Double Robust Estimators Research Articles

Double-Robust Estimation in Difference-in-Differences with an Application to Traffic Safety Evaluation

Difference-in-differences (DID) is a widely used approach for drawing causal inference from observational panel data. Two common estimation strategies for DID are outcome regression and propensity score weighting. In this paper, motivated by a real application in traffic safety research, we propose a new double-robust DID estimator that hybridizes regression and propensity score weighting. We particularly focus on the case of discrete outcomes. We show that the proposed double-robust estimator possesses the desirable large-sample robustness property. We conduct a simulation study to examine its finite-sample performance and compare with alternative methods. Our empirical results from a Pennsylvania Department of Transportation data suggest that rumble strips are marginally effective in reducing vehicle crashes.

PROPERTIES OF DOUBLY ROBUST ESTIMATORS WHEN NUISANCE FUNCTIONS ARE ESTIMATED NONPARAMETRICALLY

An estimator of a finite-dimensional parameter is said to be doubly robust (DR) if it imposes parametric specifications on two unknown nuisance functions, but only requires that one of these two specifications is correct in order for the estimator to be consistent for the object of interest. In this article, we study versions of such estimators that use local polynomial smoothing for estimating the nuisance functions. We show that such semiparametric two-step (STS) versions of DR estimators have favorable theoretical and practical properties relative to other commonly used STS estimators. We also show that these gains are not generated by the DR property alone. Instead, it needs to be combined with an orthogonality condition on the estimation residuals from the nonparametric first stage, which we show to be satisfied in a wide range of models.

Foreign connections and the difference they make: how migrant ties influence political interest and attitudes in Mexico

Beyond the economic and social effects of international migration researchers show regular exchanges between immigrants and stay-at-homes produce political spillovers in sending countries. As a broad body of literature demonstrates, most migrants maintain at least some form of contact with key connections back home, whether through long-distance communication, remittance sending, or in person visits. We investigate if exposure to international migration affects non-migrant citizens political interest, awareness, and attitudes about the efficacy of elections using longitudinal survey data from the Mexico 2006 Panel Study. We use a novel statistical approach combining Double Robust estimation technique with propensity score weighting. Our results suggest that Mexican non-migrant citizens exposed to international migration through social connections and remittances are more likely to be politically aware than those without. We also offer theoretical pathways to explain how ideational and material resources embedded in migrant social networks influences the political interest of stay-at-home citizens.

Multi-valued Double Robust quantile treatment effect

An empirical approach for the analysis of treatment effect at various quantiles in the case of multiple treatment conditions is here proposed. Outcome changes under multiple treatment conditions are computed using (a) inverse propensity score weights and (b) unconditional outcome distribution within each group. Through (a) and (b), the standard double robust estimator is extended to evaluate treatment effect not only on average but also in the tails (quantiles). A Monte Carlo study designed to examine and assess the performance of the proposed approach and two empirical applications conclude the analysis.

Doubly robust augmented-estimating-equations estimation with nonignorable nonresponse data

The problem of nonignorable nonresponse data is ubiquitous in medical and social science studies. Analyses focused only on the missing-at-random assumption may lead to biased results. Various debias methods have been extensively studied in the literature, particularly the doubly robust (DR) estimators. We propose DR augmented-estimating-equations (AEE) estimators of the mean response which enjoy the double-robustness property under correct specification of the log odds ratio model. An advantage of DR AEE estimators is that they can efficiently use the completely observed covariates to improve estimation efficiency of existing DR estimators with nonignorable nonresponse data. We propose a model selection criterion that can consistently select the correct parametric model of the log odds ratio model from a group of candidate models. Moreover, the correctness of the required working models can be evaluated via straightforward goodness-of-fit tests. Simulation results indicate that doubly robust augmented-estimating-equations estimators are very robust to a misspecification of the baseline outcome density model or the baseline response model and dominate other competitors in the sense of having smaller mean-square errors. The analysis of a real dataset illustrates the flexibility and usefulness of the proposed methods.

Variants of double robust estimators for two-stage dynamic treatment regimes

Certain conditions and illnesses may necessitate multiple stages of treatment and thus require unique study designs to compare the efficacy of these interventions. Such studies are characterized by two or more stages of treatment punctuated by decision points where intermediate outcomes inform the choice for the next stage of treatment. The algorithm that dictates what treatments to take based on intermediate outcomes is referred to as a dynamic regime. This paper describes an efficient method of building double robust estimators of the treatment effect of different regimes. A double robust estimator utilizes both modeling of the outcome and weighting based on the modeled probability of receiving treatment in such a way that the resulting estimator is a consistent estimate of the desired population parameter under the condition that at least one of those models is correct. This new and more efficient double robust estimator is compared to another double robust estimator as well as classical regression and inverse probability weighted estimators. The methods are applied to estimate the regime effects in the STAR*D anti-depression treatment trial.

