We consider identification and inference for the average treatment effect and heterogeneous treatment effect conditional on observable covariates in the presence of unmeasured confounding. Since point identification of average treatment effect and heterogeneous treatment effect is not achievable without strong assumptions, we obtain bounds on both average and heterogeneous treatment effects by leveraging differential effects, a tool that allows for using a second treatment to learn the effect of the first treatment. The differential effect is the effect of using one treatment in lieu of the other, and it could be identified in some observational studies in which treatments are not randomly assigned to units, where differences in outcomes may be due to biased assignments rather than treatment effects. With differential effects, we develop a flexible and easy-to-implement semi-parametric framework to estimate bounds and establish asymptotic properties over the support for conducting statistical inference. We provide conditions under which causal estimands are point identifiable as well in the proposed framework. The proposed method is examined by a simulation study and two case studies using datasets from National Health and Nutrition Examination Survey and Youth Risk Behavior Surveillance System.
相關內容
When an exposure of interest is confounded by unmeasured factors, an instrumental variable (IV) can be used to identify and estimate certain causal contrasts. Identification of the marginal average treatment effect (ATE) from IVs relies on strong untestable structural assumptions. When one is unwilling to assert such structure, IVs can nonetheless be used to construct bounds on the ATE. Famously, Balke and Pearl (1997) proved tight bounds on the ATE for a binary outcome, in a randomized trial with noncompliance and no covariate information. We demonstrate how these bounds remain useful in observational settings with baseline confounders of the IV, as well as randomized trials with measured baseline covariates. The resulting bounds on the ATE are non-smooth functionals, and thus standard nonparametric efficiency theory is not immediately applicable. To remedy this, we propose (1) under a novel margin condition, influence function-based estimators of the bounds that can attain parametric convergence rates when the nuisance functions are modeled flexibly, and (2) estimators of smooth approximations of these bounds. We propose extensions to continuous outcomes, explore finite sample properties in simulations, and illustrate the proposed estimators in a randomized experiment studying the effects of vaccination encouragement on flu-related hospital visits.
We propose principled prediction intervals to quantify the uncertainty of a large class of synthetic control predictions or estimators in settings with staggered treatment adoption, offering precise non-asymptotic coverage probability guarantees. From a methodological perspective, we provide a detailed discussion of different causal quantities to be predicted, which we call `causal predictands', allowing for multiple treated units with treatment adoption at possibly different points in time. From a theoretical perspective, our uncertainty quantification methods improve on prior literature by (i) covering a large class of causal predictands in staggered adoption settings, (ii) allowing for synthetic control methods with possibly nonlinear constraints, (iii) proposing scalable robust conic optimization methods and principled data-driven tuning parameter selection, and (iv) offering valid uniform inference across post-treatment periods. We illustrate our methodology with an empirical application studying the effects of economic liberalization in the 1990s on GDP for emerging European countries. Companion general-purpose software packages are provided in Python, R and Stata.
In empirical studies with time-to-event outcomes, investigators often leverage observational data to conduct causal inference on the effect of exposure when randomized controlled trial data is unavailable. Model misspecification and lack of overlap are common issues in observational studies, and they often lead to inconsistent and inefficient estimators of the average treatment effect. Estimators targeting overlap weighted effects have been proposed to address the challenge of poor overlap, and methods enabling flexible machine learning for nuisance models address model misspecification. However, the approaches that allow machine learning for nuisance models have not been extended to the setting of weighted average treatment effects for time-to-event outcomes when there is poor overlap. In this work, we propose a class of one-step cross-fitted double/debiased machine learning estimators for the weighted cumulative causal effect as a function of restriction time. We prove that the proposed estimators are consistent, asymptotically linear, and reach semiparametric efficiency bounds under regularity conditions. Our simulations show that the proposed estimators using nonparametric machine learning nuisance models perform as well as established methods that require correctly-specified parametric nuisance models, illustrating that our estimators mitigate the need for oracle parametric nuisance models. We apply the proposed methods to real-world observational data from a UK primary care database to compare the effects of anti-diabetic drugs on cancer clinical outcomes.
