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Advances in anticancer therapies have significantly contributed to declining death rates in certain disease and clinical settings. However, they have also made it difficult to power a clinical trial in these settings with overall survival (OS) as the primary efficacy endpoint. A number of statistical approaches have therefore been recently proposed for the pre-specified analysis of OS as a safety endpoint (Shan, 2023; Fleming et al., 2024; Rodriguez et al., 2024). In this paper, we provide a simple, unifying framework that includes the aforementioned approaches (and a couple others) as special cases. By highlighting each approach's focus, priority, tolerance for risk, and strengths or challenges for practical implementation, this framework can help to facilitate discussions between stakeholders on "fit-for-purpose OS data collection and assessment of harm" (American Association for Cancer Research, 2024). We apply this framework to a real clinical trial in large B-cell lymphoma to illustrate its application and value. Several recommendations and open questions are also raised., Comment: 19 pages, 1 table

Interim analyses are prevalent in clinical trials. Although methodology is well established, there are aspects of how to operationalize and interpret interim analyses which remain unclear to many stakeholders. In this paper, a team of statisticians from the pharmaceutical industry, academia, and regulatory agencies provide a multi-stakeholder perspective on the key concepts behind interim analyses and considerations on terminology. We illustrate our proposals using a hypothetical clinical trial., Comment: 25 pages, 6 tables

In randomized controlled trials, forest plots are frequently used to investigate the homogeneity of treatment effect estimates in subgroups. However, the interpretation of subgroup-specific treatment effect estimates requires great care due to the smaller sample size of subgroups and the large number of investigated subgroups. Bayesian shrinkage methods have been proposed to address these issues, but they often focus on disjoint subgroups while subgroups displayed in forest plots are overlapping, i.e., each subject appears in multiple subgroups. In our approach, we first build a flexible Cox model based on all available observations, including categorical covariates that identify the subgroups of interest and their interactions with the treatment group variable. We explore both penalized partial likelihood estimation with a lasso or ridge penalty for treatment-by-covariate interaction terms, and Bayesian estimation with a regularized horseshoe prior. One advantage of the Bayesian approach is the ability to derive credible intervals for shrunken subgroup-specific estimates. In a second step, the Cox model is marginalized to obtain treatment effect estimates for all subgroups. We illustrate these methods using data from a randomized clinical trial in follicular lymphoma and evaluate their properties in a simulation study. In all simulation scenarios, the overall mean-squared error is substantially smaller for penalized and shrinkage estimators compared to the standard subgroup-specific treatment effect estimator but leads to some bias for heterogeneous subgroups. We recommend that subgroup-specific estimators, which are typically displayed in forest plots, are more routinely complemented by treatment effect estimators based on shrinkage methods. The proposed methods are implemented in the R package bonsaiforest., Comment: 23 pages, 2 figures and 1 table (main text), 6 figures and 2 tables (supplementary materials)

We revisit the question of what randomization ratio (RR) maximizes power of the logrank test in event-driven survival trials under proportional hazards (PH). By comparing three approximations of the logrank test (Schoenfeld, Freedman, Rubinstein) to empirical simulations, we find that the RR that maximizes power is the RR that balances number of events across treatment arms at the end of the trial. This contradicts the common misconception implied by Schoenfeld's approximation that 1:1 randomization maximizes power. Besides power, we consider other factors that might influence the choice of RR (accrual, trial duration, sample size, etc.). We perform simulations to better understand how unequal randomization might impact these factors in practice. Altogether, we derive 6 insights to guide statisticians in the design of survival trials considering unequal randomization., Comment: 17 pages, 3 figures, 2 tables

The SAVVY project aims to improve the analyses of adverse events (AEs) in clinical trials through the use of survival techniques appropriately dealing with varying follow-up times and competing events (CEs). This paper summarizes key features and conclusions from the various SAVVY papers. Through theoretical investigations using simulations and in an empirical study including randomized clinical trials from several sponsor organisations, biases from ignoring varying follow-up times or CEs are investigated. The bias of commonly used estimators of the absolute and relative AE risk is quantified. Furthermore, we provide a cursory assessment of how pertinent guidelines for the analysis of safety data deal with the features of varying follow-up time and CEs. SAVVY finds that for both, avoiding bias and categorization of evidence with respect to treatment effect on AE risk into categories, the choice of the estimator is key and more important than features of the underlying data such as percentage of censoring, CEs, amount of follow-up, or value of the gold-standard. The choice of the estimator of the cumulative AE probability and the definition of CEs are crucial. SAVVY recommends using the Aalen-Johansen estimator (AJE) with an appropriate definition of CEs whenever the risk for AEs is to be quantified. There is an urgent need to improve the guidelines of reporting AEs so that incidence proportions or one minus Kaplan-Meier estimators are finally replaced by the AJE with appropriate definition of CEs., Comment: 17 pages, 1 Figure, 4 Tables. arXiv admin note: text overlap with arXiv:2008.07881

