Semiparametric Efficiency Research Articles

A stableness of resistance model for nonresponse adjustment with callback data

Abstract Nonresponse arises frequently in surveys, and follow-ups are routinely made to increase the response rate. In order to monitor the follow-up process, callback data have been used in social sciences and survey studies for decades. In modern surveys, the availability of callback data is increasing because the response rate is decreasing, and follow-ups are essential to collect maximum information. Although callback data are helpful to reduce the bias in surveys, such data have not been widely used in statistical analysis until recently. We propose a stableness of resistance assumption for nonresponse adjustment with callback data. We establish the identification and the semiparametric efficiency theory under this assumption, and propose a suite of semiparametric estimation methods including doubly robust estimators, which generalize existing parametric approaches for callback data analysis. We apply the approach to a Consumer Expenditure Survey dataset. The results suggest an association between nonresponse and high housing expenditures.

Estimating causal effects for binary outcomes using per-decision inverse probability weighting.

Micro-randomized trials are commonly conducted for optimizing mobile health interventions such as push notifications for behavior change. In analyzing such trials, causal excursion effects are often of primary interest, and their estimation typically involves inverse probability weighting (IPW). However, in a micro-randomized trial, additional treatments can often occur during the time window over which an outcome is defined, and this can greatly inflate the variance of the causal effect estimator because IPW would involve a product of numerous weights. To reduce variance and improve estimation efficiency, we propose two new estimators using a modified version of IPW, which we call "per-decision IPW." The second estimator further improves efficiency using the projection idea from the semiparametric efficiency theory. These estimators are applicable when the outcome is binary and can be expressed as the maximum of a series of sub-outcomes defined over sub-intervals of time. We establish the estimators' consistency and asymptotic normality. Through simulation studies and real data applications, we demonstrate substantial efficiency improvement of the proposed estimator over existing estimators. The new estimators can be used to improve the precision of primary and secondary analyses for micro-randomized trials with binary outcomes.

Semiparametric efficiency gains from parametric restrictions on propensity scores

Abstract We explore how much knowing a parametric restriction on propensity scores improves semiparametric efficiency bounds in the potential outcome framework. For stratified propensity scores, considered as a parametric model, we derive explicit formulas for the efficiency gain from knowing how the covariate space is split. Based on these, we find that the efficiency gain decreases as the partition of the stratification becomes finer. For general parametric models, where it is hard to obtain explicit representations of efficiency bounds, we propose a novel framework that enables us to see whether knowing a parametric model is valuable in terms of efficiency even when it is high dimensional. In addition to the intuitive fact that knowing the parametric model does not help much if it is sufficiently flexible, we discover that the efficiency gain can be nearly zero even though the parametric assumption significantly restricts the space of possible propensity scores.

On propensity score matching with a diverging number of matches

Abstract This paper re-examines the work of Abadie & Imbens (2016) on propensity score matching for average treatment effect estimation. We explore the asymptotic behaviour of these estimators when the number of nearest neighbours, M, grows with the sample size. It is shown, while not surprising, but technically nontrivial, that the modified estimators can improve upon the original fixed M-estimators in terms of efficiency. Additionally, we demonstrate the potential to attain the semiparametric efficiency lower bound when the propensity score admits some special structures, echoing the insight of Hahn (1998).

Valid instrumental variable selection method using negative control outcomes and constructing efficient estimator.

In observational studies, instrumental variable (IV) methods are commonly applied when there are unmeasured covariates. In Mendelian randomization, constructing an allele score using many single nucleotide polymorphisms is often implemented; however, estimating biased causal effects by including some invalid IVs poses some risks. Invalid IVs are those IV candidates that are associated with unobserved variables. To solve this problem, we developed a novel strategy using negative control outcomes (NCOs) as auxiliary variables. Using NCOs, we are able to select only valid IVs and exclude invalid IVs without knowing which of the instruments are invalid. We also developed a new two-step estimation procedure and proved the semiparametric efficiency of our estimator. The performance of our proposed method was superior to some previous methods through simulations. Subsequently, we applied the proposed method to the UK Biobank dataset. Our results demonstrate that the use of an auxiliary variable, such as an NCO, enables the selection of valid IVs with assumptions different from those used in previousmethods.

