Double Robust Estimators Research Articles

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.

Use of a machine learning framework to predict substance use disorder treatment success.

There are several methods for building prediction models. The wealth of currently available modeling techniques usually forces the researcher to judge, a priori, what will likely be the best method. Super learning (SL) is a methodology that facilitates this decision by combining all identified prediction algorithms pertinent for a particular prediction problem. SL generates a final model that is at least as good as any of the other models considered for predicting the outcome. The overarching aim of this work is to introduce SL to analysts and practitioners. This work compares the performance of logistic regression, penalized regression, random forests, deep learning neural networks, and SL to predict successful substance use disorders (SUD) treatment. A nationwide database including 99,013 SUD treatment patients was used. All algorithms were evaluated using the area under the receiver operating characteristic curve (AUC) in a test sample that was not included in the training sample used to fit the prediction models. AUC for the models ranged between 0.793 and 0.820. SL was superior to all but one of the algorithms compared. An explanation of SL steps is provided. SL is the first step in targeted learning, an analytic framework that yields double robust effect estimation and inference with fewer assumptions than the usual parametric methods. Different aspects of SL depending on the context, its function within the targeted learning framework, and the benefits of this methodology in the addiction field are discussed.

Model averaged double robust estimation.

Researchers estimating causal effects are increasingly challenged with decisions on how to best control for a potentially high-dimensional set of confounders. Typically, a single propensity score model is chosen and used to adjust for confounding, while the uncertainty surrounding which covariates to include into the propensity score model is often ignored, and failure to include even one important confounder will results in bias. We propose a practical and generalizable approach that overcomes the limitations described above through the use of model averaging. We develop and evaluate this approach in the context of double robust estimation. More specifically, we introduce the model averaged double robust (MA-DR) estimators, which account for model uncertainty in both the propensity score and outcome model through the use of model averaging. The MA-DR estimators are defined as weighted averages of double robust estimators, where each double robust estimator corresponds to a specific choice of the outcome model and the propensity score model. The MA-DR estimators extend the desirable double robustness property by achieving consistency under the much weaker assumption that either the true propensity score model or the true outcome model be within a specified, possibly large, class of models. Using simulation studies, we also assessed small sample properties, and found that MA-DR estimators can reduce mean squared error substantially, particularly when the set of potential confounders is large relative to the sample size. We apply the methodology to estimate the average causal effect of temozolomide plus radiotherapy versus radiotherapy alone on one-year survival in a cohort of 1887 Medicare enrollees who were diagnosed with glioblastoma between June 2005 and December 2009.

Outcome of Patients with Multiple Myeloma and CKS1B Gene Amplification after Autologous Hematopoietic Stem Cell Transplantation

The gain/amplification of the CKS1B gene on chromosome 1q21 region is associated with a poor outcome in patients with multiple myeloma (MM). However, there are limited data on the outcome of patients with CKS1B amplification after a single high-dose chemotherapy and autologous hematopoietic stem cell transplantation (auto-HCT). We retrospectively evaluated the outcome of patients with CKS1B amplification who received an auto-HCT between June 2012 and July 2014 at our institution. We identified 58 patients with MM and CKS1B gene amplification detected by fluorescent in situ hybridization (FISH). We compared their outcomes with a propensity score–matched control group of 58 patients without CKS1B amplification who were treated at approximately the same time. The primary objective was to compare the progression-free (PFS) and overall survival (OS) between the CKS1B and the control groups. Stratified log-rank test with the matched pairs as strata and double robust estimation under the Cox model were used to assess the effect of CKS1B gene amplification on PFS or OS in the matched cohort. Patients in the CKS1B and control groups were well matched for age, gender, disease status, year of auto-HCT, response to pretransplantation therapy, and baseline hemoglobin level. In both groups, 57% patients were in first remission and 43% had relapsed disease at auto-HCT. Twenty-seven (47%) patients with CKS1B amplification had concurrent monosomy 13 or 13q deletion; 6 (10%) by conventional cytogenetics only, 16 (28%) by FISH only, and 5 (9%) by both. Median follow-up after auto-HCT was 25.4 months. The median PFS of the CKS1B and the control groups were 15.0 months and 33.0 months (P = .002), respectively. The median OS have not been reached yet. The 2-year OS rates in the CKS1B and the control groups were 62% and 91% (P = .02), respectively. In conclusion, Patients with CKS1B amplification are more likely to have additional high-risk cytogenetic abnormalities and a shorter PFS and OS after an auto-HCT.

