Methodology (stat.ME)

Abstract: This tutorial discusses a recently developed methodology for causal inference based on longitudinal modified treatment policies (LMTPs). LMTPs generalize many commonly used parameters for causal inference including average treatment effects, and facilitate the mathematical formalization, identification, and estimation of many novel parameters. LMTPs apply to a wide variety of exposures, including binary, multivariate, and continuous, as well as interventions that result in violations of the positivity assumption. LMTPs can accommodate time-varying treatments and confounders, competing risks, loss-to-follow-up, as well as survival, binary, or continuous outcomes. This tutorial aims to illustrate several practical uses of the LMTP framework, including describing different estimation strategies and their corresponding advantages and disadvantages. We provide numerous examples of types of research questions which can be answered within the proposed framework. We go into more depth with one of these examples -- specifically, estimating the effect of delaying intubation on critically ill COVID-19 patients' mortality. We demonstrate the use of the open source R package lmtp to estimate the effects, and we provide code on https://github.com/kathoffman/lmtp-tutorial.

Abstract: In this article, we consider statistical inference based on dependent competing risks data from Marshall-Olkin bivariate Weibull distribution. The maximum likelihood estimates of the unknown model parameters have been computed by using the Newton-Raphson method under adaptive Type II progressive hybrid censoring with partially observed failure causes. The existence and uniqueness of maximum likelihood estimates are derived. Approximate confidence intervals have been constructed via the observed Fisher information matrix using the asymptotic normality property of the maximum likelihood estimates. Bayes estimates and highest posterior density credible intervals have been calculated under gamma-Dirichlet prior distribution by using the Markov chain Monte Carlo technique. Convergence of Markov chain Monte Carlo samples is tested. In addition, a Monte Carlo simulation is carried out to compare the effectiveness of the proposed methods. Further, three different optimality criteria have been taken into account to obtain the most effective censoring plans. Finally, a real-life data set has been analyzed to illustrate the operability and applicability of the proposed methods.

Abstract: We propose a continuous optimization algorithm for the Column Subset Selection Problem (CSSP) and Nystr\"om approximation. The CSSP and Nystr\"om method construct low-rank approximations of matrices based on a predetermined subset of columns. It is well known that choosing the best column subset of size $k$ is a difficult combinatorial problem. In this work, we show how one can approximate the optimal solution by defining a penalized continuous loss function which is minimized via stochastic gradient descent. We show that the gradients of this loss function can be estimated efficiently using matrix-vector products with a data matrix $X$ in the case of the CSSP or a kernel matrix $K$ in the case of the Nystr\"om approximation. We provide numerical results for a number of real datasets showing that this continuous optimization is competitive against existing methods.

Abstract: Sequential, multiple assignment randomized trials (SMARTs), which assist in the optimization of adaptive interventions, are growing in popularity in education and behavioral sciences. This is unsurprising, as adaptive interventions reflect the sequential, tailored nature of learning in a classroom or school. Nonetheless, as is true elsewhere in education research, observed effect sizes in education-based SMARTs are frequently small. As a consequence, statistical efficiency is of paramount importance in their analysis. The contributions of this manuscript are two-fold. First, we provide an overview of adaptive interventions and SMART designs for researchers in education science. Second, we propose four techniques that have the potential to improve statistical efficiency in the analysis of SMARTs. We demonstrate the benefits of these techniques in SMART settings both through the analysis of a SMART designed to optimize an adaptive intervention for increasing cognitive behavioral therapy delivery in school settings and through a comprehensive simulation study. Each of the proposed techniques is easily implementable, either with over-the-counter statistical software or through R code provided in an online supplement.