Semiparametric double robust and efficient estimation for mean functionals with response missing at random

Under dimension reduction structure, several semiparametric estimators for the mean of missing response are proposed, which can efficiently deal with the dimensionality problem. Specifically, a generalized version of Augmented Inverse Probability Weighting estimator (AIPW) is proposed and its double robustness, estimation consistency and asymptotic efficiency are investigated. A generalized version of Inverse Probability Weighting (IPW) estimator is also introduced. An asymptotic efficiency reduction phenomenon occurs in the sense that the IPW estimator with the true selection probability is asymptotically less efficient than the one with an estimated selection probability. Besides, two partial imputation and two complete imputation estimators are discussed. We further systematically investigate the comparisons among these estimators in theory. Several simulation studies and a real data analysis are conducted for performance examination and illustration.

Multiple robust estimation of marginal structural mean models for unconstrained outcomes.

We consider estimation, from longitudinal observational data, of the parameters of marginal structural mean models for unconstrained outcomes. Current proposals include inverse probability of treatment weighted and double robust (DR) estimators. A difficulty with DR estimation is that it requires postulating a sequence of models, one for the each mean of the counterfactual outcome given covariate and treatment history up to each exposure time point. Most natural models for such means are often incompatible. Robins et al., (2000b) proposed a parameterization of the likelihood which implies compatible parametric models for such means. Their parameterization has not been exploited to construct DR estimators and one goal of this article is to fill this gap. More importantly, exploiting this parameterization we propose a multiple robust (MR) estimator that confers even more protection against model misspecification than DR estimators. Our methods are easy to implement as they are based on the iterative fit of a sequence of weighted regressions.

Doubly-Robust Estimation of Effect of Imaging Resource Utilization on Discharge Decisions in Emergency Departments.

Cluster analysis provides a data-driven multidimensional approach for identifying distinct subgroups of patients in a cohort. Each of the clusters represents a particular health condition with specific clinical trajectory and medical needs. Patients visiting emergency rooms do not share the same health condition, therefore discriminating between groups may have implications for diagnostic testing and resource utilization. We carried out this retrospective cohort study on 13825 patients who visited the emergency rooms in three Emory hospitals presenting with head trauma and non-stroke-like non-specific neurologic symptoms from January 2010 to September 2015. We utilized k-means clustering to find five distinct subgroups. Then, we investigated if getting an emergency head CT scan could have a statistically significant effect on getting discharged from the hospital. Adjusted effect estimation method was applied on each cluster to estimate the association between receiving a diagnostic test (e.g., head CT scan) on the disposition status. Out of five patient subgroups in the cohort, the chance of getting discharged for two clusters were significantly affected by getting a head CT scan. They both include comparatively older, African American or black patients who arrived in the ER with EMS, the latter suggesting critical health conditions.

Comparing Costs of Traditional and Specialty Probation for People With Serious Mental Illness.

Specialty mental health probation reduces the likelihood of rearrest for people with mental illness, who are overrepresented in the justice system. This study tested whether specialty probation was associated with lower costs than traditional probation during the two years after placement in probation. A longitudinal, matched study compared costs of behavioral health care and criminal justice contacts among 359 probationers with mental illness at prototypic specialty or traditional agencies. Compared with traditional officers, specialty officers supervised smaller caseloads, established better relationships with supervisees, and participated more in treatment. Participants and officers were interviewed, and administrative databases were integrated to capture service use and criminal justice contacts. Unit costs were attached to these data to estimate costs incurred by each participant over two years. Cost differences were estimated by using machine-learning algorithms combined with targeted maximum-likelihood estimation (TMLE), a double-robust estimator that accounts for associations between confounders and both treatment assignment and outcomes. Specialty probation cost $11,826 (p<.001) less per participant than traditional probation, with overall savings of about 51%. Specialty and traditional probation did not differ in criminal justice costs because the additional costs for supervision of specialty caseloads were offset by reduced recidivism. However, for behavioral health care, specialty probation cost an estimated $14,049 (p<.001) less per client than traditional probation. Greater outpatient costs were more than offset by reduced emergency, inpatient, and residential costs. Well-implemented specialty probation yielded substantial savings-and should be considered in justice reform efforts for people with mental illness.

