While well-established methods for time-to-event data are available when the proportional hazards assumption holds, there is no consensus on the best inferential approach under non-proportional hazards (NPH). However, a wide range of parametric and non-parametric methods for testing and estimation in this scenario have been proposed. To provide recommendations on the statistical analysis of clinical trials where non proportional hazards are expected, we conducted a comprehensive simulation study under different scenarios of non-proportional hazards, including delayed onset of treatment effect, crossing hazard curves, subgroups with different treatment effect and changing hazards after disease progression. We assessed type I error rate control, power and confidence interval coverage, where applicable, for a wide range of methods including weighted log-rank tests, the MaxCombo test, summary measures such as the restricted mean survival time (RMST), average hazard ratios, and milestone survival probabilities as well as accelerated failure time regression models. We found a trade-off between interpretability and power when choosing an analysis strategy under NPH scenarios. While analysis methods based on weighted logrank tests typically were favorable in terms of power, they do not provide an easily interpretable treatment effect estimate. Also, depending on the weight function, they test a narrow null hypothesis of equal hazard functions and rejection of this null hypothesis may not allow for a direct conclusion of treatment benefit in terms of the survival function. In contrast, non-parametric procedures based on well interpretable measures as the RMST difference had lower power in most scenarios. Model based methods based on specific survival distributions had larger power, however often gave biased estimates and lower than nominal confidence interval coverage.
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Sequential neural posterior estimation (SNPE) techniques have been recently proposed for dealing with simulation-based models with intractable likelihoods. Unlike approximate Bayesian computation, SNPE techniques learn the posterior from sequential simulation using neural network-based conditional density estimators by minimizing a specific loss function. The SNPE method proposed by Lueckmann et al. (2017) used a calibration kernel to boost the sample weights around the observed data, resulting in a concentrated loss function. However, the use of calibration kernels may increase the variances of both the empirical loss and its gradient, making the training inefficient. To improve the stability of SNPE, this paper proposes to use an adaptive calibration kernel and several variance reduction techniques. The proposed method greatly speeds up the process of training, and provides a better approximation of the posterior than the original SNPE method and some existing competitors as confirmed by numerical experiments.
A simple way of obtaining robust estimates of the "center" (or the "location") and of the "scatter" of a dataset is to use the maximum likelihood estimate with a class of heavy-tailed distributions, regardless of the "true" distribution generating the data. We observe that the maximum likelihood problem for the Cauchy distributions, which have particularly heavy tails, is geodesically convex and therefore efficiently solvable (Cauchy distributions are parametrized by the upper half plane, i.e. by the hyperbolic plane). Moreover, it has an appealing geometrical meaning: the datapoints, living on the boundary of the hyperbolic plane, are attracting the parameter by unit forces, and we search the point where these forces are in equilibrium. This picture generalizes to several classes of multivariate distributions with heavy tails, including, in particular, the multivariate Cauchy distributions. The hyperbolic plane gets replaced by symmetric spaces of noncompact type. Geodesic convexity gives us an efficient numerical solution of the maximum likelihood problem for these distribution classes. This can then be used for robust estimates of location and spread, thanks to the heavy tails of these distributions.
Appendicitis is among the most frequent reasons for pediatric abdominal surgeries. Previous decision support systems for appendicitis have focused on clinical, laboratory, scoring, and computed tomography data and have ignored abdominal ultrasound, despite its noninvasive nature and widespread availability. In this work, we present interpretable machine learning models for predicting the diagnosis, management and severity of suspected appendicitis using ultrasound images. Our approach utilizes concept bottleneck models (CBM) that facilitate interpretation and interaction with high-level concepts understandable to clinicians. Furthermore, we extend CBMs to prediction problems with multiple views and incomplete concept sets. Our models were trained on a dataset comprising 579 pediatric patients with 1709 ultrasound images accompanied by clinical and laboratory data. Results show that our proposed method enables clinicians to utilize a human-understandable and intervenable predictive model without compromising performance or requiring time-consuming image annotation when deployed. For predicting the diagnosis, the extended multiview CBM attained an AUROC of 0.80 and an AUPR of 0.92, performing comparably to similar black-box neural networks trained and tested on the same dataset.
Refinement calculus provides a structured framework for the progressive and modular development of programs, ensuring their correctness throughout the refinement process. This paper introduces a refinement calculus tailored for quantum programs. To this end, we first study the partial correctness of nondeterministic programs within a quantum while language featuring prescription statements. Orthogonal projectors, which are equivalent to subspaces of the state Hilbert space, are taken as assertions for quantum states. In addition to the denotational semantics where a nondeterministic program is associated with a set of trace-nonincreasing super-operators, we also present their semantics in transforming a postcondition to the weakest liberal postconditions and, conversely, transforming a precondition to the strongest postconditions. Subsequently, refinement rules are introduced based on these dual semantics, offering a systematic approach to the incremental development of quantum programs applicable in various contexts. To illustrate the practical application of the refinement calculus, we examine examples such as the implementation of a $Z$-rotation gate, the repetition code, and the quantum-to-quantum Bernoulli factory. Furthermore, we present Quire, a Python-based interactive prototype tool that provides practical support to programmers engaged in the stepwise development of correct quantum programs.
Personalized adaptive interventions offer the opportunity to increase patient benefits, however, there are challenges in their planning and implementation. Once implemented, it is an important question whether personalized adaptive interventions are indeed clinically more effective compared to a fixed gold standard intervention. In this paper, we present an innovative N-of-1 trial study design testing whether implementing a personalized intervention by an online reinforcement learning agent is feasible and effective. Throughout, we use a new study on physical exercise recommendations to reduce pain in endometriosis for illustration. We describe the design of a contextual bandit recommendation agent and evaluate the agent in simulation studies. The results show that, first, implementing a personalized intervention by an online reinforcement learning agent is feasible. Second, such adaptive interventions have the potential to improve patients' benefits even if only few observations are available. As one challenge, they add complexity to the design and implementation process. In order to quantify the expected benefit, data from previous interventional studies is required. We expect our approach to be transferable to other interventions and clinical interventions.