When planning an oncology clinical trial, the usual approach is to assume proportional hazards and even an exponential distribution for time-to-event endpoints. Often, besides the gold-standard endpoint overall survival (OS), progression-free survival (PFS) is considered as a second confirmatory endpoint. We use a survival multistate model to jointly model these two endpoints and find that neither exponential distribution nor proportional hazards will typically hold for both endpoints simultaneously. The multistate model provides a stochastic process approach to model the dependency of such endpoints neither requiring latent failure times nor explicit dependency modelling such as copulae. We use the multistate model framework to simulate clinical trials with endpoints OS and PFS and show how design planning questions can be answered using this approach. In particular, non-proportional hazards for at least one of the endpoints are naturally modelled as well as their dependency to improve planning. We consider an oncology trial on non-small-cell lung cancer as a motivating example from which we derive relevant trial design questions. We then illustrate how clinical trial design can be based on simulations from a multistate model. Key applications are co-primary endpoints and group-sequential designs. Simulations for these applications show that the standard simplifying approach may very well lead to underpowered or overpowered clinical trials. Our approach is quite general and can be extended to more complex trial designs, further endpoints, and other therapeutic areas. An R package is available on CRAN., Comment: 28 pages, 3 tables, 12 figures

In causal inference, the correct formulation of the scientific question of interest is a crucial step. Here we apply the estimand framework to a comparison of the outcomes of patient-level clinical trials and observational data to help structure the clinical question. In addition, we complement the estimand framework with the target trial framework to address specific issues in defining the estimand attributes using observational data and discuss synergies and differences of the two frameworks. Whereas the estimand framework proves useful to address the challenge that in clinical trials and routine clinical practice patients may switch to subsequent systemic therapies after the initially assigned systematic treatment, the target trial framework supports addressing challenges around baseline confounding and the index date. We apply the combined framework to compare long-term outcomes of a pooled set of three previously reported randomized phase 3 trials studying patients with metastatic non-small cell lung cancer receiving front-line chemotherapy (randomized clinical trial cohort) and similar patients treated with front-line chemotherapy as part of routine clinical care (observational comparative cohort). We illustrate the process to define the estimand attributes and select the estimator to estimate the estimand of interest while accounting for key baseline confounders, index date, and receipt of subsequent therapies. The proposed combined framework provides more clarity on the causal contrast of interest and the estimator to adopt and thus facilitates design and interpretation of the analyses.

For the analysis of a time-to-event endpoint in a single-arm or randomized clinical trial it is generally perceived that interpretation of a given estimate of the survival function, or the comparison between two groups, hinges on some quantification of the amount of follow-up. Typically, a median of some loosely defined quantity is reported. However, whatever median is reported, is typically not answering the question(s) trialists actually have in terms of follow-up quantification. In this paper, inspired by the estimand framework, we formulate a comprehensive list of relevant scientific questions that trialists have when reporting time-to-event data. We illustrate how these questions should be answered, and that reference to an unclearly defined follow-up quantity is not needed at all. In drug development, key decisions are made based on randomized controlled trials, and we therefore also discuss relevant scientific questions not only when looking at a time-to-event endpoint in one group, but also for comparisons. We find that different thinking about some of the relevant scientific questions around follow-up is required depending on whether a proportional hazards assumption can be made or other patterns of survival functions are anticipated, e.g. delayed separation, crossing survival functions, or the potential for cure. We conclude the paper with practical recommendations., Comment: 31 pages

