Weak Convergence of the Wild Bootstrap for the Aalen-Johansen Estimator of the Cumulative Incidence Function of a Competing Risk

Incompletely Observed Nonparametric Factorial Designs With Repeated Measurements: A Wild Bootstrap Approach

In many life science experiments or medical studies, subjects are repeatedly observed and measurements are collected in factorial designs with multivariate data. The analysis of such multivariate data is typically based on multivariate analysis of variance (MANOVA) or mixed models, requiring complete data, and certain assumption on the underlying parametric distribution such as continuity or a specific covariance structure, for example, compound symmetry. However, these methods are usually not applicable when discrete data or even ordered categorical data are present. In such cases, nonparametric rank-based methods that do not require stringent distributional assumptions are the preferred choice. However, in the multivariate case, most rank-based approaches have only been developed for complete observations. It is the aim of this work to develop asymptotic correct procedures that are capable of handling missing values, allowing for singular covariance matrices and are applicable for ordinal or ordered categorical data. This is achieved by applying a wild bootstrap procedure in combination with quadratic form-type test statistics. Beyond proving their asymptotic correctness, extensive simulation studies validate their applicability for small samples. Finally, two real data examples are analyzed.

Racial and ethnic differences in second primary lung cancer risk among lung cancer survivors

BACKGROUND

Recent therapeutic advances have improved survival among lung cancer (LC) patients, who are now at high risk of second primary lung cancer (SPLC). Hispanics comprise the largest minority in the United States, who have shown a lower LC incidence and mortality than other races, and yet their SPLC risk is poorly understood. We quantified the SPLC incidence patterns among Hispanics vs other races.

METHODS

We used data from the Multiethnic Cohort, a population-based cohort of 5 races (African American, Japanese American, Hispanic, Native Hawaiian, and White), recruited between 1993 and 1996 and followed through 2017. We identified patients diagnosed with initial primary lung cancer (IPLC) and SPLC via linkage to Surveillance, Epidemiology, and End Results registries. We estimated the 10-year cumulative incidence of IPLC (in the entire cohort) and SPLC (among IPLC patients). A standardized incidence ratio (SIR) was calculated as the ratio of SPLC-to-IPLC incidence by race and ethnicity.

RESULTS

Among 202 692 participants, 6788 (3.3%) developed IPLC over 3 871  417 person-years. The 10-year cumulative IPLC incidence was lower among Hispanics (0.80%, 0.72 to 0.88) vs Whites (1.67%, 1.56 to 1.78) or Blacks (2.44%, 2.28 to 2.60). However, the 10-year SPLC incidence following IPLC was higher among Hispanics (3.11%, 1.62 to 4.61) vs Whites (2.80%, 1.94 to 3.66) or Blacks (2.29%, 1.48 to 3.10), resulting in a significantly higher SIR for Hispanics (SIR = 8.27, 5.05 to 12.78) vs Whites (SIR = 5.60, 4.11 to 7.45) or Blacks (SIR = 3.48, 2.42 to 4.84; P < .001).

CONCLUSION

Hispanics have a higher SPLC incidence following IPLC than other races, which may be potentially due to better survival after IPLC and extended duration for SPLC development. Continuing surveillance is warranted to reduce racial disparities among LC survivors.

General independent censoring in event-driven trials with staggered entry

Randomized clinical trials with time-to-event endpoints are frequently stopped after a prespecified number of events has been observed. This practice leads to dependent data and nonrandom censoring, which can in general not be solved by conditioning on the underlying baseline information. In case of staggered study entry, matters are complicated substantially. The present paper demonstrates that the study design at hand entails general independent censoring in the counting process sense, provided that the analysis is based on study time information only. To illustrate that the filtrations must not use abundant information, we simulated data of event-driven trials and evaluated them by means of Cox regression models with covariates for the calendar times. The Breslow curves of the cumulative baseline hazard showed considerable deviations, which implies that the analysis is disturbed by conditioning on the calendar time variables. A second simulation study further revealed that Efron's classical bootstrap, unlike the (martingale-based) wild bootstrap, may lead to biased results in the given setting, as the assumption of random censoring is violated. This is exemplified by an analysis of data on immunotherapy in patients with advanced, previously treated nonsmall cell lung cancer.

Confidence bands in survival analysis

BACKGROUND

Providing estimates of uncertainty for statistical quantities is important for statistical inference. When the statistical quantity of interest is a survival curve, which is a function over time, the appropriate type of uncertainty estimate is a confidence band constructed to account for the correlation between points on the curve, we will call this a simultaneous confidence band. This, however, is not the type of confidence band provided in standard software, which is constructed by joining the confidence intervals at given time points.

METHODS

We show that this type of band does not have desirable joint/simultaneous coverage properties in comparison to simultaneous bands.

