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SummaryIn this discussion, we examine the contributions of Qian et al. (2021) and potential applications of the newly developed estimator for the causal excursion effect in binary outcome data. Specifically, we consider extension of their method to count outcomes and observational data, propose an alternative use of their method for analysing excursion effect trajectories and discuss ways of improving estimator efficiency.

In prevalent cohort studies where subjects are recruited at a cross-section, the time to an event may be subject to length-biased sampling, with the observed data being either the forward recurrence time, or the backward recurrence time, or their sum. In the regression setting, assuming a semiparametric accelerated failure time model for the underlying event time, where the intercept parameter is absorbed into the nuisance parameter, it has been shown that the model remains invariant under these observed data set-ups and can be fitted using standard methodology for accelerated failure time model estimation, ignoring the length-bias. However, the efficiency of these estimators is unclear, owing to the fact that the observed covariate distribution, which is also length-biased, may contain information about the regression parameter in the accelerated life model. We demonstrate that if the true covariate distribution is completely unspecified, then the naive estimator based on the conditional likelihood given the covariates is fully efficient for the slope.

We propose a multithreshold change plane regression model which naturally partitions the observed subjects into subgroups with different covariate effects. The underlying grouping variable is a linear function of observed covariates and thus multiple thresholds produce change planes in the covariate space. We contribute a novel two-stage estimation approach to determine the number of subgroups, the location of thresholds, and all other regression parameters. In the first stage we adopt a group selection principle to consistently identify the number of subgroups, while in the second stage change point locations and model parameter estimates are refined by a penalized induced smoothing technique. Our procedure allows sparse solutions for relatively moderate- or high-dimensional covariates. We further establish the asymptotic properties of our proposed estimators under appropriate technical conditions. We evaluate the performance of the proposed methods by simulation studies and provide illustrations using two medical data examples. Our proposal for subgroup identification may lead to an immediate application in personalized medicine.

Summary Left-truncation poses extra challenges for the analysis of complex time-to-event data. We propose a general semiparametric regression model for left-truncated and right-censored competing risks data that is based on a novel weighted conditional likelihood function. Targeting the subdistribution hazard, our parameter estimates are directly interpretable with regard to the cumulative incidence function. We compare different weights from recent literature and develop a heuristic interpretation from a cure model perspective that is based on pseudo risk sets. Our approach accommodates external time-dependent covariate effects on the subdistribution hazard. We establish consistency and asymptotic normality of the estimators and propose a sandwich estimator of the variance. In comprehensive simulation studies we demonstrate solid performance of the proposed method. Comparing the sandwich estimator with the inverse Fisher information matrix, we observe a bias for the inverse Fisher information matrix and diminished coverage probabilities in settings with a higher percentage of left-truncation. To illustrate the practical utility of the proposed method, we study its application to a large HIV vaccine efficacy trial dataset.

Random survival forest and survival trees are popular models in statistics and machine learning. However, there is a lack of general understanding regarding consistency, splitting rules and influence of the censoring mechanism. In this paper, we investigate the statistical properties of existing methods from several interesting perspectives. First, we show that traditional splitting rules with censored outcomes rely on a biased estimation of the within-node failure distribution. To exactly quantify this bias, we develop a concentration bound of the within-node estimation based on non i.i.d. samples and apply it to the entire forest. Second, we analyze the entanglement between the failure and censoring distributions caused by univariate splits, and show that without correcting the bias at an internal node, survival tree and forest models can still enjoy consistency under suitable conditions. In particular, we demonstrate this property under two cases: a finite-dimensional case where the splitting variables and cutting points are chosen randomly, and a high-dimensional case where the covariates are weakly correlated. Our results can also degenerate into an independent covariate setting, which is commonly used in the random forest literature for high-dimensional sparse models. However, it may not be avoidable that the convergence rate depends on the total number of variables in the failure and censoring distributions. Third, we propose a new splitting rule that compares bias-corrected cumulative hazard functions at each internal node. We show that the rate of consistency of this new model depends only on the number of failure variables, which improves from non-bias-corrected versions. We perform simulation studies to confirm that this can substantially benefit the prediction error.

We propose interval censored recursive forests (ICRF), an iterative tree ensemble method for interval censored survival data. This nonparametric regression estimator addresses the splitting bias problem of existing tree-based methods and iteratively updates survival estimates in a self-consistent manner. Consistent splitting rules are developed for interval censored data, convergence is monitored using out-of-bag samples, and kernel-smoothing is applied. The ICRF is uniformly consistent and displays high prediction accuracy in both simulations and applications to avalanche and national mortality data. An R package icrf is available on CRAN and Supplementary Materials for this article are available online.