Adapting the final sample size of a trial to the evidence accruing during the trial is a natural way to address planning uncertainty. Designs with adaptive sample size need to account for their optional stopping to guarantee strict type-I error-rate control. A variety of different methods to maintain type-I error-rate control after unplanned changes of the initial sample size have been proposed in the literature. This makes interim analyses for the purpose of sample size recalculation feasible in a regulatory context. Since the sample size is usually determined via an argument based on the power of the trial, an interim analysis raises the question of how the final sample size should be determined conditional on the accrued information. Conditional power is a concept often put forward in this context. Since it depends on the unknown effect size, we take a strict estimation perspective and compare assumed conditional power, observed conditional power, and predictive power with respect to their properties as estimators of the unknown conditional power. We then demonstrate that pre-planning an interim analysis using methodology for unplanned interim analyses is ineffective and naturally leads to the concept of optimal two-stage designs. We conclude that unplanned design adaptations should only be conducted as reaction to trial-external new evidence, operational needs to violate the originally chosen design, or post hoc changes in the objective criterion. Finally, we show that commonly discussed sample size recalculation rules can lead to paradoxical outcomes and propose two alternative ways of reacting to newly emerging trial-external evidence.

The estimand framework included in the addendum to the ICH E9 guideline facilitates discussions to ensure alignment between the key question of interest, the analysis, and interpretation. Therapeutic knowledge and drug mechanism play a crucial role in determining the strategy and defining the estimand for clinical trial designs. Clinical trials in patients with hematological malignancies often present unique challenges for trial design due to complexity of treatment options and existence of potential curative but highly risky procedures, e.g. stem cell transplant or treatment sequence across different phases (induction, consolidation, maintenance). Here, we illustrate how to apply the estimand framework in hematological clinical trials and how the estimand framework can address potential difficulties in trial result interpretation. This paper is a result of a cross-industry collaboration to connect the International Conference on Harmonisation (ICH) E9 addendum concepts to applications. Three randomized phase 3 trials will be used to consider common challenges including intercurrent events in hematologic oncology trials to illustrate different scientific questions and the consequences of the estimand choice for trial design, data collection, analysis, and interpretation. Template language for describing estimand in both study protocols and statistical analysis plans is suggested for statisticians' reference., Comment: 5 tables, 1 figure

Analyses of adverse events (AEs) are an important aspect of the evaluation of experimental therapies. The SAVVY (Survival analysis for AdVerse events with Varying follow-up times) project aims to improve the analyses of AE data in clinical trials through the use of survival techniques appropriately dealing with varying follow-up times, censoring, and competing events (CE). In an empirical study including seventeen randomized clinical trials the effect of varying follow-up times, censoring, and competing events on comparisons of two treatment arms with respect to AE risks is investigated. The comparisons of relative risks (RR) of standard probability-based estimators to the gold-standard Aalen-Johansen estimator or hazard-based estimators to an estimated hazard ratio (HR) from Cox regression are done descriptively, with graphical displays, and using a random effects meta-analysis on AE level. The influence of different factors on the size of the bias is investigated in a meta-regression. We find that for both, avoiding bias and categorization of evidence with respect to treatment effect on AE risk into categories, the choice of the estimator is key and more important than features of the underlying data such as percentage of censoring, CEs, amount of follow-up, or value of the gold-standard RR. There is an urgent need to improve the guidelines of reporting AEs so that incidence proportions are finally replaced by the Aalen-Johansen estimator - rather than by Kaplan-Meier - with appropriate definition of CEs. For RRs based on hazards, the HR based on Cox regression has better properties than the ratio of incidence densities.

The SAVVY project aims to improve the analyses of adverse event (AE) data 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 (KME) are used, which either ignore censoring or CEs. In an empirical study including randomized clinical trials from several sponsor organisations, these potential sources of bias are investigated. The main aim is to compare the estimators that are typically used in AE analysis to the Aalen-Johansen estimator (AJE) as the gold-standard. Here, one-sample findings are reported, while a companion paper considers consequences when comparing treatment groups. Estimators are compared with descriptive statistics, graphical displays and with a random effects meta-analysis. The influence of different factors on the size of the bias is investigated in a meta-regression. Comparisons are conducted at the maximum follow-up time and at earlier evaluation time points. CEs definition does not only include death before AE but also end of follow-up for AEs due to events possibly related to the disease course or the treatment. Ten sponsor organisations provided 17 trials including 186 types of AEs. The one minus KME was on average about 1.2-fold larger than the AJE. Leading forces influencing bias were the amount of censoring and of CEs. As a consequence, 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. There is a need to improve the guidelines of reporting risks of AEs so that the KME and the incidence proportion are replaced by the AJE with an appropriate definition of CEs.

