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We establish an identity that relates B-spline functions to linear combinations of gamma density functions. Utilizing this connection, we illustrate that, for exponentially distributed lifetimes, the distribution of the MLE in various hybrid censoring schemes can be expressed in terms of gamma density functions with simple weights. As an example, representations for the density functions in the case of Type-I sequential, Type-II progressive hybrid, generalized Type-I progressive hybrid, and generalized Type-II progressive hybrid censoring schemes are presented. It turns out that the representations arising from the spacings’ based approach introduced in Cramer and Balakrishnan (Stat Methodol 10:128–150, 2013) are more compact than those available in the literature so far.

In this paper, structural properties of (progressive) hybrid censoring schemes are established by studying the possible data scenarios resulting from the hybrid censoring scheme. The results illustrate that the distributions of hybrid censored random variables can be immediately derived from the cases of Type-I and Type-II censored data. Furthermore, it turns out that results in likelihood and Bayesian inference are also obtained directly which explains the similarities present in the probabilistic and statistical analysis of these censoring schemes. The power of the approach is illustrated by applying the approach to the quite complex unified Type-II (progressive) hybrid censoring scheme. Finally, it is shown that the approach is not restricted to (progressively Type-II censored) order statistics and that it can be extended to almost any kind of ordered data.

Background Eurotransplant liver transplant candidates are prioritized by Model for End-stage Liver Disease (MELD), a 90-day waitlist survival risk score based on the INR, creatinine and bilirubin. Several studies revised the original MELD score, UNOS-MELD, with transplant candidate data by modelling 90-day waitlist mortality from waitlist registration, censoring patients at delisting or transplantation. This approach ignores biomarkers reported after registration, and ignores informative censoring by transplantation and delisting. Methods We study how MELD revision is affected by revision from calendar-time cross-sections and correction for informative censoring with inverse probability censoring weighting (IPCW). For this, we revised UNOS-MELD on patients with chronic liver cirrhosis on the Eurotransplant waitlist between 2007 and 2019 ( n  = 13,274) with Cox models with as endpoints 90-day survival (a) from registration and (b) from weekly drawn calendar-time cross-sections. We refer to the revised score from cross-section with IPCW as DynReMELD , and compare DynReMELD to UNOS-MELD and ReMELD, a prior revision of UNOS-MELD for Eurotransplant, in geographical validation. Results Revising MELD from calendar-time cross-sections leads to significantly different MELD coefficients. IPCW increases estimates of absolute 90-day waitlist mortality risks by approximately 10 percentage points. DynReMELD has improved discrimination over UNOS-MELD (delta c-index: 0.0040, p  < 0.001) and ReMELD (delta c-index: 0.0015, p  < 0.01), with differences comparable in magnitude to the addition of an extra biomarker to MELD (delta c-index: ± 0.0030). Conclusion Correcting for selection bias by transplantation/delisting does not improve discrimination of revised MELD scores, but substantially increases estimated absolute 90-day mortality risks. Revision from cross-section uses waitlist data more efficiently, and improves discrimination compared to revision of MELD exclusively based on information available at listing.

Background Precision medicine is an emerging field that involves the selection of treatments based on patients’ individual prognostic data. It is formalized through the identification of individualized treatment rules (ITRs) that maximize a clinical outcome. When the type of outcome is time-to-event, the correct handling of censoring is crucial for estimating reliable optimal ITRs. Methods We propose a jackknife estimator of the value function to allow for right-censored data for a binary treatment. The jackknife estimator or leave-one-out-cross-validation approach can be used to estimate the value function and select optimal ITRs using existing machine learning methods. We address the issue of censoring in survival data by introducing an inverse probability of censoring weighted (IPCW) adjustment in the expression of the jackknife estimator of the value function. In this paper, we estimate the optimal ITR by using random survival forest (RSF) and Cox proportional hazards model (COX). We use a Z-test to compare the optimal ITRs learned by RSF and COX with the zero-order model (or one-size-fits-all). Through simulation studies, we investigate the asymptotic properties and the performance of our proposed estimator under different censoring rates. We illustrate our proposed method on a phase III clinical trial of non-small cell lung cancer data. Results Our simulations show that COX outperforms RSF for small sample sizes. As sample sizes increase, the performance of RSF improves, in particular when the expected log failure time is not linear in the covariates. The estimator is fairly normally distributed across different combinations of simulation scenarios and censoring rates. When applied to a non-small-cell lung cancer data set, our method determines the zero-order model (ZOM) as the best performing model. This finding highlights the possibility that tailoring may not be needed for this cancer data set. Conclusion The jackknife approach for estimating the value function in the presence of right-censored data shows satisfactory performance when there is small to moderate censoring. Winsorizing the upper and lower percentiles of the estimated survival weights for computing the IPCWs stabilizes the estimator.