RESULTS

There are different ways of constructing simultaneous confidence bands, and we find that bands based on the likelihood ratio appear to have the most desirable properties. Although there is no standard software available in the three major statistical packages to compute likelihood-based simultaneous bands, we summarise and give code to use available statistical software to construct other simultaneous forms of bands, which we illustrate using a study of colon cancer.

CONCLUSIONS

There is a need for more user-friendly statistical software to compute simultaneous confidence bands using the available methods.

Nonsmooth backfitting for the excess risk additive regression model with two survival time scales

We consider an extension of Aalen’s additive regression model that allows covariates to have effects that vary on two different time scales. The two time scales considered are equal up to a constant for each individual and vary across individuals, such as follow-up time and age in medical studies or calendar time and age in longitudinal studies. The model was introduced in Scheike (2001), where it was solved using smoothing techniques. We present a new backfitting algorithm for estimating the structured model without having to use smoothing. Estimators of the cumulative regression functions on the two time scales are suggested by solving local estimating equations jointly on the two time scales. We provide large-sample properties and simultaneous confidence bands. The model is applied to data on myocardial infarction, providing a separation of the two effects stemming from time since diagnosis and age.

Causal inference methods for small non-randomized studies: Methods and an application in COVID-19

The usual development cycles are too slow for the development of vaccines, diagnostics and treatments in pandemics such as the ongoing SARS-CoV-2 pandemic. Given the pressure in such a situation, there is a risk that findings of early clinical trials are overinterpreted despite their limitations in terms of size and design. Motivated by a non-randomized open-label study investigating the efficacy of hydroxychloroquine in patients with COVID-19, we describe in a unified fashion various alternative approaches to the analysis of non-randomized studies. A widely used tool to reduce the impact of treatment-selection bias are so-called propensity score (PS) methods. Conditioning on the propensity score allows one to replicate the design of a randomized controlled trial, conditional on observed covariates. Extensions include the g-computation approach, which is less frequently applied, in particular in clinical studies. Moreover, doubly robust estimators provide additional advantages. Here, we investigate the properties of propensity score based methods including three variations of doubly robust estimators in small sample settings, typical for early trials, in a simulation study. R code for the simulations is provided.

General tests of the Markov property in multi-state models

Multi-state models for event history analysis most commonly assume the process is Markov. This article considers tests of the Markov assumption that are applicable to general multi-state models. Two approaches using existing methodology are considered; a simple method based on including time of entry into each state as a covariate in Cox models for the transition intensities and a method involving detecting a shared frailty through a stratified Commenges–Andersen test. In addition, using the principle that under a Markov process the future rate of transitions of the process at times $t &gt; s$ should not be influenced by the state occupied at time $s$, a new class of general tests is developed by considering summaries from families of log-rank statistics where patients are grouped by the state occupied at varying initial time $s$. An extended form of the test applicable to models that are Markov conditional on observed covariates is also derived. The null distribution of the proposed test statistics are approximated by using wild bootstrap sampling. The approaches are compared in simulation and applied to a dataset on sleeping behavior. The most powerful test depends on the particular departure from a Markov process, although the Cox-based method maintained good power in a wide range of scenarios. The proposed class of log-rank statistic based tests are most useful in situations where the non-Markov behavior does not persist, or is not uniform in nature across patient time.

Bootstrap and permutation rank tests for proportional hazards under right censoring

We address the testing problem of proportional hazards in the two-sample survival setting allowing right censoring, i.e., we check whether the famous Cox model is underlying. Although there are many test proposals for this problem, only a few papers suggest how to improve the performance for small sample sizes. In this paper, we do exactly this by carrying out our test as a permutation as well as a wild bootstrap test. The asymptotic properties of our test, namely asymptotic exactness under the null and consistency, can be transferred to both resampling versions. Various simulations for small sample sizes reveal an actual improvement of the empirical size and a reasonable power performance when using the resampling versions. Moreover, the resampling tests perform better than the existing tests of Gill and Schumacher and Grambsch and Therneau . The tests' practical applicability is illustrated by discussing real data examples.

Dynamic inference in general nested case-control designs

Nested case-control designs are attractive in studies with a time-to-event endpoint if the outcome is rare or if interest lies in evaluating expensive covariates. The appeal is that these designs restrict to small subsets of all patients at risk just prior to the observed event times. Only these small subsets need to be evaluated. Typically, the controls are selected at random and methods for time-simultaneous inference have been proposed in the literature. However, the martingale structure behind nested case-control designs allows for more powerful and flexible non-standard sampling designs. We exploit that structure to find simultaneous confidence bands based on wild bootstrap resampling procedures within this general class of designs. We show in a simulation study that the intended coverage probability is obtained for confidence bands for cumulative baseline hazard functions. We apply our methods to observational data about hospital-acquired infections.