We propose a projection pursuit technique in survival analysis for finding lower-dimensional projections that exhibit differentiated survival outcome. This idea is formally introduced as the change-plane Cox model, a non-regular Cox model with a change-plane in the covariate space dividing the population into two subgroups whose hazards are proportional. The proposed technique offers a potential framework for principled subgroup discovery. Estimation of the change-plane is accomplished via likelihood maximization over a data-driven sieve constructed using sliced inverse regression. Consistency of the sieve procedure for the change-plane parameters is established. In simulations the sieve estimator demonstrates better classification performance for subgroup identification than alternatives.

We develop an approach for feature elimination in statistical learning with kernel machines, based on recursive elimination of features.We present theoretical properties of this method and show that it is uniformly consistent in finding the correct feature space under certain generalized assumptions.We present four case studies to show that the assumptions are met in most practical situations and present simulation results to demonstrate performance of the proposed approach., 50 pages, 5 figures, submitted to Annals of Statistics

Direct regression modeling of the subdistribution has become popular for analyzing data with multiple, competing event types. All general approaches so far are based on non-likelihood based procedures and target covariate effects on the subdistribution. We introduce a novel weighted likelihood function that allows for a direct extension of the Fine-Gray model to a broad class of semiparametric regression models. The model accommodates time-dependent covariate effects on the subdistribution hazard. To motivate the proposed likelihood method, we derive standard nonparametric estimators and discuss a new interpretation based on pseudo risk sets. We establish consistency and asymptotic normality of the estimators and propose a sandwich estimator of the variance. In comprehensive simulation studies we demonstrate the solid performance of the weighted NPMLE in the presence of independent right censoring. We provide an application to a very large bone marrow transplant dataset, thereby illustrating its practical utility.

Nonparametric regression models do not require the specification of the functional form between the outcome and the covariates. Despite their popularity, the amount of diagnostic statistics, in comparison to their parametric counter-parts, is small. We propose a goodness-of-fit test for nonparametric regression models with linear smoother form. In particular, we apply this testing framework to smoothing spline ANOVA models. The test can consider two sources of lack-of-fit: whether covariates that are not currently in the model need to be included, and whether the current model fits the data well. The proposed method derives estimated residuals from the model. Then, statistical dependence is assessed between the estimated residuals and the covariates using the HSIC. If dependence exists, the model does not capture all the variability in the outcome associated with the covariates, otherwise the model fits the data well. The bootstrap is used to obtain p-values. Application of the method is demonstrated with a neonatal mental development data analysis. We demonstrate correct type I error as well as power performance through simulations.

A personalized treatment strategy formalizes evidence-based treatment selection by mapping patient information to a recommended treatment. Personalized treatment strategies can produce better patient outcomes while reducing cost and treatment burden. Thus, among clinical and intervention scientists, there is a growing interest in conducting randomized clinical trials when one of the primary aims is estimation of a personalized treatment strategy. However, at present, there are no appropriate sample size formulae to assist in the design of such a trial. Furthermore, because the sampling distribution of the estimated outcome under an estimated optimal treatment strategy can be highly sensitive to small perturbations in the underlying generative model, sample size calculations based on standard (uncorrected) asymptotic approximations or computer simulations may not be reliable. We offer a simple and robust method for powering a single stage, two-armed randomized clinical trial when the primary aim is estimating the optimal single stage personalized treatment strategy. The proposed method is based on inverting a plugin projection confidence interval and is thereby regular and robust to small perturbations of the underlying generative model. The proposed method requires elicitation of two clinically meaningful parameters from clinical scientists and uses data from a small pilot study to estimate nuisance parameters, which are not easily elicited. The method performs well in simulated experiments and is illustrated using data from a pilot study of time to conception and fertility awareness.

As a new strategy for treatment which takes individual heterogeneity into consideration, personalized medicine is of growing interest. Discovering individualized treatment rules (ITRs) for patients who have heterogeneous responses to treatment is one of the important areas in developing personalized medicine. As more and more information per individual is being collected in clinical studies and not all of the information is relevant for treatment discovery, variable selection becomes increasingly important in discovering individualized treatment rules. In this article, we develop a variable selection method based on penalized outcome weighted learning through which an optimal treatment rule is considered as a classification problem where each subject is weighted proportional to his or her clinical outcome. We show that the resulting estimator of the treatment rule is consistent and establish variable selection consistency and the asymptotic distribution of the estimators. The performance of the proposed approach is demonstrated via simulation studies and an analysis of chronic depression data.