A quantile regression model for failure-time data with time-dependent covariates.

Since survival data occur over time, often important covariates that we wish to consider also change over time. Such covariates are referred as time-dependent covariates. Quantile regression offers flexible modeling of survival data by allowing the covariates to vary with quantiles. This article provides a novel quantile regression model accommodating time-dependent covariates, for analyzing survival data subject to right censoring. Our simple estimation technique assumes the existence of instrumental variables. In addition, we present a doubly-robust estimator in the sense of Robins and Rotnitzky (1992, Recovery of information and adjustment for dependent censoring using surrogate markers. In: Jewell, N. P., Dietz, K. and Farewell, V. T. (editors), AIDS Epidemiology. Boston: Birkhaäuser, pp. 297-331.). The asymptotic properties of the estimators are rigorously studied. Finite-sample properties are demonstrated by a simulation study. The utility of the proposed methodology is demonstrated using the Stanford heart transplant dataset.

Optimal treatment regimes for survival endpoints using a locally-efficient doubly-robust estimator from a classification perspective.

A treatment regime at a single decision point is a rule that assigns a treatment, among the available options, to a patient based on the patient's baseline characteristics. The value of a treatment regime is the average outcome of a population of patients if they were all treated in accordance to the treatment regime, where large values are desirable. The optimal treatment regime is a regime which results in the greatest value. Typically, the optimal treatment regime is estimated by positing a regression relationship for the outcome of interest as a function of treatment and baseline characteristics. However, this can lead to suboptimal treatment regimes when the regression model is misspecified. We instead consider value search estimators for the optimal treatment regime where we directly estimate the value for any treatment regime and then maximize this estimator over a class of regimes. For many studies the primary outcome of interest is survival time which is often censored. We derive a locally efficient, doubly robust, augmented inverse probability weighted complete case estimator for the value function with censored survival data and study the large sample properties of this estimator. The optimization is realized from a weighted classification perspective that allows us to use available off the shelf software. In some studies one treatment may have greater toxicity or side effects, thus we also consider estimating a quality adjusted optimal treatment regime that allows a patient to trade some additional risk of death in order to avoid the more invasive treatment.

Mean response estimation with missing response in the presence of high-dimensional covariates

ABSTRACTThis paper studies the problem of mean response estimation where missingness occurs to the response but multiple-dimensional covariates are observable. Two main challenges occur in this situation: curse of dimensionality and model specification. The non parametric imputation method relieves model specification but suffers curse of dimensionality, while some model-based methods such as inverse probability weighting (IPW) and augmented inverse probability weighting (AIPW) methods are the opposite. We propose a unified non parametric method to overcome the two challenges with the aiding of sufficient dimension reduction. It imposes no parametric structure on propensity score or conditional mean response, and thus retains the non parametric flavor. Moreover, the estimator achieves the optimal efficiency that a double robust estimator can attain. Simulations were conducted and it demonstrates the excellent performances of our method in various situations.

Double robust estimator of average causal treatment effect for censored medical cost data.

In observational studies, estimation of average causal treatment effect on a patient's response should adjust for confounders that are associated with both treatment exposure and response. In addition, the response, such as medical cost, may have incomplete follow-up. In this article, a double robust estimator is proposed for average causal treatment effect for right censored medical cost data. The estimator is double robust in the sense that it remains consistent when either the model for the treatment assignment or the regression model for the response is correctly specified. Double robust estimators increase the likelihood the results will represent a valid inference. Asymptotic normality is obtained for the proposed estimator, and an estimator for the asymptotic variance is also derived. Simulation studies show good finite sample performance of the proposed estimator and a real data analysis using the proposed method is provided as illustration. Copyright © 2016 John Wiley & Sons, Ltd.

The Value of Knowing the Propensity Score for Estimating Average Treatment Effects

In a treatment effect model with unconfoundedness, treatment assignments are not only independent of potential outcomes given the covariates, but also given the score alone. Despite this powerful dimension reduction property, adjusting for the score is known to lead to an estimator of the average treatment effect with lower asymptotic efficiency than one based on adjusting for all covariates. Moreover, knowledge of the score does not change the efficiency bound for estimating average treatment effects, and many empirical strategies are more efficient when an estimate of the score is used instead of its true value. Here, we resolve this propensity score paradox by demonstrating the value of knowledge of the score.We show that by exploiting such knowledge properly, it is possible to construct an efficient treatment effect estimator that is not affected by the curse of dimensionality, which yields desirable second order asymptotic properties and finite sample performance. The method combines knowledge of the score with a nonparametric adjustment for covariates, building on ideas from the literature on double robust estimation. It is straightforward to implement, and performs well in simulations. We also show that confidence intervals based on our estimator and a simple variance estimate have remarkably robust coverage properties with respect to the implementation details of the nonparametric adjustment step.