Confidence interval estimation for the changepoint of treatment stratification in the presence of a qualitative covariate-treatment interaction

The goal in stratified medicine is to administer the "best" treatment to a patient. Not all patients might benefit from the same treatment; the choice of best treatment can depend on certain patient characteristics. In this article, it is assumed that a time-to-event outcome is considered as a patient-relevant outcome and a qualitative interaction between a continuous covariate and treatment exists, ie, that patients with different values of one specific covariate should be treated differently. We suggest and investigate different methods for confidence interval estimation for the covariate value, where the treatment recommendation should be changed based on data collected in a randomized clinical trial. An adaptation of Fieller's theorem, the delta method, and different bootstrap approaches (normal, percentile-based, wild bootstrap) are investigated and compared in a simulation study. Extensions to multivariable problems are presented and evaluated. We observed appropriate confidence interval coverage following Fieller's theorem irrespective of sample size but at the cost of very wide or even infinite confidence intervals. The delta method and the wild bootstrap approach provided the smallest intervals but inadequate coverage for small to moderate event numbers, also depending on the location of the true changepoint. For the percentile-based bootstrap, wide intervals were observed, and it was slightly conservative regarding coverage, whereas the normal bootstrap did not provide acceptable results for many scenarios. The described methods were also applied to data from a randomized clinical trial comparing two treatments for patients with symptomatic, severe carotid artery stenosis, considering patient's age as predictive marker.

Optimizing the Design and Analysis of Clinical Trials for Antibacterials Against Multidrug-resistant Organisms: A White Paper From COMBACTE's STAT-Net

Innovations are urgently required for clinical development of antibacterials against multidrug-resistant organisms. Therefore, a European, public-private working group (STAT-Net; part of Combatting Bacterial Resistance in Europe [COMBACTE]), has reviewed and tested several innovative trials designs and analytical methods for randomized clinical trials, which has resulted in 8 recommendations. The first 3 focus on pharmacokinetic and pharmacodynamic modeling, emphasizing the pertinence of population-based pharmacokinetic models, regulatory procedures for the reassessment of old antibiotics, and rigorous quality improvement. Recommendations 4 and 5 address the need for more sensitive primary end points through the use of rank-based or time-dependent composite end points. Recommendation 6 relates to the applicability of hierarchical nested-trial designs, and the last 2 recommendations propose the incorporation of historical or concomitant trial data through Bayesian methods and/or platform trials. Although not all of these recommendations are directly applicable, they provide a solid, evidence-based approach to develop new, and established, antibacterials and address this public health challenge.

Confidence bands for multiplicative hazards models: Flexible resampling approaches

We propose new resampling‐based approaches to construct asymptotically valid time‐simultaneous confidence bands for cumulative hazard functions in multistate Cox models. In particular, we exemplify the methodology in detail for the simple Cox model with time‐dependent covariates, where the data may be subject to independent right‐censoring or left‐truncation. We use simulations to investigate their finite sample behavior. Finally, the methods are utilized to analyze two empirical examples with survival and competing risks data.

On estimands and the analysis of adverse events in the presence of varying follow-up times within the benefit assessment of therapies

The analysis of adverse events (AEs) is a key component in the assessment of a drug's safety profile. Inappropriate analysis methods may result in misleading conclusions about a therapy's safety and consequently its benefit-risk ratio. The statistical analysis of AEs is complicated by the fact that the follow-up times can vary between the patients included in a clinical trial. This paper takes as its focus the analysis of AE data in the presence of varying follow-up times within the benefit assessment of therapeutic interventions. Instead of approaching this issue directly and solely from an analysis point of view, we first discuss what should be estimated in the context of safety data, leading to the concept of estimands. Although the current discussion on estimands is mainly related to efficacy evaluation, the concept is applicable to safety endpoints as well. Within the framework of estimands, we present statistical methods for analysing AEs with the focus being on the time to the occurrence of the first AE of a specific type. We give recommendations which estimators should be used for the estimands described. Furthermore, we state practical implications of the analysis of AEs in clinical trials and give an overview of examples across different indications. We also provide a review of current practices of health technology assessment (HTA) agencies with respect to the evaluation of safety data. Finally, we describe problems with meta-analyses of AE data and sketch possible solutions.

The wild bootstrap for multivariate Nelson-Aalen estimators

We rigorously extend the widely used wild bootstrap resampling technique to the multivariate Nelson-Aalen estimator under Aalen's multiplicative intensity model. Aalen's model covers general Markovian multistate models including competing risks subject to independent left-truncation and right-censoring. This leads to various statistical applications such as asymptotically valid confidence bands or tests for equivalence and proportional hazards. This is exemplified in a data analysis examining the impact of ventilation on the duration of intensive care unit stay. The finite sample properties of the new procedures are investigated in a simulation study.