Individualized treatment rules recommend treatments based on individual patient characteristics in order to maximize clinical benefit. When the clinical outcome of interest is survival time, estimation is often complicated by censoring. We develop nonparametric methods for estimating an optimal individualized treatment rule in the presence of censored data. To adjust for censoring, we propose a doubly robust estimator which requires correct specification of either the censoring model or survival model, but not both; the method is shown to be Fisher consistent when either model is correct. Furthermore, we establish the convergence rate of the expected survival under the estimated optimal individualized treatment rule to the expected survival under the optimal individualized treatment rule. We illustrate the proposed methods using simulation study and data from a Phase III clinical trial on non-small cell lung cancer.

This article introduces a new machine learning task, called latent supervised learning, where the goal is to learn a binary classifier from continuous training labels that serve as surrogates for the unobserved class labels. We investigate a specific model where the surrogate variable arises from a two-component Gaussian mixture with unknown means and variances, and the component membership is determined by a hyperplane in the covariate space. The estimation of the separating hyperplane and the Gaussian mixture parameters forms what shall be referred to as the change-line classification problem. We propose a data-driven sieve maximum likelihood estimator for the hyperplane, which in turn can be used to estimate the parameters of the Gaussian mixture. The estimator is shown to be consistent. Simulations as well as empirical data show the estimator has high classification accuracy.

SUMMARY In single hypothesis testing, power is a nondecreasing function of Type I error rate; hence it is desirable to test at the nominal level exactly to achieve optimal power. The optimal power puzzle arises from the fact that for multiple testing under the false discovery rate paradigm, such a monotonic relationship may not hold. In particular, exact false discovery rate control may lead to a less powerful testing procedure if a test statistic fails to fulfil the monotone likelihood ratio condition. In this article, we identify different scenarios wherein the condition fails and give caveats for conducting multiple testing in practical settings. We study an important assumption that has been used implicitly in the multiple testing literature. In the context of false discovery rate analysis (Benjamini & Hochberg, 1995), we show that the assumption can be violated in many important settings. The goal of this article is to explicitly state the assumption to bridge the gap in conventional methodological development, rigorously investigate the legitimacy of the assumption in various settings, and give caveats for conducting multiple testing in practice. To identify this assumption, it is helpful to first examine closely the framework of single hypothesis testing. Suppose we want to test H0 versus H1 based on the observed value of a continuous random variable X. A binary decision rule δ ∈{ 0,1} divides the sample space S into two regions, S = S0 ∪ S1, such that δ =0 when X ∈ S0 and δ =1 when X ∈ S1 .L etT (·) be a function of X, with small values indicating evidence against H0. The critical region S1 can be expressed as S1 ={ x ∈ S : T (x ) t }= 1 − G1(t), respectively. Since α(t) increasesint and β(t)decreasesint,weconcludethat β(t)decreasesin α(t).Thereforetheoptimalchoice

Estimating individualized treatment rules is a central task for personalized medicine. [zhao2012estimating] and [zhang2012robust] proposed outcome weighted learning to estimate individualized treatment rules directly through maximizing the expected outcome without modeling the response directly. In this paper, we extend the outcome weighted learning to right censored survival data without requiring either an inverse probability of censoring weighting or a semiparametric modeling of the censoring and failure times as done in [zhao2015doubly]. To accomplish this, we take advantage of the tree based approach proposed in [zhu2012recursively] to nonparametrically impute the survival time in two different ways. The first approach replaces the reward of each individual by the expected survival time, while in the second approach only the censored observations are imputed by their conditional expected failure times. We establish consistency and convergence rates for both estimators. In simulation studies, our estimators demonstrate improved performance compared to existing methods. We also illustrate the proposed method on a phase III clinical trial of non-small cell lung cancer., Comment: Accepted by EJS

