the ordinary least-squares estimator in a simulation study and show that the maximum likelihood estimator has smaller mean squared error than the ordinary least-squares estimator. An illustration with data on patients with carcinoma of the oropharynx is provided.     

Study duration for clinical trials with survival response and early stopping rule

A comparative clinical trial with built-in sequential stopping rules allows earlier-than-scheduled stopping, should there be a significant indication of treatment difference. In a clinical trial where the major outcome is time (survival time or response) to a certain event such as failure, the design of the study should determine how long one needs to accrue patients and follow through until there is a sufficient number of events observed during the entire study duration. This paper proposes a unified design procedure for group sequential clinical trials with survival response. The time to event is assumed to be exponentially distributed, but the arguments extend naturally to the proportional hazards model after suitable transformation on the time scale. An example from the Eastern Cooperative Oncology Group (ECOG) is given to illustrate how this procedure can be implemented. The same example is used to explore the overall operating characteristics and the robustness of the proposed group sequential design.     

Natural history of tuberculosis: duration and fatality of untreated pulmonary tuberculosis in HIV negative patients: a systematic review

Background

The prognosis, specifically the case fatality and duration, of untreated tuberculosis is important as many patients are not correctly diagnosed and therefore receive inadequate or no treatment. Furthermore, duration and case fatality of tuberculosis are key parameters in interpreting epidemiological data.

Methodology and Principal Findings

To estimate the duration and case fatality of untreated pulmonary tuberculosis in HIV negative patients we reviewed studies from the pre-chemotherapy era. Untreated smear-positive tuberculosis among HIV negative individuals has a 10-year case fatality variously reported between 53% and 86%, with a weighted mean of 70%. Ten-year case fatality of culture-positive smear-negative tuberculosis was nowhere reported directly but can be indirectly estimated to be approximately 20%. The duration of tuberculosis from onset to cure or death is approximately 3 years and appears to be similar for smear-positive and smear-negative tuberculosis.

Conclusions

Current models of untreated tuberculosis that assume a total duration of 2 years until self-cure or death underestimate the duration of disease by about one year, but their case fatality estimates of 70% for smear-positive and 20% for culture-positive smear-negative tuberculosis appear to be satisfactory.     

Genotype accounts for intraspecific variation in the timing and duration of multiple,sequential life-cycle events in a willow species

Premise

Phenological variation among individuals within populations is common and has a variety of ecological and evolutionary consequences, including forming the basis for population-level responses to environmental change. Although the timing of life-cycle events has genetic underpinnings, whether intraspecific variation in the duration of life-cycle events reflects genetic differences among individuals is poorly understood.

Methods

We used a common garden experiment with 10 genotypes of Salix hookeriana (coastal willow) from northern California, United States to investigate the extent to which genetic variation explains intraspecific variation in the timing and duration of multiple, sequential life-cycle events: flowering, leaf budbreak, leaf expansion, fruiting, and fall leaf coloration. We used seven clones of each genotype, for a total of 70 individual trees.

Results

Genotype affected each sequential life-cycle event independently and explained on average 62% of the variation in the timing and duration of vegetative and reproductive life-cycle events. All events were significantly heritable. A single genotype tended to be “early” or “late” across life-cycle events, but for event durations, there was no consistent response within genotypes.

Conclusions

This research demonstrates that genetic variation can be a major component underlying intraspecific variation in the timing and duration of life-cycle events. It is often assumed that the environment affects durations, but we show that genetic factors also play a role. Because the timing and duration of events are independent of one another, our results suggest that the effects of environmental change on one event will not necessarily cascade to subsequent events.     

Climate change impacts on seabirds and marine mammals: The importance of study duration,thermal tolerance and generation time

Understanding climate change impacts on top predators is fundamental to marine biodiversity conservation, due to their increasingly threatened populations and their importance in marine ecosystems. We conducted a systematic review of the effects of climate change (prolonged, directional change) and climate variability on seabirds and marine mammals. We extracted data from 484 studies (4808 published studies were reviewed), comprising 2215 observations on demography, phenology, distribution, diet, behaviour, body condition and physiology. The likelihood of concluding that climate change had an impact increased with study duration. However, the temporal thresholds for the effects of climate change to be discernibly varied from 10 to 29 years depending on the species, the biological response and the oceanic study region. Species with narrow thermal ranges and relatively long generation times were more often reported to be affected by climate change. This provides an important framework for future assessments, with guidance on response- and region-specific temporal dimensions that need to be considered when reporting effects of climate change. Finally, we found that tropical regions and non-breeding life stages were poorly covered in the literature, a concern that should be addressed to enable a better understanding of the vulnerability of marine predators to climate change.     

Experimental duration and predator satiation levels systematically affect functional response parameters

Empirical feeding studies where density‐dependent consumption rates are fitted to functional response models are often used to parameterize the interaction strengths in models of population or food‐web dynamics. However, the relationship between functional response parameter estimates from short‐term feeding studies and real‐world, long‐term, trophic interaction strengths remains largely unexamined. In a critical first step to address this void, we tested for systematic effects of experimental duration and predator satiation on the estimate of functional response parameters, namely attack rate and handling time. Analyzing a large data set covering a wide range of predator taxa and body masses, we show that attack rates decrease with increasing experimental duration, and that handling times of starved predators are consistently shorter than those of satiated predators. Therefore, both the experimental duration and the predator satiation level have a strong and systematic impact on the predictions of population dynamics and food‐web stability. Our study highlights potential pitfalls at the intersection of empirical and theoretical applications of functional responses. We conclude our study with some practical suggestions for how these implications should be addressed in the future to improve predictive abilities and realism in models of predator–prey interactions.