Comparison of nonparametric estimators of the expected number of recurrent events

We compare the performance of nonparametric estimators for the mean number of recurrent events and provide a systematic overview for different recurrent event settings. The mean number of recurrent events is an easily interpreted marginal feature often used for treatment comparisons in clinical trials. Incomplete observations, dependencies between successive events, terminating events acting as competing risk, or gaps between at risk periods complicate the estimation. We use survival multistate models to represent different complex recurrent event situations, profiting from recent advances in nonparametric estimation for non-Markov multistate models, and explain several estimators by using multistate intensity processes, including the common Nelson-Aalen-type estimators with and without competing mortality. In addition to building on estimation of state occupation probabilities in non-Markov models, we consider a simple extension of the Nelson-Aalen estimator by allowing for dependence on the number of prior recurrent events. We pay particular attention to the assumptions required for the censoring mechanism, one issue being that some settings require the censoring process to be entirely unrelated while others allow for state-dependent or event-driven censoring. We conducted extensive simulation studies to compare the estimators in various complex situations with recurrent events. Our practical example deals with recurrent chronic obstructive pulmonary disease exacerbations in a clinical study, which will also be used to illustrate two-sample-inference using resampling.

Inferences for current chronic graft-versus-host-disease free and relapse free survival

This paper provides the methodologies of a new summary curve that measures the dynamic outcome following allogenic hematopoietic cell transplantation. This new summary curve computes the probabilities that a patient is alive in remission and free of severe-to-moderate chronic graft-versus-host disease (GVHD) over time. The probability is called Current chronic GVHD-free, Relapse-Free Survival (CGRFS). Based on a multistate model depicting the possible states that a patient may experience after transplant, CGRFS can be formulated as a linear combination of five survival functions. This method is known as the model-free approach. In this paper we provide the inferences of the model-free approach, including estimation of CGRFS, precision evaluation and comparison of CGRFS between two independent samples.

Robust estimation and variable selection for the accelerated failure time model

This article concerns robust modeling of the survival time for cancer patients. Accurate prediction of patient survival time is crucial to the development of effective therapeutic strategies. To this goal, we propose a unified Expectation-Maximization approach combined with the L1 -norm penalty to perform variable selection and parameter estimation simultaneously in the accelerated failure time model with right-censored survival data of moderate sizes. Our approach accommodates general loss functions, and reduces to the well-known Buckley-James method when the squared-error loss is used without regularization. To mitigate the effects of outliers and heavy-tailed noise in real applications, we recommend the use of robust loss functions under the general framework. Furthermore, our approach can be extended to incorporate group structure among covariates. We conduct extensive simulation studies to assess the performance of the proposed methods with different loss functions and apply them to an ovarian carcinoma study as an illustration.

Summary concordance index for meta-analysis of prognosis studies with a survival outcome

In prognosis studies to evaluate association between a continuous biomarker and a survival outcome, investigators often classify subjects into two subclasses of the high- and low-expression groups and apply simple survival analysis techniques of the Kaplan-Meier method and the logrank test. The high- and low-expressions are defined according to whether or not the observation of the biomarker is higher than the cut-off value, which is heterogeneous across studies. The heterogeneous definitions of the cut-off value make it difficult to apply the standard meta-analysis techniques. We propose a method to estimate the concordance index for a survival outcome synthesizing published prognosis studies, in which the Kaplan-Meier estimates for the high- and low-expression groups are reported. We illustrate our proposed method with a real dataset for meta-analysis of prognosis studies evaluating Ki-67 in early breast cancer and evaluate its performance with a simulation study.

Survival analysis for AdVerse events with VarYing follow-up times (SAVVY)-estimation of adverse event risks

Background

The SAVVY project aims to improve the analyses of adverse events (AEs), whether prespecified or emerging, in clinical trials through the use of survival techniques appropriately dealing with varying follow-up times and competing events (CEs). Although statistical methodologies have advanced, in AE analyses, often the incidence proportion, the incidence density, or a non-parametric Kaplan-Meier estimator are used, which ignore either censoring or CEs. In an empirical study including randomized clinical trials from several sponsor organizations, these potential sources of bias are investigated. The main purpose is to compare the estimators that are typically used to quantify AE risk within trial arms to the non-parametric Aalen-Johansen estimator as the gold-standard for estimating cumulative AE probabilities. A follow-up paper will consider consequences when comparing safety between treatment groups.