In recent years, there has been much interest in data-driven decision making. Although this can unlock tremendous value across industries, it is very important to remember that data-driven decisions are uncertain quantities with error bars associated with them. Despite the obvious importance, error bars of data-driven decisions have been underinvestigated. The paper “Confidence Intervals for Data-Driven Inventory Policies with Demand Censoring” bridges this gap in knowledge for inventory problems. Specifically, Gah-Yi Ban derives approximate analytic formulas for confidence intervals of data-driven solutions to the classical dynamic inventory management problem of Scarf (1959b) . Both censored and uncensored data scenarios are considered, and the analyses extend to other commonly studied problems, such as the repeated newsvendor problem and base stock policy problem. Extensive computations on realistic simulated data validate the approximate analytic formulas, which are based on asymptotic theory, establishing their practical value. We revisit the classical dynamic inventory management problem of Scarf [Scarf H (1959b) The optimality of ( s , S ) policies in the dynamic inventory problem. Arrow KJ, Karlin S, Suppes P, eds. Mathematical Methods in the Social Science (Stanford University Press, Stanford, CA), 196–202.] from the perspective of a decision maker who has n historical selling seasons of data and must make ordering decisions for the upcoming season. We develop a nonparametric estimation procedure for the (S, s) policy that is consistent and then characterize the finite sample properties of the estimated (S, s) levels by deriving their asymptotic confidence intervals. We also consider having at least some of the past selling seasons of data censored from the absence of backlogging and show that the intuitive procedure of first correcting for censoring in the demand data yields inconsistent estimates. We then show how to correctly use the censored data to obtain consistent estimates and derive asymptotic confidence intervals for this policy using Stein’s method. We further show the confidence intervals can be used to effectively bound the difference between the expected total cost of an estimated policy and that of the optimal policy. We validate our results with extensive computations on simulated data. Our results extend to the repeated newsvendor problem and the base stock policy problem by appropriate parameter choices.

Background Evaluating the performance of predictive models for survival is essential before they can be trusted for real-world applications and decision making. While good measures such as the C-index are available for model discrimination, the toolbox for model calibration is much more limited in the time-to-event setting. The method of D-calibration was therefore an important contribution that yields a single numeric value for calibration across the available follow-up time. D-calibration consists of performing a Pearson’s goodness-of-fit test on transformed survival times. Censored survival times are handled using an imputation approach which however tends to yield a conservative test and loss of power. Methods In this paper, we introduce A-calibration based on Akritas’s goodness-of-fit test which is designed specifically for censored time-to-event data. Through theoretical arguments, simulations, and a case study, we compare A- and D-calibration as measures of calibration. In the simulation study, the power of each test to reject a false null-hypothesis was assessed for varying censoring mechanisms (memoryless, uniform and zero censoring), censoring rates, and parameter values of the predictive model considered. Results The simulation study demonstrated that A-calibration had similar or superior power to D-calibration in all considered cases, and that D-calibration, unlike A-calibration, was particularly sensitive to censoring. Conclusions Advantages of A-calibration compared to D-calibration have been demonstrated through theoretical considerations, a simulation study, and a case study, while no disadvantages relative to D-calibration were identified.