A randomized trial allows estimation of the causal effect of an intervention compared to a control in the overall population and in subpopulations defined by baseline characteristics. Often, however, clinical questions also arise regarding the treatment effect in subpopulations of patients, which would experience clinical or disease related events post-randomization. Events that occur after treatment initiation and potentially affect the interpretation or the existence of the measurements are called {\it intercurrent events} in the ICH E9(R1) guideline. If the intercurrent event is a consequence of treatment, randomization alone is no longer sufficient to meaningfully estimate the treatment effect. Analyses comparing the subgroups of patients without the intercurrent events for intervention and control will not estimate a causal effect. This is well known, but post-hoc analyses of this kind are commonly performed in drug development. An alternative approach is the principal stratum strategy, which classifies subjects according to their potential occurrence of an intercurrent event on both study arms. We illustrate with examples that questions formulated through principal strata occur naturally in drug development and argue that approaching these questions with the ICH E9(R1) estimand framework has the potential to lead to more transparent assumptions as well as more adequate analyses and conclusions. In addition, we provide an overview of assumptions required for estimation of effects in principal strata. Most of these assumptions are unverifiable and should hence be based on solid scientific understanding. Sensitivity analyses are needed to assess robustness of conclusions.

Sample size derivation is a crucial element of the planning phase of any confirmatory trial. A sample size is typically derived based on constraints on the maximal acceptable type I error rate and a minimal desired power. Here, power depends on the unknown true effect size. In practice, power is typically calculated either for the smallest relevant effect size or a likely point alternative. The former might be problematic if the minimal relevant effect is close to the null, thus requiring an excessively large sample size. The latter is dubious since it does not account for the a priori uncertainty about the likely alternative effect size. A Bayesian perspective on the sample size derivation for a frequentist trial naturally emerges as a way of reconciling arguments about the relative a priori plausibility of alternative effect sizes with ideas based on the relevance of effect sizes. Many suggestions as to how such `hybrid' approaches could be implemented in practice have been put forward in the literature. However, key quantities such as assurance, probability of success, or expected power are often defined in subtly different ways in the literature. Starting from the traditional and entirely frequentist approach to sample size derivation, we derive consistent definitions for the most commonly used `hybrid' quantities and highlight connections, before discussing and demonstrating their use in the context of sample size derivation for clinical trials.

COVID-19 outbreak has rapidly evolved into a global pandemic. The impact of COVID-19 on patient journeys in oncology represents a new risk to interpretation of trial results and its broad applicability for future clinical practice. We identify key intercurrent events that may occur due to COVID-19 in oncology clinical trials with a focus on time-to-event endpoints and discuss considerations pertaining to the other estimand attributes introduced in the ICH E9 addendum. We propose strategies to handle COVID-19 related intercurrent events, depending on their relationship with malignancy and treatment and the interpretability of data after them. We argue that the clinical trial objective from a world without COVID-19 pandemic remains valid. The estimand framework provides a common language to discuss the impact of COVID-19 in a structured and transparent manner. This demonstrates that the applicability of the framework may even go beyond what it was initially intended for., Comment: Paper written on behalf of the industry working group on estimands in oncology (www.oncoestimand.org). Accepted for publication in a special issue of Statistics in Biopharmaceutical Research

The assessment of safety is an important aspect of the evaluation of new therapies in clinical trials, with analyses of adverse events being an essential part of this. Standard methods for the analysis of adverse events such as the incidence proportion, i.e. the number of patients with a specific adverse event out of all patients in the treatment groups, do not account for both varying follow-up times and competing risks. Alternative approaches such as the Aalen-Johansen estimator of the cumulative incidence function have been suggested. Theoretical arguments and numerical evaluations support the application of these more advanced methodology, but as yet there is to our knowledge only insufficient empirical evidence whether these methods would lead to different conclusions in safety evaluations. The Survival analysis for AdVerse events with VarYing follow-up times (SAVVY) project strives to close this gap in evidence by conducting a meta-analytical study to assess the impact of the methodology on the conclusion of the safety assessment empirically. Here we present the rationale and statistical concept of the empirical study conducted as part of the SAVVY project. The statistical methods are presented in unified notation and examples of their implementation in R and SAS are provided.