Methods

Estimators are compared with descriptive statistics, graphical displays, and a more formal assessment using a random effects meta-analysis. The influence of different factors on the size of deviations from the gold-standard is investigated in a meta-regression. Comparisons are conducted at the maximum follow-up time and at earlier evaluation times. CEs definition does not only include death before AE but also end of follow-up for AEs due to events related to the disease course or safety of the treatment.

Results

Ten sponsor organizations provided 17 clinical trials including 186 types of investigated AEs. The one minus Kaplan-Meier estimator was on average about 1.2-fold larger than the Aalen-Johansen estimator and the probability transform of the incidence density ignoring CEs was even 2-fold larger. The average bias using the incidence proportion was less than 5%. Assuming constant hazards using incidence densities was hardly an issue provided that CEs were accounted for. The meta-regression showed that the bias depended mainly on the amount of censoring and on the amount of CEs.

Conclusions

The choice of the estimator of the cumulative AE probability and the definition of CEs are crucial. We recommend using the Aalen-Johansen estimator with an appropriate definition of CEs whenever the risk for AEs is to be quantified and to change the guidelines accordingly.

Real-time forecasting of COVID-19 bed occupancy in wards and Intensive Care Units

This paper presents a mathematical model that provides a real-time forecast of the number of COVID-19 patients admitted to the ward and the Intensive Care Unit (ICU) of a hospital based on the predicted inflow of patients, their Length of Stay (LoS) in both the ward and the ICU as well as transfer of patients between the ward and the ICU. The data required for this forecast is obtained directly from the hospital's data warehouse. The resulting algorithm is tested on data from the first COVID-19 peak in the Netherlands, showing that the forecast is very accurate. The forecast may be visualised in real-time in the hospital's control centre and is used in several Dutch hospitals during the second COVID-19 peak.

Sample size calculations for single-arm survival studies using transformations of the Kaplan-Meier estimator

In single-arm clinical trials with survival outcomes, the Kaplan-Meier estimator and its confidence interval are widely used to assess survival probability and median survival time. Since the asymptotic normality of the Kaplan-Meier estimator is a common result, the sample size calculation methods have not been studied in depth. An existing sample size calculation method is founded on the asymptotic normality of the Kaplan-Meier estimator using the log transformation. However, the small sample properties of the log transformed estimator are quite poor in small sample sizes (which are typical situations in single-arm trials), and the existing method uses an inappropriate standard normal approximation to calculate sample sizes. These issues can seriously influence the accuracy of results. In this paper, we propose alternative methods to determine sample sizes based on a valid standard normal approximation with several transformations that may give an accurate normal approximation even with small sample sizes. In numerical evaluations via simulations, some of the proposed methods provided more accurate results, and the empirical power of the proposed method with the arcsine square-root transformation tended to be closer to a prescribed power than the other transformations. These results were supported when methods were applied to data from three clinical trials.

Regression with a right-censored predictor using inverse probability weighting methods

In a longitudinal study, measures of key variables might be incomplete or partially recorded due to drop-out, loss to follow-up, or early termination of the study occurring before the advent of the event of interest. In this paper, we focus primarily on the implementation of a regression model with a randomly censored predictor. We examine, particularly, the use of inverse probability weighting methods in a generalized linear model (GLM), when the predictor of interest is right-censored, to adjust for censoring. To improve the performance of the complete-case analysis and prevent selection bias, we consider three different weighting schemes: inverse censoring probability weights, Kaplan-Meier weights, and Cox proportional hazards weights. We use Monte Carlo simulation studies to evaluate and compare the empirical properties of different weighting estimation methods. Finally, we apply these methods to the Framingham Heart Study data as an illustrative example to estimate the relationship between age of onset of a clinically diagnosed cardiovascular event and low-density lipoprotein among cigarette smokers.