Personalized medicine has received increasing attention among statisticians, computer scientists, and clinical practitioners. A major component of personalized medicine is the estimation of individualized treatment rules (ITRs). Recently, Zhao et al. (2012) proposed outcome weighted learning (OWL) to construct ITRs that directly optimize the clinical outcome. Although OWL opens the door to introducing machine learning techniques to optimal treatment regimes, it still has some problems in performance. In this article, we propose a general framework, called Residual Weighted Learning (RWL), to improve finite sample performance. Unlike OWL which weights misclassification errors by clinical outcomes, RWL weights these errors by residuals of the outcome from a regression fit on clinical covariates excluding treatment assignment. We utilize the smoothed ramp loss function in RWL, and provide a difference of convex (d.c.) algorithm to solve the corresponding non-convex optimization problem. By estimating residuals with linear models or generalized linear models, RWL can effectively deal with different types of outcomes, such as continuous, binary and count outcomes. We also propose variable selection methods for linear and nonlinear rules, respectively, to further improve the performance. We show that the resulting estimator of the treatment rule is consistent. We further obtain a rate of convergence for the difference between the expected outcome using the estimated ITR and that of the optimal treatment rule. The performance of the proposed RWL methods is illustrated in simulation studies and in an analysis of cystic fibrosis clinical trial data., 48 pages, 3 figures

We develop a unified approach for classification and regression support vector machines for data subject to right censoring. We provide finite sample bounds on the generalization error of the algorithm, prove risk consistency for a wide class of probability measures, and study the associated learning rates. We apply the general methodology to estimation of the (truncated) mean, median, quantiles, and for classification problems. We present a simulation study that demonstrates the performance of the proposed approach., Comment: In this version, we strengthened the theoretical results and corrected a few mistakes

We present an asymptotic exponential bound for the deviation of the survival function estimator of the Cox model. We show that the bound holds even when the proportional hazards assumption does not hold.

Inspired by the non-regular framework studied in Laber and Murphy (2011), we propose a family of adaptive classifiers. We discuss briefly their asymptotic properties and show that under the non-regular framework these classifiers have an “oracle property,” and consequently have smaller asymptotic variance and smaller asymptotic test error variance than those of the original classifier. We also show that confidence intervals for the test error of the adaptive classifiers, based on either normal approximation or centered percentile bootstrap, are consistent.

Outcome-dependent sampling designs have been shown to be a cost-effective way to enhance study efficiency. We show that the outcome-dependent sampling design with a continuous outcome can be viewed as an extension of the two-stage case-control designs to the continuous-outcome case. We further show that the two-stage outcome-dependent sampling has a natural link with the missing-data and biased-sampling frameworks. Through the use of semiparametric inference and missing-data techniques, we show that a certain semiparametric maximum-likelihood estimator is computationally convenient and achieves the semiparametric efficient information bound. We demonstrate this both theoretically and through simulation. Copyright 2009, Oxford University Press.

High dimensional classification problems are prevalent in a wide range of modern scientific applications. Despite a large number of candidate classification techniques available to use, practitioners often face a dilemma of choosing between linear and general nonlinear classifiers. Specifically, simple linear classifiers have good interpretability, but may have limitations in handling data with complex structures. In contrast, general nonlinear classifiers are more flexible, but may lose interpretability and have higher tendency for overfitting. In this paper, we consider data with potential latent subgroups in the classes of interest. We propose a new method, namely the Composite Large Margin Classifier (CLM), to address the issue of classification with latent subclasses. The CLM aims to find three linear functions simultaneously: one linear function to split the data into two parts, with each part being classified by a different linear classifier. Our method has comparable prediction accuracy to a general nonlinear classifier, and it maintains the interpretability of traditional linear classifiers. We demonstrate the competitive performance of the CLM through comparisons with several existing linear and nonlinear classifiers by Monte Carlo experiments. Analysis of the Alzheimer's disease classification problem using CLM not only provides a lower classification error in discriminating cases and controls, but also identifies subclasses in controls that are more likely to develop the disease in the future.

Change-point models are widely used by statisticians to model drastic changes in the pattern of observed data. Least squares/maximum likelihood based estimation of change-points leads to curious asymptotic phenomena. When the change-point model is correctly specified, such estimates generally converge at a fast rate ($n$) and are asymptotically described by minimizers of a jump process. Under complete mis-specification by a smooth curve, that is, when a change-point model is fitted to data described by a smooth curve, the rate of convergence slows down to $n^{1/3}$ and the limit distribution changes to that of the minimizer of a continuous Gaussian process. In this paper, we provide a bridge between these two extreme scenarios by studying the limit behavior of change-point estimates under varying degrees of model mis-specification by smooth curves, which can be viewed as local alternatives. We find that the limiting regime depends on how quickly the alternatives approach a change-point model. We unravel a family of “intermediate” limits that can transition, at least qualitatively, to the limits in the two extreme scenarios. The theoretical results are illustrated via a set of carefully designed simulations. We also demonstrate how inference for the change-point parameter can be performed in absence of knowledge of the underlying scenario by resorting to sub-sampling techniques that involve estimation of the convergence rate.