For a trial with primary endpoint overall survival for a molecule with curative potential, statistical methods that rely on the proportional hazards assumption may underestimate the power and the time to final analysis. We show how a cure proportion model can be used to get the necessary number of events and appropriate timing via simulation. If Phase 1 results for the new drug are exceptional and/or the medical need in the target population is high, a Phase 3 trial might be initiated after Phase 1. Building in a futility interim analysis into such a pivotal trial may mitigate the uncertainty of moving directly to Phase 3. However, if cure is possible, overall survival might not be mature enough at the interim to support a futility decision. We propose to base this decision on an intermediate endpoint that is sufficiently associated with survival. Planning for such an interim can be interpreted as making a randomized Phase 2 trial a part of the pivotal trial: if stopped at the interim, the trial data would be analyzed and a decision on a subsequent Phase 3 trial would be made. If the trial continues at the interim then the Phase 3 trial is already underway. To select a futility boundary, a mechanistic simulation model that connects the intermediate endpoint and survival is proposed. We illustrate how this approach was used to design a pivotal randomized trial in acute myeloid leukemia, discuss historical data that informed the simulation model, and operational challenges when implementing it., Comment: 23 pages, 3 figures, 3 tables. All code is available on github: https://github.com/numbersman77/integratePhase2.git

In this paper, we derive the joint distribution of progression-free and overall survival as a function of transition probabilities in a multistate model. No assumptions on copulae or latent event times are needed and the model is allowed to be non-Markov. From the joint distribution, statistics of interest can then be computed. As an example, we provide closed formulas and statistical inference for Pearson's correlation coefficient between progression-free and overall survival in a parametric framework. The example is inspired by recent approaches to quantify the dependence between progression-free survival, a common primary outcome in phase III trials in oncology, and overall survival. We complement these approaches by providing methods of statistical inference while at the same time working within a much more parsimonious modelling framework. Our approach is completely general and can be applied to other measures of dependence. We also discuss extensions to nonparametric inference. Our analytical results are illustrated using a large randomized clinical trial in breast cancer.

The development of oncology drugs progresses through multiple phases, where after each phase a decision is made about whether to move a molecule forward. Early phase efficacy decisions are often made on the basis of single arm studies based on RECIST tumor response as endpoint. This decision rules are implicitly assuming some form of surrogacy between tumor response and long-term endpoints like progression-free survival (PFS) or overall survival (OS). The surrogacy is most often assessed as weak, but sufficient to allow a rapid decision making as early phase studies lack the survival follow up and number of patients to properly assess PFS or OS. With the emergence of therapies with new mechanisms of action, for which the link between RECIST tumor response and long-term endpoints is either not accessible yet because not enough data is available to perform a meta-regression, or the link is weaker than with classical chemotherapies, tumor response based rules may not be optimal. In this paper, we explore the use of a multistate model for decision making based on single-arm early phase trials. The multistate model allows to account for more information than the simple RECIST response status, namely, the time to get to response, the duration of response, the PFS time and time to death. We propose to base the decision on efficacy on the OS hazard ratio (HR), predicted from a multistate model based on early phase data with limited survival follow-up, combined with historical control data. Using three case studies and simulations, we illustrate the feasibility of the estimation of the OS HR using a multistate model based on limited data from early phase studies. We argue that, in the presence of limited follow up and small sample size, and on assumptions within the multistate model, the OS prediction is acceptable and may lead to better decisions for continuing the development of a drug.

A draft addendum to ICH E9 has been released for public consultation in August 2017. The addendum focuses on two topics particularly relevant for randomized confirmatory clinical trials: estimands and sensitivity analyses. The need to amend ICH E9 grew out of the realization of a lack of alignment between the objectives of a clinical trial stated in the protocol and the accompanying quantification of the "treatment effect" reported in a regulatory submission. We embed time-to-event endpoints in the estimand framework, and discuss how the four estimand attributes described in the addendum apply to time-to-event endpoints. We point out that if the proportional hazards assumption is not met, the estimand targeted by the most prevalent methods used to analyze time-to-event endpoints, logrank test and Cox regression, depends on the censoring distribution. We discuss for a large randomized clinical trial how the analyses for the primary and secondary endpoints as well as the sensitivity analyses actually performed in the trial can be seen in the context of the addendum. To the best of our knowledge, this is the first attempt to do so for a trial with a time-to-event endpoint. Questions that remain open with the addendum for time-to-event endpoints and beyond are formulated, and recommendations for planning of future trials are given. We hope that this will provide a contribution to developing a common framework based on the final version of the addendum that can be applied to design, protocols, statistical analysis plans, and clinical study reports in the future., Comment: 37 pages