Antifungal de-escalation was not associated with adverse outcome in critically ill patients treated for invasive candidiasis: post hoc analyses of the AmarCAND2 study data.

Systemic antifungal therapy (SAT) of invasive candidiasis needs to be initiated immediately upon clinical suspicion. Controversies exist about adequate time and potential harm of antifungal de-escalation (DE) in documented and suspected candidiasis in ICU patients. Our objective was to investigate whether de-escalation within 5 days of antifungal initiation is associated with an increase of the 28-day mortality in SAT-treated non-neutropenic adult ICU patients. From the 835 non-neutropenic adults recruited in the multicenter prospective observational AmarCAND2 study, we selected the patients receiving systemic antifungal therapy for a documented or suspected invasive candidiasis in the ICU and who were still alive 5 days after SAT initiation. They were included into two groups according to the occurrence of observed SAT de-escalation before day 6. The average causal SAT de-escalation effect on 28-day mortality was evaluated by using a double robust estimation. Among the 647 included patients, early de-escalation at day 5 after antifungal initiation occurred in 142 patients (22%), including 48 (34%) patients whose SAT was stopped before day 6. After adjustment for the baseline confounders, early SAT de-escalation was the solely factor not associated with increased 28-day mortality (RR 1.12, 95% CI 0.76-1.66). In non-neutropenic critically ill adult patients with documented or suspected invasive candidiasis, SAT de-escalation within 5 days was not related to increased day-28 mortality but it was associated with decreased SAT consumption. These results suggest for the first time that SAT de-escalation may be safe in these patients.

Double robust and efficient estimation of a prognostic model for events in the presence of dependent censoring.

In longitudinal data arising from observational or experimental studies, dependent subject drop-out is a common occurrence. If the goal is estimation of the parameters of a marginal complete-data model for the outcome, biased inference will result from fitting the model of interest with only uncensored subjects. For example, investigators are interested in estimating a prognostic model for clinical events in HIV-positive patients, under the counterfactual scenario in which everyone remained on ART (when in reality, only a subset had). Inverse probability of censoring weighting (IPCW) is a popular method that relies on correct estimation of the probability of censoring to produce consistent estimation, but is an inefficient estimator in its standard form. We introduce sequentially augmented regression (SAR), an adaptation of the Bang and Robins (2005. Doubly robust estimation in missing data and causal inference models. Biometrics 61, 962-972.) method to estimate a complete-data prediction model, adjusting for longitudinal missing at random censoring. In addition, we propose a closely related non-parametric approach using targeted maximum likelihood estimation (TMLE; van der Laan and Rubin, 2006. Targeted maximum likelihood learning. The International Journal of Biostatistics 2 (1), Article 11). We compare IPCW, SAR, and TMLE (implemented parametrically and with Super Learner) through simulation and the above-mentioned case study.

Robust Inference on Average Treatment Effects with Possibly More Covariates than Observations

This paper concerns robust inference on average treatment effects following model selection. In the selection on observables framework, we show how to construct confidence intervals based on a doubly-robust estimator that are robust to model selection errors and prove that they are valid uniformly over a large class of treatment effect models. The class allows for multivalued treatments with heterogeneous effects (in observables), general heteroskedasticity, and selection amongst (possibly) more covariates than observations. Our estimator attains the semiparametric efficiency bound under appropriate conditions. Precise conditions are given for any model selector to yield these results, and we show how to combine data-driven selection with economic theory. For implementation, we give a specific proposal for selection based on the group lasso and derive new technical results for high-dimensional, sparse multinomial logistic regression. A simulation study shows our estimator performs very well in finite samples over a wide range of models. Revisiting the National Supported Work demonstration data, our method yields accurate estimates and tight confidence intervals.