This study considers a panel data analysis to examine the heterogeneity in treatment effects with respect to a pre-treatment covariate of interest in the staggered difference-in-differences setting in Callaway and Sant'Anna (2021). Under a set of standard identification conditions, a doubly robust estimand conditional on the covariate identifies the group-time conditional average treatment effect given the covariate. Given this identification result, we propose a three-step estimation procedure based on nonparametric local linear regressions and parametric estimation methods, and develop a doubly robust inference method to construct a uniform confidence band of the group-time conditional average treatment effect function.
Estimation of heterogeneous treatment effects is an active area of research in causal inference. Most of the existing methods, however, focus on estimating the conditional average treatment effects of a single, binary treatment given a set of pre-treatment covariates. In this paper, we propose a method to estimate the heterogeneous causal effects of high-dimensional treatments, which poses unique challenges in terms of estimation and interpretation. The proposed approach is based on a Bayesian mixture of regularized logistic regressions to identify groups of units who exhibit similar patterns of treatment effects. By directly modeling cluster membership with covariates, the proposed methodology allows one to explore the unit characteristics that are associated with different patterns of treatment effects. Our motivating application is conjoint analysis, which is a popular survey experiment in social science and marketing research and is based on a high-dimensional factorial design. We apply the proposed methodology to the conjoint data, where survey respondents are asked to select one of two immigrant profiles with randomly selected attributes. We find that a group of respondents with a relatively high degree of prejudice appears to discriminate against immigrants from non-European countries like Iraq. An open-source software package is available for implementing the proposed methodology.
In many fields, including environmental epidemiology, researchers strive to understand the joint impact of a mixture of exposures. This involves analyzing a vector of exposures rather than a single exposure, with the most significant exposure sets being unknown. Examining every possible interaction or effect modification in a high-dimensional vector of candidates can be challenging or even impossible. To address this challenge, we propose a method for the automatic identification and estimation of exposure sets in a mixture with explanatory power, baseline covariates that modify the impact of an exposure and sets of exposures that have synergistic non-additive relationships. We define these parameters in a realistic nonparametric statistical model and use machine learning methods to identify variables sets and estimate nuisance parameters for our target parameters to avoid model misspecification. We establish a prespecified target parameter applied to variable sets when identified and use cross-validation to train efficient estimators employing targeted maximum likelihood estimation for our target parameter. Our approach applies a shift intervention targeting individual variable importance, interaction, and effect modification based on the data-adaptively determined sets of variables. Our methodology is implemented in the open-source SuperNOVA package in R. We demonstrate the utility of our method through simulations, showing that our estimator is efficient and asymptotically linear under conditions requiring fast convergence of certain regression functions. We apply our method to the National Institute of Environmental Health Science mixtures workshop data, revealing correct identification of antagonistic and agonistic interactions built into the data. Additionally, we investigate the association between exposure to persistent organic pollutants and longer leukocyte telomere length.
Regression can be really difficult in case of big datasets, since we have to dealt with huge volumes of data. The demand of computational resources for the modeling process increases as the scale of the datasets does, since traditional approaches for regression involve inverting huge data matrices. The main problem relies on the large data size, and so a standard approach is subsampling that aims at obtaining the most informative portion of the big data. In the current paper we consider an approach based on leverages scores, already existing in the current literature. The aforementioned approach proposed in order to select subdata for linear model discrimination. However, we highlight its importance on the selection of data points that are the most informative for estimating unknown parameters. We conclude that the approach based on leverage scores improves existing approaches, providing simulation experiments as well as a real data application.
Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence, the underlying rational transfer functions are allowed to have poles in general domains in the complex plane. In particular, this covers the case of specific conservative partial differential equations such as the linear Schr\"odinger and the undamped linear wave equation with spectra on the imaginary axis. By an appropriate modification of the classical continuous time Hardy space $\mathcal{H}_2$, a new $\mathcal{H}_2$ like optimal model reduction problem is introduced and first order optimality conditions are derived. As in the classical $\mathcal{H}_2$ case, these conditions exhibit a rational Hermite interpolation structure for which an iterative model reduction algorithm is proposed. Numerical examples demonstrate the effectiveness of the new method.
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.
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