On the empirical choice of the time window for restricted mean survival time

The t-year mean survival or restricted mean survival time (RMST) has been used as an appealing summary of the survival distribution within a time window [0, t]. RMST is the patient's life expectancy until time t and can be estimated nonparametrically by the area under the Kaplan-Meier curve up to t. In a comparative study, the difference or ratio of two RMSTs has been utilized to quantify the between-group-difference as a clinically interpretable alternative summary to the hazard ratio. The choice of the time window [0, t] may be prespecified at the design stage of the study based on clinical considerations. On the other hand, after the survival data have been collected, the choice of time point t could be data-dependent. The standard inferential procedures for the corresponding RMST, which is also data-dependent, ignore this subtle yet important issue. In this paper, we clarify how to make inference about a random "parameter." Moreover, we demonstrate that under a rather mild condition on the censoring distribution, one can make inference about the RMST up to t, where t is less than or even equal to the largest follow-up time (either observed or censored) in the study. This finding reduces the subjectivity of the choice of t empirically. The proposal is illustrated with the survival data from a primary biliary cirrhosis study, and its finite sample properties are investigated via an extensive simulation study.

A resampling-based test for two crossing survival curves

The area between two survival curves is an intuitive test statistic for the classical two-sample testing problem. We propose a bootstrap version of it for assessing the overall homogeneity of these curves. Our approach allows ties in the data as well as independent right censoring, which may differ between the groups. The asymptotic distribution of the test statistic as well as of its bootstrap counterpart are derived under the null hypothesis, and their consistency is proven for general alternatives. We demonstrate the finite sample superiority of the proposed test over some existing methods in a simulation study and illustrate its application by a real-data example.

Identification of Breast Cancer Subtype Specific MicroRNAs Using Survival Analysis to Find Their Role in Transcriptomic Regulation

The microRNA (miRNA) biomolecules have a significant role in the development of breast cancer, and their expression profiles are different in each subtype of breast cancer. Thus, our goal is to use the Next Generation Sequencing provided high-throughput miRNA expression and clinical data in an integrated fashion to perform survival analysis in order to identify breast cancer subtype specific miRNAs, and analyze associated genes and transcription factors. We select top 100 miRNAs for each of the four subtypes, based on the value of hazard ratio and p-value, thereafter, identify 44 miRNAs that are related to all four subtypes, which we call as four-star miRNAs. Moreover, 12, 14, 9, and 15 subtype specific, viz. one-star miRNAs, are also identified. The resulting miRNAs are validated by using machine learning methods to differentiate tumor cases from controls (for four-star miRNAs), and subtypes (for one-star miRNAs). The four-star miRNAs provide 95% average accuracy, while in case of one-star miRNAs 81% accuracy is achieved for HER2-Enriched. Differences in expression of miRNAs between cancer stages is also analyzed, and a subset of eight miRNAs is found, for which expression is increased in stage II relative to stage I, including hsa-miR-10b-5p, which contributes to breast cancer metastasis. Subsequently we prepare regulatory networks in order to identify the interactions among miRNAs, their targeted genes and transcription factors (TFs), that are targeting those miRNAs. In this way, key regulatory circuits are identified, where genes such as TP53, ESR1, BRCA1, MYC, and others, that are known to be important genetic factors for the cause of breast cancer, produce transcription factors that target the same genes as well as interact with the selected miRNAs. To provide further biological validation the Protein-Protein Interaction (PPI) networks are prepared and Kyoto Encyclopedia of Genes and Genomes pathway and gene ontology (GO) enrichment analysis are performed. Among the enriched pathways many are breast cancer-related, such as PI3K-Akt or p53 signaling pathways, and contain proteins such as TP53, also present in the regulatory networks. Moreover, we find that the genes are enriched in GO terms associated with breast cancer. Our results provide detailed analysis of selected miRNAs and their regulatory networks.