Robust inference on average treatment effects with possibly more covariates than observations

This paper concerns robust inference on average treatment effects following model selection. Under selection on observables, we construct confidence intervals using a doubly-robust estimator that are robust to model selection errors and prove their uniform validity over a large class of models that allows for multivalued treatments with heterogeneous effects and selection amongst (possibly) more covariates than observations. The semiparametric efficiency bound is attained under appropriate conditions. Precise conditions are given for any model selector to yield these results, and we specifically propose the group lasso, which is apt for treatment effects, and derive new results for high-dimensional, sparse multinomial logistic regression. Both a simulation study and revisiting the National Supported Work demonstration show our estimator performs well in finite samples.

Do debit cards increase household spending? Evidence from a semiparametric causal analysis of a survey

Motivated by recent findings in the field of consumer science, this paper evaluates the causal effect of debit cards on household consumption using population-based data from the Italy Survey on Household Income and Wealth (SHIW). Within the Rubin Causal Model, we focus on the estimand of population average treatment effect for the treated (PATT). We consider three existing estimators, based on regression, mixed matching and regression, propensity score weighting, and propose a new doubly-robust estimator. Semiparametric specification based on power series for the potential outcomes and the propensity score is adopted. Cross-validation is used to select the order of the power series. We conduct a simulation study to compare the performance of the estimators. The key assumptions, overlap and unconfoundedness, are systematically assessed and validated in the application. Our empirical results suggest statistically significant positive effects of debit cards on the monthly household spending in Italy.

Does Physician Dispensing Increase Drug Expenditures? Empirical Evidence from Switzerland.

This paper analyzes whether the opportunity for physicians to dispense drugs increases healthcare expenditures. We study the case of Switzerland, where dispensing physicians face financial incentives to overprescribe and sell more expensive pharmaceuticals. Using comprehensive physician-level data, we exploit the regional variation in the dispensing regime to estimate causal effects. The empirical strategy consists of a doubly-robust estimation that combines inverse probability weighting with regression. Our main finding suggests that dispensing leads to higher drug costs on the order of 34% per patient.

A comparison of doubly robust estimators of the mean with missing data

We consider data with a continuous outcome that is missing at random and a fully observed set of covariates. We compare by simulation a variety of doubly-robust (DR) estimators for estimating the mean of the outcome. An estimator is DR if it is consistent when either the regression model for the mean function or the propensity to respond is correctly specified. Performance of different methods is compared in terms of root mean squared error of the estimates and width and coverage of confidence intervals or posterior credibility intervals in repeated samples. Overall, the DR methods tended to yield better inference than the incorrect model when either the propensity or mean model is correctly specified, but were less successful for small sample sizes, where the asymptotic DR property is less consequential. Two methods tended to outperform the other DR methods: penalized spline of propensity prediction [Little RJA, An H. Robust likelihood-based analysis of multivariate data with missing values. Statist Sinica. 2004;14:949–968] and the robust method proposed in [Cao W, Tsiatis AA, Davidian M. Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data. Biometrika. 2009;96:723–734].

Coarsened Propensity Scores and Hybrid Estimators for Missing Data and Causal Inference

SummaryIn the areas of missing data and causal inference, there is great interest in doubly robust (DR) estimators that involve both an outcome regression (RG) model and a propensity score (PS) model. These DR estimators are consistent and asymptotically normal if either model is correctly specified. Despite their theoretical appeal, the practical utility of DR estimators has been disputed (e.g. Kang and Schaffer, Statistical Science 2007; 22: 523–539). One of the major concerns is the possibility of erratic estimates resulting from near‐zero denominators due to extreme values of the estimated PS. In contrast, the usual RG estimator based on the RG model alone is efficient when the RG model is correct and generally more stable than the DR estimators, although it can be biased when the RG model is incorrect. In light of the unique advantages of the RG and DR estimators, we propose a class of hybrid estimators that attempt to strike a reasonable balance between the RG and DR estimators. These hybrid estimators are motivated by heuristic arguments that coarsened PS estimates are less likely to take extreme values and less sensitive to misspecification of the PS model than the original model‐based PS estimates. The proposed estimators are compared with existing estimators in simulation studies and illustrated with real data from a large observational study on obstetric labour progression and birth outcomes.

Effect of hormone receptor status and local treatment on overall survival for early-stage breast cancer.