Linking Clicks to Bricks: Spillover Benefits of Online Advertising

Businesses have widely used email ads to directly send promotional information to consumers. While email ads serve as a convenient channel that allows firms to target consumers online, are they effective in increasing offline revenues for firms that predominantly sell in brick-and-mortar stores? Is the effect of email ads, if any, heterogeneous across different consumer segments? If so, on which consumers is the effect highest? In our paper, we address these questions using a unique high-dimensional dataset from one of the largest retailers in the United States, which links each consumer’s online behaviors to the item-level purchase records in physical stores. We use a doubly robust estimator (DRE) that incorporates nonparametric machine learning methods and allows us to perform causal estimation on observational data. Using the DRE we find that on average receiving email ads can increase a consumer’s spending in physical stores by approximately $0.47. Additionally, we find that the increased offline sales result from increased purchase probability and a wider variety of products being purchased by the consumer. Further, we use a data-driven approach to demonstrate that the effect of email ads is heterogeneous across different consumer segments. Overall, our results provide empirical evidence for the positive effect of email ads on consumers’ offline purchase. Receiving email ads from the retailer can generate awareness and remind the consumer of the retailer’s offerings of various products and services, which could increase the consumer’s spending in the retailer’s physical stores. These findings have direct implications for managers to improve their digital marketing strategy design and for policy makers who are interested in evaluating the economic impact of prevalent email advertising.

Introduction to Double Robust Methods for Incomplete Data.

Most methods for handling incomplete data can be broadly classified as inverse probability weighting (IPW) strategies or imputation strategies. The former model the occurrence of incomplete data; the latter, the distribution of the missing variables given observed variables in each missingness pattern. Imputation strategies are typically more efficient, but they can involve extrapolation, which is difficult to diagnose and can lead to large bias. Double robust (DR) methods combine the two approaches. They are typically more efficient than IPW and more robust to model misspecification than imputation. We give a formal introduction to DR estimation of the mean of a partially observed variable, before moving to more general incomplete-data scenarios. We review strategies to improve the performance of DR estimators under model misspecification, reveal connections between DR estimators for incomplete data and 'design-consistent' estimators used in sample surveys, and explain the value of double robustness when using flexible data-adaptive methods for IPW or imputation.

Factors associated with improved outcomes for metastatic inflammatory breast cancer patients.

Controversy exists regarding the role oflocoregional therapy for stage IV inflammatory breast cancer (IBC). This study aims to determine indicators of prognosis, including primary tumor resection, for stage IV IBC patients. Using the National Cancer Data Base, female patients diagnosed 2010-2013 with unilateral a priori metastatic T4d invasive adenocarcinoma of the breast were identified. We conducted propensity score matched analysis to balance confounders of surgery versus no-surgery. Stratified log-rank test and double-robust estimation under the Cox model were used to assess the effect of surgery, and margins, on overall survival (OS) in the propensity score matched cohort. Of 1266 patients, 41% underwent surgery. In the unmatched cohort, median OS of the surgery and no-surgery groups was 36 and 20months, respectively (p<0.001). In the matched cohort (n=588), the median OS of surgery and no-surgery groups was 29 and 27months, respectively (p=0.052). Patients with negative margin surgery (p=0.024), hormone receptor-positive (p=0.019), HER2-positive disease (p<0.0001), treated with chemotherapy (p<0.0001) and hormonal therapy (p<0.0001), had better survival. Those with brain metastases had increased risk of death (p<0.0001). This study represents the largest cohort of metastatic IBC patients, and identified negative margin surgery, systemic therapy, hormone receptor and HER2-positive disease as factors associated with improved outcomes. While these findings should be interpreted cautiously, they may be used to guide further investigations into local control and quality of life in this patient population with limited treatment options.

Data-Adaptive Estimation for Double-Robust Methods in Population-Based Cancer Epidemiology: Risk Differences for Lung Cancer Mortality by Emergency Presentation.