Analysis of Survival Curves: Statistical Methods Accounting for the Presence of Long-Term Survivors

Some anti-cancer treatments (e. g., immunotherapies) determine, on the long term, a durable survival in a small percentage of treated patients; in graphical terms, long-term survivors typically give rise to a plateau in the right tail of the survival curve. In analysing these datasets, medians are unable to recognize the presence of this plateau. To account for long-term survivors, both value-frameworks of ASCO and ESMO have incorporated post-hoc corrections that upgrade the framework scores when a survival plateau is present. However, the empiric nature of these post-hoc corrections is self-evident. To capture the presence of a survival plateau by quantitative methods, two approaches have thus far been proposed: the milestone method and the area-under-the-curve (AUC) method. The first approach identifies a long-term time-point in the follow-up (“milestone”) at which survival percentages are extracted. The second approach, which is based on the measurement of AUC of survival curves, essentially is the rearrangement of previous methods determining mean lifetime survival; similarly to the milestone method, the application of AUC can be “restricted” to a pre-specified time-point of the follow-up. This Mini-Review examines the literature published on this topic. The main characteristics of these two methods are highlighted along with their advantages and disadvantages. The conclusion is that both the milestone method and the AUC method are able to capture the presence of a survival plateau.

Pitfalls in graft survival analysis

The reliability of a scientific work depends on the accuracy of the analysis. Scientific publications in the field of kidney transplantation still contain methodical errors that may lead to wrong conclusions and result in severe consequences for the patients. Using the data from the Collaborative Transplant Study, we are presenting in this study six examples of the erroneous usage of statistical methods and show how these mistakes can be avoided.

Threshold regression to accommodate a censored covariate

In several common study designs, regression modeling is complicated by the presence of censored covariates. Examples of such covariates include maternal age of onset of dementia that may be right censored in an Alzheimer's amyloid imaging study of healthy subjects, metabolite measurements that are subject to limit of detection censoring in a case-control study of cardiovascular disease, and progressive biomarkers whose baseline values are of interest, but are measured post-baseline in longitudinal neuropsychological studies of Alzheimer's disease. We propose threshold regression approaches for linear regression models with a covariate that is subject to random censoring. Threshold regression methods allow for immediate testing of the significance of the effect of a censored covariate. In addition, they provide for unbiased estimation of the regression coefficient of the censored covariate. We derive the asymptotic properties of the resulting estimators under mild regularity conditions. Simulations demonstrate that the proposed estimators have good finite-sample performance, and often offer improved efficiency over existing methods. We also derive a principled method for selection of the threshold. We illustrate the approach in application to an Alzheimer's disease study that investigated brain amyloid levels in older individuals, as measured through positron emission tomography scans, as a function of maternal age of dementia onset, with adjustment for other covariates. We have developed an R package, censCov, for implementation of our method, available at CRAN.

Statistical Methods for Conditional Survival Analysis

We investigate the survival distribution of the patients who have survived over a certain time period. This is called a conditional survival distribution. In this paper, we show that one-sample estimation, two-sample comparison and regression analysis of conditional survival distributions can be conducted using the regular methods for unconditional survival distributions that are provided by the standard statistical software, such as SAS and SPSS. We conduct extensive simulations to evaluate the finite sample property of these conditional survival analysis methods. We illustrate these methods with real clinical data.

On Estimation of Optimal Treatment Regimes For Maximizing t-Year Survival Probability

A treatment regime is a deterministic function that dictates personalized treatment based on patients' individual prognostic information. There is increasing interest in finding optimal treatment regimes, which determine treatment at one or more treatment decision points so as to maximize expected long-term clinical outcome, where larger outcomes are preferred. For chronic diseases such as cancer or HIV infection, survival time is often the outcome of interest, and the goal is to select treatment to maximize survival probability. We propose two nonparametric estimators for the survival function of patients following a given treatment regime involving one or more decisions, i.e., the so-called value. Based on data from a clinical or observational study, we estimate an optimal regime by maximizing these estimators for the value over a prespecified class of regimes. Because the value function is very jagged, we introduce kernel smoothing within the estimator to improve performance. Asymptotic properties of the proposed estimators of value functions are established under suitable regularity conditions, and simulations studies evaluate the finite-sample performance of the proposed regime estimators. The methods are illustrated by application to data from an AIDS clinical trial.