60 Background: Mastectomy and breast conserving therapy (BCT) have been established as interventions with equivalent survival for early stage breast cancer. However, trials comparing these approaches pre-date the understanding of breast cancer heterogeneity. We hypothesized that if heterogeneity is considered, the surgical approach may impact survival. Methods: Using the National Cancer Database (NCDB) from 2004 to 2005, we evaluated the overall survival (OS) of women with Stage I breast cancer who underwent mastectomy, BCT (surgery with radiation), or breast conserving surgery (BCS, surgery without radiation). Since only ER and PR data were available, we categorized tumors as ER and/or PR positive (HR+) or ER and PR negative (HR-). We performed propensity score-matched analysis using the covariates associated with the choice of surgical therapy. We used the Cox proportional hazards model for analyses of OS in pre-matched and matched cohorts. Double robust estimation under the Cox model was used in the analyses of the matched cohort. Results: A total of 16,646 female patients met study criteria: 1,845 (11%) received BCS, 11,214 (67%) received BCT, and 3,587 (22%) underwent mastectomy. Patients undergoing BCT had superior survival outcomes compared to those undergoing mastectomy or BCS (5-year OS was 96% vs 90% vs 87% respectively, p<0.001). After adjusting for other risk factors, BCT remained significantly associated with OS (HR 0.57 [95% CI 0.50, 0.66] for BCT vs BCS; HR 0.67 [95% CI 0.6, 0.76] for BCT vs mastectomy). In the matched cohort (1706 patients in each treatment group), comparison of OS in multivariate analysis confirmed the survival benefit associated with BCT over mastectomy (HR 0.73 [95% CI, 0.59, 0.89]) in the HR+ subset but not in the HR- subset (HR 0.91 [95%CI 0.62, 1.34]) of patients. BCT showed better OS than BCS in both HR+ and HR- subsets (HR 0.63 [95% CI, 0.52, 0.77], HR 0.67 [95%CI 0.46, 0.98] respectively). No differences were seen in OS between mastectomy and BCS in either HR+ or HR- cohorts (HR 0.87 [95%CI. 0.73, 1.03]), HR 0.73 [95%CI 0.51, 1.06] respectively). Conclusions: When tumor heterogeneity is considered, type of local therapy appears to impact the survival of women with Stage I breast cancer.

Extending the Bayesian Adjustment for Confounding algorithm to binary treatment covariates to estimate the effect of smoking on carotid intima-media thickness: the Multi-Ethnic Study of Atherosclerosis.

We illustrate the application of the Bayesian Adjustment for Confounding (BAC) algorithm when the treatment covariate is binary. Using data from the Multi-Ethnic Study of Atherosclerosis, we estimate the effect of ever smoking on common carotid artery intimal medial thickness among adult Caucasian participants (n=1378). Our novel implementation of the BAC algorithm is performed first from an outcome model perspective and second from a treatment model perspective with both inverse probability weighting and doubly-robust estimation techniques. The BAC results are compared with the results obtained using standard model averaging and full model strategies, giving a range of adjusted estimates between 45.50 and 65.30 μm for increased common carotid artery intimal medial thickness among ever smokers. For both perspectives, we observe that BAC offers similar performance to using the fully specified outcome and/or treatment model (the full outcome model ever smoking effect is 48.61 μm; 95% CI: (0.62, 96.60)). We then redo the analyses for the African American, Hispanic, and Chinese adult participants to study the robustness of these findings with reduced sample size. For the Chinese subcohort, which corresponds to the smallest sample size (n=436), we find that, from a treatment model perspective, BAC reduces the variability of the estimates in comparison with using a full model approach. This suggests that the use of BAC in conjunction with inverse probability weighting and doubly-robust estimation can be advantageous when applied to relatively small sample sizes. This conjecture is subsequently verified on the basis of three simulated experiments.

Test the reliability of doubly robust estimation with missing response data

In statistical inference, one has to make sure that the underlying regression model is correctly specified otherwise the resulting estimation may be biased. Model checking is an important method to detect any departure of the regression model from the true one. Missing data are a ubiquitous problem in social and medical studies. If the underlying regression model is correctly specified, recent researches show great popularity of the doubly robust (DR) estimates method for handling missing data because of its robustness to the misspecification of either the missing data model or the conditional mean model, that is, the model for the conditional expectation of true regression model conditioning on the observed quantities. However, little work has been devoted to the goodness of fit test for DR estimates method. In this article, we propose a testing method to assess the reliability of the estimator derived from the DR estimating equation with possibly missing response and always observed auxiliary variables. Numerical studies demonstrate that the proposed test can control type I errors well. Furthermore the proposed method can detect departures from model assumptions in the marginal mean model of interest powerfully. A real dementia data set is used to illustrate the method for the diagnosis of model misspecification in the problem of missing response with an always observed auxiliary variable for cross-sectional data.