Current status data arises when a continuous response is reduced to an indicator of whether the response is greater or less than a random threshold value. In this article we consider adaptive penalized M-estimators (including the penalized least squares estimators and the penalized maximum likelihood estimators) for nonparametric and semiparametric models with current status data, under the assumption that the unknown nonparametric parameters belong to unknown Sobolev spaces. The Cox model is used as a representative of the semiparametric models. It is shown that the modified penalized M-estimators of the nonparametric parameters can achieve adaptive convergence rates, even when the degrees of smoothing are not known in advance. $$\sqrt{n}$$ consistency, asymptotic normality and inference based on the weighted bootstrap for the estimators of the regression parameter in the Cox model are also established. A simulation study is conducted for the Cox model to evaluate the finite sample efficacy of the proposed approach and to compare it with the ordinary maximum likelihood estimator. It is demonstrated that the proposed method is computationally superior.We apply the proposed approach to the California Partner Study analysis.

In this note, we consider weighted log-rank statistics applied to clustered survival data with variable cluster sizes and arbitrary treatment assignments within clusters. Specifically, we verify that the contribution over the time interval for which the risk set proportion is arbitrarily small (the so-called “tail instability”) is asymptotically negligible. These results were claimed but not proven by Gangnon and Kosorok [2004. Sample-size formula for clustered survival data using weighted log-rank statistics. Biometrika 91, 263–275.] who developed sample size formulas in this context. The main difficulty is that standard martingale methods cannot be used on account of the dependencies within clusters, and new methods are required.

We consider frequentist inference for the parametric component θ separately from the nuisance parameter η in semiparametric models based on sampling from the posterior of the profile likelihood. We prove that this procedure gives a first-order–correct approximation to the maximum likelihood estimator and consistent estimation of the efficient Fisher information for θ, without computing derivatives or using complicated numerical approximations. An exact Bayesian interpretation is established under a certain data-dependent prior. The sampler is useful in particular when the nuisance parameter is not estimable at the rate, where neither bootstrap validity nor general automatic variance estimation has been theoretically justified. Even when the nuisance parameter is consistent and the bootstrap is known to be valid, the proposed Markov chain Monte Carlo procedure can yield computational savings, because maximization of the likelihood is not required. The theory is verified for three examples. The methods are ...

M-estimation is a widely used technique for statistical inference. In this paper, we study properties of ordinary and weighted M-estimators for semiparametric models, especially when there exist parameters that cannot be estimated at the n convergence rate. Results on consistency, rates of convergence for all parameters, and n consistency and asymptotic normality for the Euclidean parameters are provided. These results, together with a generic paradigm for studying semiparametric M-estimators, provide a valuable extension to previous related research on semiparametric maximum-likelihood estimators (MLEs). Although penalized M-estimation does not in general fit in the framework we discuss here, it is shown for a great variety of models that many of the forgoing results still hold, including the n consistency and asymptotic normality of the Euclidean parameters. For semiparametric M-estimators that are not likelihood based, general inference procedures for the Euclidean parameters have not previously been developed. We demonstrate that our paradigm leads naturally to verification of the validity of the weighted bootstrap in this setting. For illustration, several examples are investigated in detail. The new M-estimation framework and accompanying weighted bootstrap technique shed light on a universal way of investigating semiparametric models.

As a useful alternative to Cox's proportional hazard model, the additive risk model assumes that the hazard function is the sum of the baseline hazard function and the regression function of covariates. This article is concerned with estimation and prediction for the additive risk models with right censored survival data, especially when the dimension of the covariates is comparable to or larger than the sample size. Principal component regression is proposed to give unique and numerically stable estimators. Asymptotic properties of the proposed estimators, component selection based on the weighted bootstrap, and model evaluation techniques are discussed. This approach is illustrated with analysis of the primary biliary cirrhosis clinical data and the diffuse large B-cell lymphoma genomic data. It is shown that this methodology is numerically stable and effective in dimension reduction, while still being able to provide satisfactory prediction and classification results.

