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Causal effects are often characterized with population summaries. These might provide an incomplete picture when there are heterogeneous treatment effects across subgroups. Since the subgroup structure is typically unknown, it is more challenging to identify and evaluate subgroup effects than population effects. We propose a new solution to this problem: Causal k-Means Clustering, which harnesses the widely-used k-means clustering algorithm to uncover the unknown subgroup structure. Our problem differs significantly from the conventional clustering setup since the variables to be clustered are unknown counterfactual functions. We present a plug-in estimator which is simple and readily implementable using off-the-shelf algorithms, and study its rate of convergence. We also develop a new bias-corrected estimator based on nonparametric efficiency theory and double machine learning, and show that this estimator achieves fast root-n rates and asymptotic normality in large nonparametric models. Our proposed methods are especially useful for modern outcome-wide studies with multiple treatment levels. Further, our framework is extensible to clustering with generic pseudo-outcomes, such as partially observed outcomes or otherwise unknown functions. Finally, we explore finite sample properties via simulation, and illustrate the proposed methods in a study of treatment programs for adolescent substance abuse.

Weighting is a general and often-used method for statistical adjustment. Weighting has two objectives: first, to balance covariate distributions, and second, to ensure that the weights have minimal dispersion and thus produce a more stable estimator. A recent, increasingly common approach directly optimizes the weights toward these two objectives. However, this approach has not yet been feasible in large-scale datasets when investigators wish to flexibly balance general basis functions in an extended feature space. For example, many balancing approaches cannot scale to national-level health services research studies. To address this practical problem, we describe a scalable and flexible approach to weighting that integrates a basis expansion in a reproducing kernel Hilbert space with state-of-the-art convex optimization techniques. Specifically, we use the rank-restricted Nystr\"{o}m method to efficiently compute a kernel basis for balancing in {nearly} linear time and space, and then use the specialized first-order alternating direction method of multipliers to rapidly find the optimal weights. In an extensive simulation study, we provide new insights into the performance of weighting estimators in large datasets, showing that the proposed approach substantially outperforms others in terms of accuracy and speed. Finally, we use this weighting approach to conduct a national study of the relationship between hospital profit status and heart attack outcomes in a comprehensive dataset of 1.27 million patients. We find that for-profit hospitals use interventional cardiology to treat heart attacks at similar rates as other hospitals, but have higher mortality and readmission rates.

We propose a simple and general framework for nonparametric estimation of heterogeneous treatment effects under fairness constraints. Under standard regularity conditions, we show that the resulting estimators possess the double robustness property. We use this framework to characterize the trade-off between fairness and the maximum welfare achievable by the optimal policy. We evaluate the methods in a simulation study and illustrate them in a real-world case study.

We study counterfactual classification as a new tool for decision-making under hypothetical (contrary to fact) scenarios. We propose a doubly-robust nonparametric estimator for a general counterfactual classifier, where we can incorporate flexible constraints by casting the classification problem as a nonlinear mathematical program involving counterfactuals. We go on to analyze the rates of convergence of the estimator and provide a closed-form expression for its asymptotic distribution. Our analysis shows that the proposed estimator is robust against nuisance model misspecification, and can attain fast $\sqrt{n}$ rates with tractable inference even when using nonparametric machine learning approaches. We study the empirical performance of our methods by simulation and apply them for recidivism risk prediction.

We study a new class of estimands in causal inference, which are the solutions to a stochastic nonlinear optimization problem that in general cannot be obtained in closed form. The optimization problem describes the counterfactual state of a system after an intervention, and the solutions represent the optimal decisions in that counterfactual state. In particular, we develop a counterfactual mean-variance optimization approach, which can be used for optimal allocation of resources after an intervention. We propose a doubly-robust nonparametric estimator for the optimal solution of the counterfactual mean-variance program. We go on to analyze rates of convergence and provide a closed-form expression for the asymptotic distribution of our estimator. Our analysis shows that the proposed estimator is robust against nuisance model misspecification, and can attain fast $\sqrt{n}$ rates with tractable inference even when using nonparametric methods. This result is applicable to general nonlinear optimization problems subject to linear constraints whose coefficients are unknown and must be estimated. In this way, our findings contribute to the literature in optimization as well as causal inference. We further discuss the problem of calibrating our counterfactual covariance estimator to improve the finite-sample properties of our proposed optimal solution estimators. Finally, we evaluate our methods via simulation, and apply them to problems in healthcare policy and portfolio construction.

We propose PLLay, a novel topological layer for general deep learning models based on persistence landscapes, in which we can efficiently exploit the underlying topological features of the input data structure. In this work, we show differentiability with respect to layer inputs, for a general persistent homology with arbitrary filtration. Thus, our proposed layer can be placed anywhere in the network and feed critical information on the topological features of input data into subsequent layers to improve the learnability of the networks toward a given task. A task-optimal structure of PLLay is learned during training via backpropagation, without requiring any input featurization or data preprocessing. We provide a novel adaptation for the DTM function-based filtration, and show that the proposed layer is robust against noise and outliers through a stability analysis. We demonstrate the effectiveness of our approach by classification experiments on various datasets., Comment: 29 pages, 7 figures

Modern longitudinal studies collect feature data at many timepoints, often of the same order of sample size. Such studies are typically affected by {dropout} and positivity violations. We tackle these problems by generalizing effects of recent incremental interventions (which shift propensity scores rather than set treatment values deterministically) to accommodate multiple outcomes and subject dropout. We give an identifying expression for incremental intervention effects when dropout is conditionally ignorable (without requiring treatment positivity), and derive the nonparametric efficiency bound for estimating such effects. Then we present efficient nonparametric estimators, showing that they converge at fast parametric rates and yield uniform inferential guarantees, even when nuisance functions are estimated flexibly at slower rates. We also study the variance ratio of incremental intervention effects relative to more conventional deterministic effects in a novel infinite time horizon setting, where the number of timepoints can grow with sample size, and show that incremental intervention effects yield near-exponential gains in statistical precision in this setup. Finally we conclude with simulations and apply our methods in a study of the effect of low-dose aspirin on pregnancy outcomes., Comment: 52 pages

We develop a novel algorithm for feature extraction in time series data by leveraging tools from topological data analysis. Our algorithm provides a simple, efficient way to successfully harness topological features of the attractor of the underlying dynamical system for an observed time series. The proposed methodology relies on the persistent landscapes and silhouette of the Rips complex obtained after a de-noising step based on principal components applied to a time-delayed embedding of a noisy, discrete time series sample. We analyze the stability properties of the proposed approach and show that the resulting TDA-based features are robust to sampling noise. Experiments on synthetic and real-world data demonstrate the effectiveness of our approach. We expect our method to provide new insights on feature extraction from granular, noisy time series data., Comment: 28 pages

In this paper we develop a framework for characterizing causal effects via distributional distances. In particular we define a causal effect in terms of the $L_1$ distance between different counterfactual outcome distributions, rather than the typical mean difference in outcome values. Comparing entire counterfactual outcome distributions can provide more nuanced and valuable measures for exploring causal effects beyond the average treatment effect. First, we propose a novel way to estimate counterfactual outcome densities, which is of independent interest. Then we develop an efficient estimator of our target causal effect. We go on to provide error bounds and asymptotic properties of the proposed estimator, along with bootstrap-based confidence intervals. Finally, we illustrate the methods via simulations and real data., Comment: 46 pages