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A theory of martingales for censoring is developed. The Doob-Meyer martingale is shown to be inadequate in general, and a repaired martingale is proposed with a non-predictable centering term. Associated martingale transforms, variation processes, and covariation processes are developed based on a measure of half-predictability that generalizes predictability. The development is applied to study the Kaplan Meier estimator.

We study causal inference and efficient estimation for the expected number of recurrent events in the presence of a terminal event. We define our estimand as the vector comprising both the expected number of recurrent events and the failure survival function evaluated along a sequence of landmark times. We identify the estimand in the presence of right-censoring and causal selection as an observed data functional under coarsening at random, derive the nonparametric efficiency bound, and propose a multiply-robust estimator that achieves the bound and permits nonparametric estimation of nuisance parameters. Throughout, no absolute continuity assumption is made on the underlying probability distributions of failure, censoring, or the observed data. Additionally, we derive the class of influence functions when the coarsening distribution is known and review how published estimators may belong to the class. Along the way, we highlight some interesting inconsistencies in the causal lifetime analysis literature.

This paper reconsiders several results of historical and current importance to nonparametric estimation of the survival distribution for failure in the presence of right-censored observation times, demonstrating in particular how Volterra integral equations of the first kind help inter-connect the resulting estimators. The paper begins by considering Efron's self-consistency equation, introduced in a seminal 1967 Berkeley symposium paper. Novel insights provided in the current work include the observations that (i) the self-consistency equation leads directly to an anticipating Volterra integral equation of the first kind whose solution is given by a product-limit estimator for the censoring survival function; (ii) a definition used in this argument immediately establishes the familiar product-limit estimator for the failure survival function; (iii) the usual Volterra integral equation for the product-limit estimator of the failure survival function leads to an immediate and simple proof that it can be represented as an inverse probability of censoring weighted estimator (i.e., under appropriate conditions). Finally, we show that the resulting inverse probability of censoring weighted estimators, attributed to a highly influential 1992 paper of Robins and Rotnitzky, were implicitly introduced in Efron's 1967 paper in its development of the redistribution-to-the-right algorithm. All results developed herein allow for ties between failure and/or censored observations., Comment: 21 pages

Constructing an optimal adaptive treatment strategy becomes complex when there are a large number of potential tailoring variables. In such scenarios, many of these extraneous variables may contribute little or no benefit to an adaptive strategy while increasing implementation costs and putting an undue burden on patients. Although existing methods allow selection of the informative prognostic factors, statistical inference is complicated by the data-driven selection process. To remedy this deficiency, we adapt the Universal Post-Selection Inference procedure to the semiparametric Robust Q-learning method and the unique challenges encountered in such multistage decision methods. In the process, we also identify a uniform improvement to confidence intervals constructed in this post-selection inference framework. Under certain rate assumptions, we provide theoretical results that demonstrate the validity of confidence regions and tests constructed from our proposed procedure. The performance of our method is compared to the Selective Inference framework through simulation studies, demonstrating the strengths of our procedure and its applicability to multiple selection mechanisms.

Researchers are often interested in learning not only the effect of treatments on outcomes, but also the pathways through which these effects operate. A mediator is a variable that is affected by treatment and subsequently affects outcome. Existing methods for penalized mediation analyses may lead to ignoring important mediators and either assume that finite-dimensional linear models are sufficient to remove confounding bias, or perform no confounding control at all. In practice, these assumptions may not hold. We propose a method that considers the confounding functions as nuisance parameters to be estimated using data-adaptive methods. We then use a novel regularization method applied to this objective function to identify a set of important mediators. We derive the asymptotic properties of our estimator and establish the oracle property under certain assumptions. Asymptotic results are also presented in a local setting which contrast the proposal with the standard adaptive lasso. We also propose a perturbation bootstrap technique to provide asymptotically valid post-selection inference for the mediated effects of interest. The performance of these methods will be discussed and demonstrated through simulation studies.

The use of cumulative incidence functions for characterizing the risk of one type of event in the presence of others has become increasingly popular over the past decade. The problems of modeling, estimation and inference have been treated using parametric, nonparametric and semi-parametric methods. Efforts to develop suitable extensions of machine learning methods, such as regression trees and related ensemble methods, have begun comparatively recently. In this paper, we propose a novel approach to estimating cumulative incidence curves in a competing risks setting using regression trees and associated ensemble estimators. The proposed methods employ augmented estimators of the Brier score risk as the primary basis for building and pruning trees, and lead to methods that are easily implemented using existing R packages. Data from the Radiation Therapy Oncology Group (trial 9410) is used to illustrate these new methods.

The use of cumulative incidence functions for characterizing the risk of one type of event in the presence of others has become increasingly popular over the past decade. The problems of modeling, estimation and inference have been treated using parametric, nonparametric and semi-parametric methods. Efforts to develop suitable extensions of machine learning methods, such as regression trees and related ensemble methods, have begun only recently. In this paper, we develop a novel approach to building regression trees for estimating cumulative incidence curves in a competing risks setting. The proposed methods employ augmented estimators of the Brier score risk as the primary basis for building and pruning trees. The proposed methods are easily implemented using the R statistical software package. Simulation studies demonstrate the utility of our approach in the competing risks setting. Data from the Radiation Therapy Oncology Group (trial 9410) is used to illustrate these new methods.

Q-learning is a regression-based approach that is widely used to formalize the development of an optimal dynamic treatment strategy. Finite dimensional working models are typically used to estimate certain nuisance parameters, and misspecification of these working models can result in residual confounding and/or efficiency loss. We propose a robust Q-learning approach which allows estimating such nuisance parameters using data-adaptive techniques. We study the asymptotic behavior of our estimators and provide simulation studies that highlight the need for and usefulness of the proposed method in practice. We use the data from the "Extending Treatment Effectiveness of Naltrexone" multi-stage randomized trial to illustrate our proposed methods.

Machine learning provides a broad framework for addressing high-dimensional prediction problems in classification and regression. While machine learning is often applied for imaging problems in medical physics, there are many efforts to apply these principles to biological data towards questions of radiation biology. Here, we provide a review of radiogenomics modeling frameworks and efforts towards genomically-guided radiotherapy. We first discuss medical oncology efforts to develop precision biomarkers. We next discuss similar efforts to create clinical assays for normal tissue or tumor radiosensitivity. We then discuss modeling frameworks for radiosensitivity and the evolution of machine learning to create predictive models for radiogenomics., Comment: 32 pages, 3 figures, 3 tables

In this paper, we propose a general class of algorithms for optimizing an extensive variety of nonsmoothly penalized objective functions that satisfy certain regularity conditions. The proposed framework utilizes the majorization-minimization (MM) algorithm as its core optimization engine. The resulting algorithms rely on iterated soft-thresholding, implemented componentwise, allowing for fast, stable updating that avoids the need for any high-dimensional matrix inversion. We establish a local convergence theory for this class of algorithms under weaker assumptions than previously considered in the statistical literature. We also demonstrate the exceptional effectiveness of new acceleration methods, originally proposed for the EM algorithm, in this class of problems. Simulation results and a microarray data example are provided to demonstrate the algorithm's capabilities and versatility., Comment: A revised version of this paper has been published in the Electronic Journal of Statistics