In this paper, we propose a structural framework for population-based cancer epidemiology and evaluate the performance of double-robust estimators for a binary exposure in cancer mortality. We conduct numerical analyses to study the bias and efficiency of these estimators. Furthermore, we compare 2 different model selection strategies based on 1) Akaike’s Information Criterion and the Bayesian Information Criterion and 2) machine learning algorithms, and we illustrate double-robust estimators’ performance in a real-world setting. In simulations with correctly specified models and near-positivity violations, all but the naive estimators had relatively good performance. However, the augmented inverse-probability-of-treatment weighting estimator showed the largest relative bias. Under dual model misspecification and near-positivity violations, all double-robust estimators were biased. Nevertheless, the targeted maximum likelihood estimator showed the best bias-variance trade-off, more precise estimates, and appropriate 95% confidence interval coverage, supporting the use of the data-adaptive model selection strategies based on machine learning algorithms. We applied these methods to estimate adjusted 1-year mortality risk differences in 183,426 lung cancer patients diagnosed after admittance to an emergency department versus persons with a nonemergency cancer diagnosis in England (2006–2013). The adjusted mortality risk (for patients diagnosed with lung cancer after admittance to an emergency department) was 16% higher in men and 18% higher in women, suggesting the importance of interventions targeting early detection of lung cancer signs and symptoms.

Double robust estimator in general treatment regimes based on Covariate-balancing

ABSTRACTDouble robust estimators have double the chance of being a consistent estimator of a causal effect in binary treatments cases. In this paper, we proposed an estimator of a causal effect for general treatment regimes based on covariate-balancing. Under parametrical situation, our estimator has double robustness.

Comparison of the ability of double-robust estimators to correct bias in propensity score matching analysis. A Monte Carlo simulation study.

As covariates are not always adequately balanced after propensity score matching and double- adjustment can be used to remove residual confounding, we compared the performance of several double-robust estimators in different scenarios. We conducted a series of Monte Carlo simulations on virtual observational studies. After estimating the propensity scores by logistic regression, we performed 1:1 optimal, nearest-neighbor, and caliper matching. We used 4 estimators on each matched sample: (1) a crude estimator without double-adjustment, (2) double-adjustment for the propensity scores, (3) double-adjustment for the unweighted unbalanced covariates, and (4) double-adjustment for the unbalanced covariates, weighted by their strength of association with the outcome. The crude estimator led to highest bias in all tested scenarios. Double-adjustment for the propensity scores effectively removed confounding only when the propensity score models were correctly specified. Double-adjustment for the unbalanced covariates was more robust to misspecification. Double-adjustment for the weighted unbalanced covariates outperformed the other approaches in every scenario and using any matching algorithm, as measured by the mean squared error. Double-adjustment can be used to remove residual confounding after propensity score matching. The unbalanced covariates with the strongest confounding effects should be adjusted.

Comparing Public Safety Outcomes for Traditional Probation vs Specialty Mental Health Probation

ImportanceProbation is a cornerstone of efforts to reduce mass incarceration. Although it is understudied, specialty probation could improve outcomes for the overrepresented group of people with mental illness.ObjectiveTo test whether specialty probation yields better public safety outcomes than traditional probation.Design, Setting, and ParticipantsA longitudinal observational study with group matching on age, sex, race/ethnicity, probation time, and offense at 2 urban agencies that exemplify specialty and traditional probation. Enrollment began October 19, 2005; follow-up data were complete January 26, 2013. Participants were 359 diverse probationers with serious mental health problems and functional impairment. Probationers and officers were assessed 3 times (for probationers, 6-month retention, 315 of 359 [88%]; 12-month retention 304 of 359 [85%]) and follow-up records were obtained. Machine learning algorithms were combined with a targeted maximum likelihood estimation, a double robust estimator that accounts for associations between confounders and both treatment assignment and outcomes. Statistical analysis was conducted from January 1, 2016 to May 5, 2017.InterventionsSpecialty probationers were assigned to small, homogeneous caseloads supervised by experts. Prior data indicate that specialty officers had better relationships with probationers, participated more in probationers’ treatment, and relied more on positive compliance strategies than traditional officers.Main Outcomes and MeasuresViolence during 1 year, determined by probationer report, officer report, and records, and rearrest during a period of 2 to 5 years, according to Federal Bureau of Investigation records.ResultsParticipants were 183 specialty (73.8% of 248 eligible) and 176 traditional (56.6% of 311 eligible) probationers (205 men and 154 women; mean [SD] age, 36.9 [10.6]). Although specialty probation had no significant effect on violence (odds ratio, 0.97; 95% CI, 0.69-1.36), the odds of rearrest were 2.68 times higher for traditional probationers than for specialty probationers (95% CI, 1.86-3.84; P < .001). At 2 years, estimated probabilities of rearrest were 28.6% for specialty probationers and 51.8% for traditional probationers. Survival analyses indicate that arrest effects endured up to 5 years.Conclusions and RelevanceAlthough it did not specifically reduce violence, well-implemented specialty probation appears to be effective in reducing general recidivism. Reform efforts for people with mental illness could leverage probation—a ubiquitous and revitalized node of the justice system.