This paper introduces the “piggyback bootstrap.” Like the weighted bootstrap, this bootstrap procedure can be used to generate random draws that approximate the joint sampling distribution of the parametric and nonparametric maximum likelihood estimators in various semiparametric models, but the dimension of the maximization problem for each bootstrapped likelihood is smaller. This reduction results in significant computational savings in comparison to the weighted bootstrap. The procedure can be stated quite simply. First obtain a valid random draw for the parametric component of the model. Then take the draw for the nonparametric component to be the maximizer of the weighted bootstrap likelihood with the parametric component fixed at the parametric draw. We prove the procedure is valid for a class of semiparametric models that includes frailty regression models airsing in survival analysis and biased sampling models that have application to vaccine efficacy trials. Bootstrap confidence sets from the piggyback, and weighted bootstraps are compared for biased sampling data from simulated vaccine efficacy trials.

In many multicenter, randomized clinical trials, the primary outcome is the time to the first of a number of possible clinical events. An event classification committee may be convened to determine whether events that have been reported by investigators meet the predetermined criteria for primary endpoint events. When interim analyses are performed in such trials, the final classification for many reported events will not be known. Failure to account for the uncertain status of these events may result in incorrect interim analysis. The probability that an unadjudicated event will be confirmed as a primary event can typically be estimated from those events for which adjudication is complete. We show that if each unadjudicated event is weighted according to the probability that it will be the first primary event, then consistent estimates of survival probabilities and regression parameters can be obtained and unbiased log-rank tests of treatment differences performed. Moderate sample consistency of point es...

SUMMARY We consider regression for response and covariates which are temporal processes observed over intervals. A functional generalised linear model is proposed which includes extensions of standard models in multi-state survival analysis. Simple nonparametric esti mators of time-indexed parameters are developed using 'working independence' estimating equations and are shown to be uniformly consistent and to converge weakly to Gaussian processes. The procedure does not require smoothing or a Markov assumption, unlike approaches based on transition intensities. The usual definition of optimal estimating equations for parametric models is then generalised to the functional model and the optimum is identified in a class of functional generalised estimating equations. Simulations demonstrate large efficiency gains relative to working independence at times where censoring is heavy. The estimators are the basis for new tests of the covariate effects and for the estimation of models in which greater structure is imposed on the parameters, providing novel goodness-of-fit tests. The methodology's practical utility is illustrated in a data analysis.

SUMMARY We present a simple sample-size formula for weighted log-rank statistics applied to clustered survival data with variable cluster sizes and arbitrary treatment assignments within clusters. This formula is based on the asymptotic normality of weighted log-rank statistics under certain local alternatives in the clustered data context. We also provide consistent variance estimators. The derived sample-size formula reduces to Schoenfeld's (1983) formula for cases of no clustering or independence within clusters. Simulation results verify control of the Type I error and accuracy of the sample-size formula. Use of the sample-size formula in an event-driven clinical trial design is illustrated using data from the Early Treatment Diabetic Retinopathy Study.

A central limit theorem is developed for sums of independent but not identically distributed stochastic processes multiplied by independent real random variables with mean zero. Weak convergence of the Hoffmann–Jørgensen–Dudley type, as described in van der Vaart and Wellner (Weak Convergence and Empirical Processes, Springer, New York, 1996), is utilized. These results allow Monte Carlo estimation of limiting probability measures obtained from application of Pollard's (Empirical Processes: Theory and Applications, IMS, Hayward, CA, 1990) functional central limit theorem for empirical processes. An application of this theory to the two-parameter Cox score process with staggered entry data is given for illustration. For this process, the proposed multiplier bootstrap appears to be the first successful method for estimating the associated limiting distribution. The results of this paper compliment previous bootstrap and multiplier central limit theorems for independent and identically distributed empirical processes.

In this article, we introduce a new type of tree-based method, reinforcement learning trees (RLT), which exhibits significantly improved performance over traditional methods such as random forests (Breiman 2001) under high-dimensional settings. The innovations are three-fold. First, the new method implements reinforcement learning at each selection of a splitting variable during the tree construction processes. By splitting on the variable that brings the greatest future improvement in later splits, rather than choosing the one with largest marginal effect from the immediate split, the constructed tree uses the available samples in a more efficient way. Moreover, such an approach enables linear combination cuts at little extra computational cost. Second, we propose a variable muting procedure that progressively eliminates noise variables during the construction of each individual tree. The muting procedure also takes advantage of reinforcement learning and prevents noise variables from being considered in the search for splitting rules, so that toward terminal nodes, where the sample size is small, the splitting rules are still constructed from only strong variables. Last, we investigate asymptotic properties of the proposed method under basic assumptions and discuss rationale in general settings. Supplementary materials for this article are available online.