Assessing Noninferiority in Treatment Trials for Severe Infectious Diseases: an Extension to the Entire Follow-Up Period Using a Cure-Death Multistate Model

In current and former clinical trials for the development of antibacterial drugs, various primary endpoints have been used, and treatment effects are evaluated mostly in noninferiority analyses at the end of follow-up, which varies between studies. A more convincing and highly patient-relevant statement would be a noninferiority assessment over the entire follow-up period with cure and death as coprimary endpoints, while preserving the desired alpha level for statistical testing. To account for the time-dynamic pattern of cure and death, we apply a cure-death multistate model. The endpoint of interest is "get cured and stay alive over time." Noninferiority between treatments over the entire follow-up period is studied by means of one-sided confidence bands provided by a flexible resampling technique. We illustrate the technique by applying it to a recently published study and establish noninferiority in being cured and alive over a time frame of interest for the entire population, patients with hospital-acquired pneumonia, but not for the subset of patients with ventilator-associated pneumonia. Our analysis improves the original results in the sense that our endpoint is more patient benefiting, a stronger noninferiority statement is demonstrated, and the time dependency of cure and death, competing events, and different follow-up times is captured. Multistate methodology combined with confidence bands adds a valuable statistical tool for clinical trials in the context of infection control. The framework is not restricted to the cure-death model but can be adapted to more complex multistate endpoints and equivalence or superiority analyses.

Understanding Mortality of Femoral Fractures Following Low-Impact Trauma in Persons With and Without Care Need

OBJECTIVES

Persons with osteoporotic fracture history are subject to an increased risk for subsequent fractures and mortality. The aim of this retrospective study was to investigate the impact of a previous osteoporotic low-impact (fragility) index fracture (eg, forearm, lower leg) on mortality of a subsequent femoral fracture.

DESIGN

Retrospective cohort study.

PARTICIPANTS/MEASUREMENTS

Claims data of a German health insurance agency including >1.2 million insurants aged 65 years or older and observed between 2004 and 2009.

METHODS

A multistate model was developed handling index fractures and care need as time-dependent exposures, while age was chosen as the underlying time scale. Excess risks were expressed as differences in cause-specific hazards. Nelson-Aalen estimates were used for their nonparametric estimation. Time-simultaneous statistical inference was based on confidence bands provided by wild bootstrap resampling.

RESULTS

Excess femoral fracture risk increased with progressive age and was highest in persons with care need. It was observed starting from an age of 79 years in women and 85 years in men onward. A prior index fracture increased mortality after a femoral fracture by increasing femoral fracture risk, while leaving the hazard of death after a subsequent femoral fracture unchanged.

CONCLUSIONS

The results indicated that increased mortality of a subsequent femoral fracture is not triggered by an intrinsically increased mortality hazard but an increased femoral fracture incidence.

A wild bootstrap approach for the selection of biomarkers in early diagnostic trials

BACKGROUND

In early diagnostic trials, particularly in biomarker studies, the aim is often to select diagnostic tests among several methods. In case of metric, discrete, or even ordered categorical data, the area under the receiver operating characteristic (ROC) curve (denoted by AUC) is an appropriate overall accuracy measure for the selection, because the AUC is independent of cut-off points.

METHODS

For selection of biomarkers the individual AUC's are compared with a pre-defined threshold. To keep the overall coverage probability or the multiple type-I error rate, simultaneous confidence intervals and multiple contrast tests are considered. We propose a purely nonparametric approach for the estimation of the AUC's with the corresponding confidence intervals and statistical tests. This approach uses the correlation among the statistics to account for multiplicity. For small sample sizes, a Wild-Bootstrap approach is presented. It is shown that the corresponding intervals and tests are asymptotically exact.

RESULTS

Extensive simulation studies indicate that the derived Wild-Bootstrap approach keeps and exploits the nominal type-I error at best, even for high accuracies and in case of small samples sizes. The strength of the correlation, the type of covariance structure, a skewed distribution, and also a moderate imbalanced case-control ratio do not have any impact on the behavior of the approach. A real data set illustrates the application of the proposed methods.

CONCLUSION

We recommend the new Wild Bootstrap approach for the selection of biomarkers in early diagnostic trials, especially for high accuracies and small samples sizes.

Evaluating hospital performance based on excess cause-specific incidence

Formal evaluation of hospital performance in specific types of care is becoming an indispensable tool for quality assurance in the health care system. When the prime concern lies in reducing the risk of a cause-specific event, we propose to evaluate performance in terms of an average excess cumulative incidence, referring to the center's observed patient mix. Its intuitive interpretation helps give meaning to the evaluation results and facilitates the determination of important benchmarks for hospital performance. We apply it to the evaluation of cerebrovascular deaths after stroke in Swedish stroke centers, using data from Riksstroke, the Swedish stroke registry. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.