Quantile treatment effect and double robust estimators

PurposeSeveral approaches have been proposed to evaluate treatment effect, relying on matching methods propensity score, quantile regression, influence function, bootstrap and various combinations of the above. This paper considers two of these approaches to define the quantile double robust (DR) estimator: the inverse propensity score weights, to compare potential output of treated and untreated groups; the Machado and Mata quantile decomposition approach to compute the unconditional quantiles within each group – treated and control. Two Monte Carlo studies and an empirical application for the Italian job labor market conclude the analysis. The paper aims to discuss these issue.Design/methodology/approachThe DR estimator is extended to analyze the tails of the distribution comparing treated and untreated groups, thus defining the quantile based DR estimator. It allows us to measure the treatment effect along the entire outcome distribution. Such a detailed analysis uncovers the presence of heterogeneous impacts of the treatment along the outcome distribution. The computation of the treatment effect at the quantiles, points out variations in the impact of treatment along the outcome distributions. Indeed it is often the case that the impact in the tails sizably differs from the average treatment effect.FindingsTwo Monte Carlo studies show that away from average, the quantile DR estimator can be profitably implemented. In the real data example, the nationwide results are compared with the analysis at a regional level. While at the median and at the upper quartile the nationwide impact is similar to the regional impacts, at the first quartile – the lower incomes – the nationwide effect is close to the North-Center impact but undervalues the impact in the South.Originality/valueThe computation of the treatment effect at various quantiles allows to point out discrepancies between treatment and control along the entire outcome distributions. The discrepancy in the tails may differ from the divergence between the average values. Treatment can be more effective at the lower/higher quantiles. The simulations show the performance at the quartiles of quantile DR estimator. In a wage equation comparing long and short term contracts, this estimator shows the presence of an heterogeneous impact of short term contracts. Their impact changes depending on the income level, the outcome quantiles, and on the geographical region.

Use of regional nodal irradiation and its association with survival for women with high-risk, early stage breast cancer: A National Cancer Database analysis

PurposeThe role of regional nodal irradiation (RNI) for patients with breast cancer remains controversial, particularly on the basis of nodal involvement. Using the National Cancer Database, we aimed to validate published data on whether expanding treatment fields from whole-breast irradiation (WBI) to encompass the regional nodes (WBI+RNI) affected overall survival (OS) for patients with node-positive (pN1-3) or high-risk node-negative (pN0) breast cancer treated with breast-conserving surgery and adjuvant chemotherapy. Methods and materialsWomen diagnosed with invasive breast cancer between 2004 and 2012 who met the selection criteria for the National Cancer Institute of Canada MA.20 trial were identified and stratified by receipt of RNI. Propensity score matching was used to compare 1:1 matched pairs of patients. Five-year OS was estimated using the Kaplan-Meier method. We used multivariate logistic regression to predict receipt of WBI+RNI and a multivariable Cox model to examine associations between patients' demographic, tumor, and treatment characteristics and OS using double robust estimation. ResultsOf 23,567 patients, 6,920 (29%) received WBI+RNI and 16,647 (71%) WBI. Median follow-up was 56 months. Use of WBI+RNI increased from 25.2% in 2004 to 32.2% in 2012 (P < .001). Patients receiving WBI+RNI more often had negative hormone-receptor status, ≥5 cm tumors and >1 involved node, and were not privately insured. For all patients, the 5-year OS rates were 90.8% with WBI+RNI versus 92.6% with WBI (P < .001). In the matched cohort (n = 10,922), the corresponding 5-year OS rates were 92% and 91.9% (P = .45), respectively. On multivariate analysis, WBI+RNI did not affect OS in the matched cohort (hazard ratio, 1.02; 95% confidence interval, 0.89-1.17, P = .76), regardless of pathologic nodal status. ConclusionsIn this large retrospective analysis, use of WBI+RNI did not affect 5-year OS rates for women with high-risk, early stage breast cancer undergoing breast-conserving surgery and adjuvant chemotherapy, regardless of nodal status, which confirms the findings of the MA.20 trial.

On Modeling and Estimation for the Relative Risk and Risk Difference

ABSTRACTA common problem in formulating models for the relative risk and risk difference is the variation dependence between these parameters and the baseline risk, which is a nuisance model. We address this problem by proposing the conditional log odds-product as a preferred nuisance model. This novel nuisance model facilitates maximum-likelihood estimation, but also permits doubly-robust estimation for the parameters of interest. Our approach is illustrated via simulations and a data analysis. An R package implementing the proposed methods is available on CRAN. Supplementary materials for this article are available online.