Finite sample pointwise confidence intervals for a survival distribution with right-censored data

We review and develop pointwise confidence intervals for a survival distribution with right-censored data for small samples, assuming only independence of censoring and survival. When there is no censoring, at each fixed time point, the problem reduces to making inferences about a binomial parameter. In this case, the recently developed beta product confidence procedure (BPCP) gives the standard exact central binomial confidence intervals of Clopper and Pearson. Additionally, the BPCP has been shown to be exact (gives guaranteed coverage at the nominal level) for progressive type II censoring and has been shown by simulation to be exact for general independent right censoring. In this paper, we modify the BPCP to create a 'mid-p' version, which reduces to the mid-p confidence interval for a binomial parameter when there is no censoring. We perform extensive simulations on both the standard and mid-p BPCP using a method of moments implementation that enforces monotonicity over time. All simulated scenarios suggest that the standard BPCP is exact. The mid-p BPCP, like other mid-p confidence intervals, has simulated coverage closer to the nominal level but may not be exact for all survival times, especially in very low censoring scenarios. In contrast, the two asymptotically-based approximations have lower than nominal coverage in many scenarios. This poor coverage is due to the extreme inflation of the lower error rates, although the upper limits are very conservative. Both the standard and the mid-p BPCP methods are available in our bpcp R package. Published 2016. This article is US Government work and is in the public domain in the USA.

Total Ankle Replacement Survival Rates Based on Kaplan-Meier Survival Analysis of National Joint Registry Data

National joint registry data provides unique information about primary total ankle replacement (TAR) survival. We sought to recreate survival curves among published national joint registry data sets using the Kaplan-Meier estimator. Overall, 5152 primary and 591 TAR revisions were included over a 2- to 13-year period with prosthesis survival for all national joint registries of 0.94 at 2-years, 0.87 at 5-years and 0.81 at 10-years. National joint registry datasets should strive for completion of data presentation including revision definitions, modes and time of failure, and patients lost to follow-up or death for complete accuracy of the Kaplan-Meier estimator.

Combining one-sample confidence procedures for inference in the two-sample case

We present a simple general method for combining two one-sample confidence procedures to obtain inferences in the two-sample problem. Some applications give striking connections to established methods; for example, combining exact binomial confidence procedures gives new confidence intervals on the difference or ratio of proportions that match inferences using Fisher's exact test, and numeric studies show the associated confidence intervals bound the type I error rate. Combining exact one-sample Poisson confidence procedures recreates standard confidence intervals on the ratio, and introduces new ones for the difference. Combining confidence procedures associated with one-sample t-tests recreates the Behrens-Fisher intervals. Other applications provide new confidence intervals with fewer assumptions than previously needed. For example, the method creates new confidence intervals on the difference in medians that do not require shift and continuity assumptions. We create a new confidence interval for the difference between two survival distributions at a fixed time point when there is independent censoring by combining the recently developed beta product confidence procedure for each single sample. The resulting interval is designed to guarantee coverage regardless of sample size or censoring distribution, and produces equivalent inferences to Fisher's exact test when there is no censoring. We show theoretically that when combining intervals asymptotically equivalent to normal intervals, our method has asymptotically accurate coverage. Importantly, all situations studied suggest guaranteed nominal coverage for our new interval whenever the original confidence procedures themselves guarantee coverage.

Analyzing survival curves at a fixed point in time for paired and clustered right-censored data

In clinical trials, information about certain time points may be of interest in making decisions about treatment effectiveness. Rather than comparing entire survival curves, researchers can focus on the comparison at fixed time points that may have a clinical utility for patients. For two independent samples of right-censored data, Klein et al. (2007) compared survival probabilities at a fixed time point by studying a number of tests based on some transformations of the Kaplan-Meier estimators of the survival function. However, to compare the survival probabilities at a fixed time point for paired right-censored data or clustered right-censored data, their approach would need to be modified. In this paper, we extend the statistics to accommodate the possible within-paired correlation and within-clustered correlation, respectively. We use simulation studies to present comparative results. Finally, we illustrate the implementation of these methods using two real data sets.