Doubly-Valid/Doubly-Sharp Sensitivity Analysis for Causal Inference with Unmeasured Confounding

We consider the problem of constructing bounds on the average treatment effect (ATE) when unmeasured confounders exist but have bounded influence. Specifically, we assume that omitted confounders could not change the odds of treatment for any unit by more than a fixed factor. We derive the sharp partial identification bounds implied by this assumption by leveraging distributionally robust optimization, and we propose estimators of these bounds with several novel robustness properties. The first is double sharpness: our estimators consistently estimate the sharp ATE bounds when one of two nuisance parameters is misspecified and achieve semiparametric efficiency when all nuisance parameters are suitably consistent. The second and more novel property is double validity: even when most nuisance parameters are misspecified, our estimators still provide valid but possibly conservative bounds for the ATE and our Wald confidence intervals remain valid even when our estimators are not asymptotically normal. As a result, our estimators provide a highly credible method for sensitivity analysis of causal inferences. Supplementary materials for this article are available online including a standardized description of the materials available for reproducing the work.

Identification and Semiparametric Efficiency Theory of Nonignorable Missing Data with a Shadow Variable

We consider identification and estimation with an outcome missing not at random (MNAR). We study an identification strategy based on a so-called shadow variable . A shadow variable is assumed to be correlated with the outcome but independent of the missingness process conditional on the outcome and fully observed covariates. We describe a general condition for nonparametric identification of the full data law under MNAR using a valid shadow variable. Our condition is satisfied by many commonly used models; moreover, it is imposed on the complete cases, and therefore has testable implications with observed data only. We characterize the semiparametric efficiency bound for the class of regular and asymptotically linear estimators and derive a closed form for the efficient influence function. We describe a doubly robust and locally efficient estimation method and evaluate its performance on both simulation data and a real data example about home pricing.

Sobolev Calibration of Imperfect Computer Models

Calibration refers to the statistical estimation of unknown model parameters in computer experiments, such that computer experiments can match underlying physical systems. This work develops a new calibration method for imperfect computer models, Sobolev calibration, which can rule out calibration parameters that generate overfitting calibrated functions. We prove that the Sobolev calibration enjoys desired theoretical properties including fast convergence rate, asymptotic normality and semiparametric efficiency. We also demonstrate an interesting property that the Sobolev calibration can bridge the gap between two influential methods: L 2 calibration and Kennedy and O’Hagan’s calibration. In addition to exploring the deterministic physical experiments, we theoretically justify that our method can transfer to the case when the physical process is indeed a Gaussian process, which follows the original idea of Kennedy and O’Hagan’s. Numerical simulations as well as a real-world example illustrate the competitive performance of the proposed method.

Deep partially linear cox model for current status data.

Deep learning has continuously attained huge success in diverse fields, while its application to survival data analysis remains limited and deserves further exploration. For the analysis of current status data, a deep partially linear Cox model is proposed to circumvent the curse of dimensionality. Modeling flexibility is attained by using deep neural networks (DNNs) to accommodate nonlinear covariate effects and monotone splines to approximate the baseline cumulative hazard function. We establish the convergence rate of the proposed maximum likelihood estimators. Moreover, we derive that the finite-dimensional estimator for treatment covariate effects is $\sqrt{n}$-consistent, asymptotically normal, and attains semiparametric efficiency. Finally, we demonstrate the performance of our procedures through extensive simulation studies and application to real-world data on news popularity.

Doubly robust calibration of prediction sets under covariate shift.

Conformal prediction has received tremendous attention in recent years and has offered new solutions to problems in missing data and causal inference; yet these advances have not leveraged modern semi-parametric efficiency theory for more efficient uncertainty quantification. We consider the problem of obtaining well-calibrated prediction regions that can data adaptively account for a shift in the distribution of covariates between training and test data. Under a covariate shift assumption analogous to the standard missing at random assumption, we propose a general framework based on efficient influence functions to construct well-calibrated prediction regions for the unobserved outcome in the test sample without compromising coverage.

Improving Estimation Efficiency via Regression-Adjustment in Covariate-Adaptive Randomizations with Imperfect Compliance

Abstract We investigate how to improve efficiency using regression adjustments with covariates in covariate-adaptive randomizations (CARs) with imperfect subject compliance. Our regression-adjusted estimators, which are based on the doubly robust moment for local average treatment effects, are consistent and asymptotically normal even with heterogeneous probabilities of assignment and misspecified regression adjustments. We propose an optimal but potentially misspecified linear adjustment and its further improvement via a nonlinear adjustment, both of which lead to more efficient estimators than the one without adjustments. We also provide conditions for nonparametric and regularized adjustments to achieve the semiparametric efficiency bound under CARs.