Biomarker data is often subject to limits of quantification and/or limits of detection. Statistically, this corresponds to left- or interval-censoring. To be able to associate a censored time-to-event endpoint to a biomarker covariate, the R package SurvRegCensCov provides software for Weibull regression for a right-censored endpoint, one interval-censored, and an arbitrary number of non-censored covariates. Furthermore, the package provides functions to estimate canonical parameters from censored samples based on several distributional assumptions, and a function to switch between different parametrizations used in R for Weibull regression. We illustrate the new software by applying it to assess Prentice's criteria for surrogacy in data simulated from a randomized clinical registration trial., Comment: 14 pages, 2 figures

We consider nonparametric maximum-likelihood estimation of a log-concave density in case of interval-censored, right-censored and binned data. We allow for the possibility of a subprobability density with an additional mass at $+\infty$, which is estimated simultaneously. The existence of the estimator is proved under mild conditions and various theoretical aspects are given, such as certain shape and consistency properties. An EM algorithm is proposed for the approximate computation of the estimator and its performance is illustrated in two examples.

The assumption of log-concavity is a flexible and appealing nonparametric shape constraint in distribution modelling. In this work, we study the log-concave maximum likelihood estimator (MLE) of a probability mass function (pmf). We show that the MLE is strongly consistent and derive its pointwise asymptotic theory under both the well- and misspecified setting. Our asymptotic results are used to calculate confidence intervals for the true log-concave pmf. Both the MLE and the associated confidence intervals may be easily computed using the R package logcondiscr. We illustrate our theoretical results using recent data from the H1N1 pandemic in Ontario, Canada., Comment: 21 pages, 7 Figures

We introduce a new smooth estimator of the ROC curve based on log-concave density estimates of the constituent distributions. We show that our estimate is asymptotically equivalent to the empirical ROC curve if the underlying densities are in fact log-concave. In addition, we empirically show that our proposed estimator exhibits an efficiency gain for finite sample sizes with respect to the standard empirical estimate in various scenarios and that it is only slightly less efficient, if at all, compared to the fully parametric binormal estimate in case the underlying distributions are normal. The estimator is also quite robust against modest deviations from the log-concavity assumption. We show that bootstrap confidence intervals for the value of the ROC curve at a fixed false positive fraction based on the new estimate are on average shorter compared to the approach by Zhou & Qin (2005), while maintaining coverage probability. Computation of our proposed estimate uses the R package logcondens that implements univariate log-concave density estimation and can be done very efficiently using only one line of code. These obtained results lead us to advocate our estimate for a wide range of scenarios., Comment: 23 pages, 6 figures, 2 tables

Publication bias, the fact that studies identified for inclusion in a meta analysis do not represent all studies on the topic of interest, is commonly recognized as a threat to the validity of the results of a meta analysis. One way to explicitly model publication bias is via selection models or weighted probability distributions. We adopt the nonparametric approach initially introduced by Dear (1992) but impose that the weight function $w$ is monotonely non-increasing as a function of the $p$-value. Since in meta analysis one typically only has few studies or "observations", regularization of the estimation problem seems sensible. In addition, virtually all parametric weight functions proposed so far in the literature are in fact decreasing. We discuss how to estimate a decreasing weight function in the above model and illustrate the new methodology on two well-known examples. The new approach potentially offers more insight in the selection process than other methods and is more flexible than parametric approaches. Some basic properties of the log-likelihood function and computation of a $p$-value quantifying the evidence against the null hypothesis of a constant weight function are indicated. In addition, we provide an approximate selection bias adjusted profile likelihood confidence interval for the treatment effect. The corresponding software and the datasets used to illustrate it are provided as the R package selectMeta. This enables full reproducibility of the results in this paper., Comment: 15 pages, 2 figures. Some minor changes according to reviewer comments