Simple extensions of the nonparametric bivariate survival estimator of Lin and Ying (Biometrika 80 (1993) 573) are proposed which permit consistent estimation over the entire support of the censored data distribution. These extensions utilize a simple class of stopping times. Large sample uniform consistency of the proposals is also established.

In many clinical studies, there are two dependent event times with one of the events being terminal, such as death, and the other being nonfatal, such as myocardial infarction or cancer relapse. Morbidity can be dependently censored by mortality, but not vice versa. Asymptotic theory is developed for simultaneous estimation of the marginal distribution functions in this semi-competing risks setting. We specify the joint distribution of the event times in the upper wedge, where the nonfatal event happens before the terminal event, with the popular gamma frailty model. The estimators are based on an adaptation of the self-consistency principle. To study their properties, we employ a modification of the functional delta-method applied to Z-estimators. This approach to weak convergence leads naturally to asymptotic validity of both the nonparametric and multiplier bootstraps, facilitating inference in spite of the complexity of the limiting distribution.

In this paper, the conservative Monte Carlo error estimation methods and theory developed in Geyer (1992a, Statist. Sci. 7, 473–483) are extended from univariate to multivariate Markov chain applications. A small simulation study demonstrates the feasibility of the proposed estimators.

Summary A simple, nonparametric two-sample test for equality of a given collection of quantiles is developed which can be applied to a variety of empirical distribution functions, including the Kaplan‐Meier estimator, a self-consistent estimator for doubly-censored data and an estimator for repeated measures data. The null hypothesis tested is that the quantiles are equal but other aspects of the distributions may diVer between the two samples. This procedure can also be applied to quantile testing in group sequential clinical trials with staggered patient entry. A simple simulation study demonstrates that the moderate sample size properties of this procedure are reasonable.

We discuss a simple simulation method for construction of exact confidence bands, having a pre-assigned confidence level simultaneously for several regression functions which are univariate polynomials in the predictors. This is accomplished by combining and extending existing results in a manner that permits both finite and infinite ranges for individual predictor variables. We illustrate the method for a logistic regression model with both dichotomous and continuous predictors.

Two-sample weighted log-rank statistics are used in the presence of right censoring to test whether failure times from two populations have different survival distributions. Kosorok has showed that large families of these statistics form stochastic processes indexed by weight functions, and that these function-indexed statistics can be used to construct versatile test procedures simultaneously sensitive to a wide array of both ordered hazards and stochastic ordering alternatives. The complexity of the asymptotic distribution of these statistics precludes obtaining p values through analytical means. In this article we develop a Monte Carlo method for accurately obtaining these p values, and we evaluate the moderate sample size properties of this method and compare the power of function-indexed statistics with previously developed weighted log-rank tests. These statistics are also examined in a data analysis of the Beta-Blocker Heart Attack Trial (BHAT). The results of this article demonstrate that...

A version of the discrete proportional hazards model is developed for psychometrical applications. In such applications, a primary covariate that influences failure times is a latent variable representing a psychological construct. The Metropolis-Hastings algorithm is studied as a method for performing marginal likelihood inference on the item parameters. The model is illustrated with a real data example that relates the age at which teenagers first experience various substances to the latent ability to avoid the onset of such behaviors.

The nonparametric monotone MLE is used to overcome the severe under-estimation of survival functions by the NPMLE for left truncated data when a monotone hazard assumption is appropriate. In this paper, we establish the consistency of the monotone MLE for interval-censored data with or without left truncation in two different ways corresponding to two different realistic conditions.

A simple Markov model is developed for assessing the predictive effect of time-dependent covariates on an intermittently observed ordinal response in continuous time. This is accomplished by reparameterizing an ergodic intensity matrix in terms of its equilibrium distribution and a parametrically independent component that assesses the rate of movement between ordinal categories. The effect of covariates on the equilibrium distribution can then be modeled using any link appropriate for ordinal data. A robust maximum likelihood estimator based on this model that is consistent and asymptotically normal is constructed. Practical data analysis issues are discussed, and a simple diagnostic tool for assessing model adequacy is developed. The utility of these methods is demonstrated with several analyses of visual acuity data, including a comparison analysis based on generalized estimating equation (GEE) methods.