Identification and Auto-Debiased Machine Learning for Outcome-Conditioned Average Structural Derivatives

This study proposes a new class of heterogeneous causal quantities, referred to as outcome-conditioned average structural derivatives (OASDs), in a general nonseparable model. An OASD is the average partial effect of a marginal change in a continuous treatment on individuals located on different parts of an outcome distribution, irrespective of individuals’ characteristics. We show that OASDs extend the unconditional quantile partial effects (UQPE) proposed by Firpo, Fortin, and Lemieux to that conditional on a set of outcome values by effectively integrating the UQPE. Exploiting such relationship brings about two merits. First, unlike UQPE that is generally not n -estimable, OASD is shown to be n -estimable. Second, our estimator achieves semiparametric efficiency bound which is a new result in the literature. We propose a novel, automatic, debiased machine-learning estimator for an OASD, and present asymptotic statistical guarantees for it. The estimator is proven to be n -consistent, asymptotically normal, and semi-parametrically efficient. We also prove the validity of the bootstrap procedure for uniform inference for the OASD process. We apply the method to Imbens, Rubin, and Sacerdote’s lottery data.

Statistical inference with regularized optimal transport

Abstract Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning and applied mathematics. However, OT distances suffer from computational and statistical scalability issues to high dimensions, which motivated the study of regularized OT methods like slicing, smoothing and entropic penalty. This work establishes a unified framework for deriving limit distributions of empirical regularized OT distances, semiparametric efficiency of the plug-in empirical estimator and bootstrap consistency. We apply the unified framework to provide a comprehensive statistical treatment of (i) average- and max-sliced $p$-Wasserstein distances, for which several gaps in existing literature are closed; (ii) smooth distances with compactly supported kernels, the analysis of which is motivated by computational considerations; and (iii) entropic OT, for which our method generalizes existing limit distribution results and establishes, for the first time, efficiency and bootstrap consistency. While our focus is on these three regularized OT distances as applications, the flexibility of the proposed framework renders it applicable to broad classes of functionals beyond these examples.

Efficient and Robust Estimation of the Generalized LATE Model

This article studies the estimation of causal parameters in the generalized local average treatment effect (GLATE) model, which expands upon the traditional LATE model to include multivalued treatments. We derive the efficient influence function (EIF) and the semiparametric efficiency bound (SPEB) for two types of causal parameters: the local average structural function (LASF) and the local average structural function for the treated (LASFT). The moment conditions generated by the EIF satisfy two robustness properties: double robustness and Neyman orthogonality. Based on the robust moment conditions, we propose the double/debiased machine learning (DML) estimator for estimating the LASF. The DML estimator is well-suited for high dimensional settings. We also propose null-restricted inference methods that are robust against weak identification issues. As an empirical application of these methods, we examine the potential health outcome across different types of health insurance plans using data from the Oregon Health Insurance Experiment.

Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes.

Regression models for continuous outcomes frequently require a transformation of the outcome, which is often specified a priori or estimated from a parametric family. Cumulative probability models (CPMs) nonparametrically estimate the transformation by treating the continuous outcome as if it is ordered categorically. They thus represent a flexible analysis approach for continuous outcomes. However, it is difficult to establish asymptotic properties for CPMs due to the potentially unbounded range of the transformation. Here we show asymptotic properties for CPMs when applied to slightly modified data where bounds, one lower and one upper, are chosen and the outcomes outside the bounds are set as two ordinal categories. We prove the uniform consistency of the estimated regression coefficients and of the estimated transformation function between the bounds. We also describe their joint asymptotic distribution, and show that the estimated regression coefficients attain the semiparametric efficiency bound. We show with simulations that results from this approach and those from using the CPM on the original data are very similar when a small fraction of the data are modified. We reanalyze a dataset of HIV-positive patients with CPMs to illustrate and compare the approaches.