In this paper, we consider the problem of finding the Least Squares estimators of two isotonic regression curves $g^\circ_1$ and $g^\circ_2$ under the additional constraint that they are ordered; e.g., $g^\circ_1 \le g^\circ_2$. Given two sets of $n$ data points $y_1, ..., y_n$ and $z_1, >...,z_n$ observed at (the same) design points, the estimates of the true curves are obtained by minimizing the weighted Least Squares criterion $L_2(a, b) = \sum_{j=1}^n (y_j - a_j)^2 w_{1,j}+ \sum_{j=1}^n (z_j - b_j)^2 w_{2,j}$ over the class of pairs of vectors $(a, b) \in \mathbb{R}^n \times \mathbb{R}^n $ such that $a_1 \le a_2 \le ...\le a_n $, $b_1 \le b_2 \le ...\le b_n $, and $a_i \le b_i, i=1, ...,n$. The characterization of the estimators is established. To compute these estimators, we use an iterative projected subgradient algorithm, where the projection is performed with a "generalized" pool-adjacent-violaters algorithm (PAVA), a byproduct of this work. Then, we apply the estimation method to real data from mechanical engineering., Comment: 23 pages, 2 figures. Second revised version according to reviewer comments

In biomedical studies, researchers are often interested in assessing the association between one or more ordinal explanatory variables and an outcome variable, at the same time adjusting for covariates of any type. The outcome variable may be continuous, binary, or represent censored survival times. In the absence of precise knowledge of the response function, using monotonicity constraints on the ordinal variables improves efficiency in estimating parameters, especially when sample sizes are small. An active set algorithm that can efficiently compute such estimators is proposed, and a characterization of the solution is provided. Having an efficient algorithm at hand is especially relevant when applying likelihood ratio tests in restricted generalized linear models, where one needs the value of the likelihood at the restricted maximizer. The algorithm is illustrated on a real life data set from oncology., Comment: 24 pages, 1 Figure, 3 Tables

We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least $(\log(n)/n)^{1/3}$ and typically $(\log(n)/n)^{2/5}$, whereas the difference between the empirical and estimated distribution function vanishes with rate $o_{\mathrm{p}}(n^{-1/2})$ under certain regularity assumptions., Comment: Published in at http://dx.doi.org/10.3150/08-BEJ141 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm), Version 3 is the extended technical report cited in version 4

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=\exp\varphi_0$ where $\varphi_0$ is a concave function on $\mathbb{R}$. The pointwise limiting distributions depend on the second and third derivatives at 0 of $H_k$, the "lower invelope" of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of $\varphi_0=\log f_0$ at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode $M(f_0)$ and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values., Comment: Published in at http://dx.doi.org/10.1214/08-AOS609 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We develop an active set algorithm for the maximum likelihood estimation of a log-concave density based on complete data. Building on this fast algorithm, we indidate an EM algorithm to treat arbitrarily censored or binned data., Comment: Changes in versions 3-4: Updated references, corrections of minor errors, shortened section on EM algorithm (which will be treated in more detail in a separate paper)

Both parametric distribution functions appearing in extreme value theory - the generalized extreme value distribution and the generalized Pareto distribution - have log-concave densities if the extreme value index gamma is in [-1,0]. Replacing the order statistics in tail index estimators by their corresponding quantiles from the distribution function that is based on the estimated log-concave density leads to novel smooth quantile and tail index estimators. These new estimators aim at estimating the tail index especially in small samples. Acting as a smoother of the empirical distribution function, the log-concave distribution function estimator reduces estimation variability to a much greater extent than it introduces bias. As a consequence, Monte Carlo simulations demonstrate that the smoothed version of the estimators are well superior to their non-smoothed counterparts, in terms of mean squared error., Comment: 17 pages, 5 figures. Slightly changed Pickand's estimator, added some more introduction and discussion

Marshall's [Nonparametric Techniques in Statistical Inference (1970) 174--176] lemma is an analytical result which implies $\sqrt{n}$--consistency of the distribution function corresponding to the Grenander [Skand. Aktuarietidskr. 39 (1956) 125--153] estimator of a non-decreasing probability density. The present paper derives analogous results for the setting of convex densities on $[0,\infty)$., Comment: Published at http://dx.doi.org/10.1214/074921707000000292 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)