The exported cases of 2019 novel coronavirus (COVID-19) infection that were confirmed outside China provide an opportunity to estimate the cumulative incidence and confirmed case fatality risk (cCFR) in mainland China. Knowledge of the cCFR is critical to characterize the severity and understand the pandemic potential of COVID-19 in the early stage of the epidemic. Using the exponential growth rate of the incidence, the present study statistically estimated the cCFR and the basic reproduction number—the average number of secondary cases generated by a single primary case in a naïve population. We modeled epidemic growth either from a single index case with illness onset on 8 December 2019 (Scenario 1), or using the growth rate fitted along with the other parameters (Scenario 2) based on data from 20 exported cases reported by 24 January 2020. The cumulative incidence in China by 24 January was estimated at 6924 cases (95% confidence interval [CI]: 4885, 9211) and 19,289 cases (95% CI: 10,901, 30,158), respectively. The latest estimated values of the cCFR were 5.3% (95% CI: 3.5%, 7.5%) for Scenario 1 and 8.4% (95% CI: 5.3%, 12.3%) for Scenario 2. The basic reproduction number was estimated to be 2.1 (95% CI: 2.0, 2.2) and 3.2 (95% CI: 2.7, 3.7) for Scenarios 1 and 2, respectively. Based on these results, we argued that the current COVID-19 epidemic has a substantial potential for causing a pandemic. The proposed approach provides insights in early risk assessment using publicly available data.

Background Chronic kidney disease (CKD) is slowly progressive, with clinically-relevant end-points of interest (e.g. kidney failure, dialysis, transplantation, death due to kidney disease) occurring many years after diagnosis, making the design of trials to evaluate treatments that slow the progression of kidney disease challenging. Recent meta-analyses have shown that the 3-year total slope of estimated glomerular filtration rate (eGFR) may serve as a reliable surrogate for these hard clinical outcomes. Existing research has focused on relaxing the linear trend assumption on the eGFR slope, accounting for informative censoring (via fitting a shared parameter model, for example), and evaluating heterogeneous treatment effects (HTEs) given predetermined subgroups. Yet, none have explored data-driven subgroup identification and HTE estimation. Methods We propose a Bayesian method that incorporates a Bayesian decision tree for HTE into a shared-parameter model that combines a survival model for censoring time with a two-slope spline model that characterizes the total eGFR slope. Our proposed approach simultaneously estimates the total eGFR slope in the presence of informative censoring and identifies interpretable subgroups of patients who experience differential treatment effects on the total eGFR slope outcome. Results Simulation studies demonstrate that our method accurately recovers treatment-effect heterogeneity with low estimation error, yielding better subgroup-specific treatment recommendations in moderate-to-large samples. Our method also controls false positives when no true heterogeneity presents. We apply our approach to the Modification of Diet in Renal Disease (MDRD) Trial, observing strong Bayesian evidence that patients with a baseline eGFR above 34.32 benefit more from the intensive systolic blood pressure control compared to patients with a baseline eGFR below 34.32. Specifically, the posterior probability that the treatment effect is larger in the higher-eGFR subgroup is 81 %. Conclusion Our proposed model can effectively capture even subtle HTEs while avoiding over-fitting when no heterogeneity exists, making it valuable for identifying HTE to inform downstream analyses such as treatment recommendations.

In this paper, a structural analysis of hybrid censoring models is presented. This new modularization approach to hybrid censoring models enables a convenient derivation of distributional results. For instance, it allows to derive the exact distribution of the MLEs under an exponential assumption for very complex hybrid scenarios. In order to illustrate the benefit of this idea, we apply it to four new unified progressive hybrid censoring schemes. They are extensions of already proposed unified Type-I/II/III/IV hybrid censoring schemes to progressively Type-II censored data. The resulting analysis shows that the modularization approach provides a powerful, efficient, and elegant tool to study even more complex hybrid censoring models.

Competing events are often ignored in epidemiological studies. Conventional methods for the analysis of survival data assume independent or noninformative censoring, which is violated when subjects that experience a competing event are censored. Because many survival studies do not apply competing risk analysis, we explain and illustrate in a nonmathematical way how to analyze and interpret survival data in the presence of competing events. Using data from the Longitudinal Aging Study Amsterdam, both marginal analyses (Kaplan–Meier method and Cox proportional-hazards regression) and competing risk analyses (cumulative incidence function [CIF], cause-specific and subdistribution hazard regression) were performed. We analyzed the association between sex and depressive symptoms, in which death before the onset of depression was a competing event. The Kaplan–Meier method overestimated the cumulative incidence of depressive symptoms. Instead, the CIF should be used. As the subdistribution hazard model has a one-to-one relation with the CIF, it is recommended for prediction research, whereas the cause-specific hazard model is recommended for etiologic research. When competing risks are present, the type of research question guides the choice of the analytical model to be used. In any case, results should be presented for all event types.