We develop methodology for a multistage decision problem with flexible number of stages in which the rewards are survival times that are subject to censoring. We present a novel Q-learning algorithm that is adjusted for censored data and allows a flexible number of stages. We provide finite sample bounds on the generalization error of the policy learned by the algorithm, and show that when the optimal Q-function belongs to the approximation space, the expected survival time for policies obtained by the algorithm converges to that of the optimal policy. We simulate a multistage clinical trial with flexible number of stages and apply the proposed censored-Q-learning algorithm to find individualized treatment regimens. The methodology presented in this paper has implications in the design of personalized medicine trials in cancer and in other life-threatening diseases., Published in at http://dx.doi.org/10.1214/12-AOS968 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper, we study the nonparametric maximum likelihood estimator for an event time distribution function at a point in the current status model with observation times supported on a grid of potentially unknown sparsity and with multiple subjects sharing the same observation time. This is of interest since observation time ties occur frequently with current status data. The grid resolution is specified as $cn^{-\gamma}$ with $c>0$ being a scaling constant and $\gamma>0$ regulating the sparsity of the grid relative to $n$, the number of subjects. The asymptotic behavior falls into three cases depending on $\gamma$: regular Gaussian-type asymptotics obtain for $\gamma1/3$ and $\gamma=1/3$ serves as a boundary at which the transition happens. The limit distribution at the boundary is different from either of the previous cases and converges weakly to those obtained with $\gamma\in(0,1/3)$ and $\gamma\in(1/3,\infty)$ as $c$ goes to $\infty$ and 0, respectively. This weak convergence allows us to develop an adaptive procedure to construct confidence intervals for the value of the event time distribution at a point of interest without needing to know or estimate $\gamma$, which is of enormous advantage from the perspective of inference. A simulation study of the adaptive procedure is presented., Comment: Published in at http://dx.doi.org/10.1214/11-AOS942 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We propose recursively imputed survival tree (RIST) regression for right-censored data. This new nonparametric regression procedure uses a novel recursive imputation approach combined with extremely randomized trees that allows significantly better use of censored data than previous tree-based methods, yielding improved model fit and reduced prediction error. The proposed method can also be viewed as a type of Monte Carlo EM algorithm, which generates extra diversity in the tree-based fitting process. Simulation studies and data analyses demonstrate the superior performance of RIST compared with previous methods.

An inverse regression methodology for assessing predictor performance in the censored data setup is developed along with inference procedures and a computational algorithm. The technique developed here allows for conditioning on the unobserved failure time along with a weighting mechanism that accounts for the censoring. The implementation is nonparametric and computationally fast. This provides an efficient methodological tool that can be used especially in cases where the usual modeling assumptions are not applicable to the data under consideration. It can also be a good diagnostic tool that can be used in the model selection process. We have provided theoretical justification of consistency and asymptotic normality of the methodology. Simulation studies and two data analyses are provided to illustrate the practical utility of the procedure.

This article considers the problem of multiple hypothesis testing using $t$-tests. The observed data are assumed to be independently generated conditional on an underlying and unknown two-state hidden model. We propose an asymptotically valid data-driven procedure to find critical values for rejection regions controlling the $k$-familywise error rate ($k$-FWER), false discovery rate (FDR) and the tail probability of false discovery proportion (FDTP) by using one-sample and two-sample $t$-statistics. We only require a finite fourth moment plus some very general conditions on the mean and variance of the population by virtue of the moderate deviations properties of $t$-statistics. A new consistent estimator for the proportion of alternative hypotheses is developed. Simulation studies support our theoretical results and demonstrate that the power of a multiple testing procedure can be substantially improved by using critical values directly, as opposed to the conventional $p$-value approach. Our method is applied in an analysis of the microarray data from a leukemia cancer study that involves testing a large number of hypotheses simultaneously., Published in at http://dx.doi.org/10.3150/10-BEJ272 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

SUMMARY In clinical trials with failure time outcomes, the cost of follow-up for the endpoint of primary interest is sometimes expensive. In many of these settings, one or more secondary or surrogate failure time outcomes are also available which either occur much earlier or are much less expensive to follow than the primary outcome. This paper presents a method for using these secondary outcomes to increase the power of detecting a treatment effect on the primary outcome only, without introducing bias. The proposed method requires that patients be randomized to either extended or limited follow-up. A test statistic is constructed by forming a minimum variance linear combination of a linear rank statistic based on the primary outcome and a mean zero weighted sum of linear rank statistics based on secondary outcomes from both follow-up arms. Large-sample distribution theory for this statistic is developed while moderate sample behaviour is explored through a simulation study.