Non-agency interventions for causal mediation in the presence of intermediate confounding

Abstract Recent approaches to causal inference have focused on causal effects defined as contrasts between the distribution of counterfactual outcomes under hypothetical interventions on the nodes of a graphical model. In this article, we develop theory for causal effects defined with respect to a different type of intervention, one which alters the information propagated through the edges of the graph. These information transfer interventions may be more useful than node interventions in settings in which causes are non-manipulable, for example when considering race or genetics as a causes. Furthermore, information transfer interventions allow us to define path-specific decompositions which are identified in the presence of treatment-induced mediator-outcome confounding, a practical problem whose general solution remains elusive. We prove that the proposed effects provide valid statistical tests of mechanisms, unlike popular methods based on randomised interventions on the mediator. We propose efficient non-parametric estimators for a covariance version of the proposed effects, using data-adaptive regression coupled with semi-parametric efficiency theory to address model misspecification bias while retaining n-consistency and asymptotic normality. We illustrate the use of our methods in two examples using publicly available data.

Semiparametric estimation of long-term treatment effects

Long-term outcomes of experimental evaluations are necessarily observed after long delays. We develop semiparametric methods for combining the short-term outcomes of experiments with observational measurements of short-term and long-term outcomes, in order to estimate long-term treatment effects. We characterize semiparametric efficiency bounds for various instances of this problem. These calculations facilitate the construction of several estimators. We analyze the finite-sample performance of these estimators with a simulation calibrated to data from an evaluation of the long-term effects of a poverty alleviation program.

Efficient calibration for imperfect epidemic models with applications to the analysis of COVID-19

Abstract The estimation of unknown parameters in simulations, also known as calibration, is crucial for practical management of epidemics and prediction of pandemic risk. A simple yet widely used approach is to estimate the parameters by minimising the sum of the squared distances between actual observations and simulation outputs. It is shown in this paper that this method is inefficient, particularly when the epidemic models are developed based on certain simplifications of reality, also known as imperfect models which are commonly used in practice. To address this issue, a new estimator is introduced that is asymptotically consistent, has a smaller estimation variance than the least-squares estimator, and achieves the semiparametric efficiency. Numerical studies are performed to examine the finite sample performance. The proposed method is applied to the analysis of the COVID-19 pandemic for 20 countries based on the susceptible-exposed-infectious-recovered model with both deterministic and stochastic simulations. The estimation of the parameters, including the basic reproduction number and the average incubation period, reveal the risk of disease outbreaks in each country and provide insights to the design of public health interventions.

Debiased inference on heterogeneous quantile treatment effects with regression rank scores

Abstract Understanding treatment effect heterogeneity is vital to many scientific fields because the same treatment may affect different individuals differently. Quantile regression provides a natural framework for modelling such heterogeneity. We propose a new method for inference on heterogeneous quantile treatment effects (HQTE) in the presence of high-dimensional covariates. Our estimator combines an ℓ1-penalised regression adjustment with a quantile-specific bias correction scheme based on rank scores. We study the theoretical properties of this estimator, including weak convergence and semi-parametric efficiency of the estimated HQTE process. We illustrate the finite-sample performance of our approach through simulations and an empirical example, dealing with the differential effect of statin usage for lowering low-density lipoprotein cholesterol levels for the Alzheimer’s disease patients who participated in the UK Biobank study.

Augmented two-step estimating equations with nuisance functionals and complex survey data

Summary Statistical inference in the presence of nuisance functionals with complex survey data is an important topic in social and economic studies. The Gini index, Lorenz curves, and quantile shares are among the commonly encountered examples. The nuisance functionals are usually handled by a plug-in nonparametric estimator and the main inferential procedure can be carried out through a two-step generalized empirical likelihood method. Unfortunately, the resulting inference is not efficient and the nonparametric version of the Wilks’ theorem breaks down even under simple random sampling. We propose an augmented estimating equations method with nuisance functionals and complex surveys. The second step augmented estimating functions obey the Neyman orthogonality condition and automatically handle the impact of the first step plug-in estimator, and the resulting estimator of the main parameters of interest is invariant to the first step method. More importantly, the generalized empirical likelihood-based Wilks’ theorem holds for the main parameters of interest under the design-based framework for commonly used survey designs, and the maximum generalized empirical likelihood estimators achieve the semiparametric efficiency bound. Performances of the proposed methods are demonstrated through simulation studies and an application using the dataset from the New York City Social Indicators Survey.