High-order methods for conservation laws can be very efficient, in particular on modern hardware. However, it can be challenging to guarantee their stability and robustness, especially for under-resolved flows. A typical approach is to combine a well-working baseline scheme with additional techniques to ensure invariant domain preservation. To obtain good results without too much dissipation, it is important to develop suitable baseline methods. In this article, we study upwind summation-by-parts operators, which have been used mostly for linear problems so far. These operators come with some built-in dissipation everywhere, not only at element interfaces as typical in discontinuous Galerkin methods. At the same time, this dissipation does not introduce additional parameters. We discuss the relation of high-order upwind summation-by-parts methods to flux vector splitting schemes and investigate their local linear/energy stability. Finally, we present some numerical examples for shock-free flows of the compressible Euler equations.
Model averaging has received much attention in the past two decades, which integrates available information by averaging over potential models. Although various model averaging methods have been developed, there are few literatures on the theoretical properties of model averaging from the perspective of stability, and the majority of these methods constrain model weights to a simplex. The aim of this paper is to introduce stability from statistical learning theory into model averaging. Thus, we define the stability, asymptotic empirical risk minimizer, generalization, and consistency of model averaging and study the relationship among them. Our results indicate that stability can ensure that model averaging has good generalization performance and consistency under reasonable conditions, where consistency means model averaging estimator can asymptotically minimize the mean squared prediction error. We also propose a L2-penalty model averaging method without limiting model weights and prove that it has stability and consistency. In order to reduce the impact of tuning parameter selection, we use 10-fold cross-validation to select a candidate set of tuning parameters and perform a weighted average of the estimators of model weights based on estimation errors. The Monte Carlo simulation and an illustrative application demonstrate the usefulness of the proposed method.
We propose and compare methods for the analysis of extreme events in complex systems governed by PDEs that involve random parameters, in situations where we are interested in quantifying the probability that a scalar function of the system's solution is above a threshold. If the threshold is large, this probability is small and its accurate estimation is challenging. To tackle this difficulty, we blend theoretical results from large deviation theory (LDT) with numerical tools from PDE-constrained optimization. Our methods first compute parameters that minimize the LDT-rate function over the set of parameters leading to extreme events, using adjoint methods to compute the gradient of this rate function. The minimizers give information about the mechanism of the extreme events as well as estimates of their probability. We then propose a series of methods to refine these estimates, either via importance sampling or geometric approximation of the extreme event sets. Results are formulated for general parameter distributions and detailed expressions are provided when Gaussian distributions. We give theoretical and numerical arguments showing that the performance of our methods is insensitive to the extremeness of the events we are interested in. We illustrate the application of our approach to quantify the probability of extreme tsunami events on shore. Tsunamis are typically caused by a sudden, unpredictable change of the ocean floor elevation during an earthquake. We model this change as a random process, which takes into account the underlying physics. We use the one-dimensional shallow water equation to model tsunamis numerically. In the context of this example, we present a comparison of our methods for extreme event probability estimation, and find which type of ocean floor elevation change leads to the largest tsunamis on shore.
Mediation analysis assesses the extent to which the exposure affects the outcome indirectly through a mediator and the extent to which it operates directly through other pathways. As the most popular method in empirical mediation analysis, the Baron-Kenny approach estimates the indirect and direct effects of the exposure on the outcome based on linear structural equation models. However, when the exposure and the mediator are not randomized, the estimates may be biased due to unmeasured confounding among the exposure, mediator, and outcome. Building on Cinelli and Hazlett (2020), we derive general omitted-variable bias formulas in linear regressions with vector responses and regressors. We then use the formulas to develop a sensitivity analysis method for the Baron-Kenny approach to mediation in the presence of unmeasured confounding. To ensure interpretability, we express the sensitivity parameters to correspond to the natural factorization of the joint distribution of the direct acyclic graph for mediation analysis. They measure the partial correlation between the unmeasured confounder and the exposure, mediator, outcome, respectively. With the sensitivity parameters, we propose a novel measure called the "robustness value for mediation" or simply the "robustness value", to assess the robustness of results based on the Baron-Kenny approach with respect to unmeasured confounding. Intuitively, the robustness value measures the minimum value of the maximum proportion of variability explained by the unmeasured confounding, for the exposure, mediator and outcome, to overturn the results of the point estimate or confidence interval for the direct and indirect effects. Importantly, we prove that all our sensitivity bounds are attainable and thus sharp.
The simulation of geological facies in an unobservable volume is essential in various geoscience applications. Given the complexity of the problem, deep generative learning is a promising approach to overcome the limitations of traditional geostatistical simulation models, in particular their lack of physical realism. This research aims to investigate the application of generative adversarial networks and deep variational inference for conditionally simulating meandering channels in underground volumes. In this paper, we review the generative deep learning approaches, in particular the adversarial ones and the stabilization techniques that aim to facilitate their training. The proposed approach is tested on 2D and 3D simulations generated by the stochastic process-based model Flumy. Morphological metrics are utilized to compare our proposed method with earlier iterations of generative adversarial networks. The results indicate that by utilizing recent stabilization techniques, generative adversarial networks can efficiently sample from target data distributions. Moreover, we demonstrate the ability to simulate conditioned simulations through the latent variable model property of the proposed approach.
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