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1.  Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data 
Biometrika  2009;96(3):723-734.
Considerable recent interest has focused on doubly robust estimators for a population mean response in the presence of incomplete data, which involve models for both the propensity score and the regression of outcome on covariates. The usual doubly robust estimator may yield severely biased inferences if neither of these models is correctly specified and can exhibit nonnegligible bias if the estimated propensity score is close to zero for some observations. We propose alternative doubly robust estimators that achieve comparable or improved performance relative to existing methods, even with some estimated propensity scores close to zero.
doi:10.1093/biomet/asp033
PMCID: PMC2798744  PMID: 20161511
Causal inference; Enhanced propensity score model; Missing at random; No unmeasured confounders; Outcome regression
2.  ESTIMATING TREATMENT EFFECTS ON HEALTHCARE COSTS UNDER EXOGENEITY: IS THERE A ‘MAGIC BULLET’? 
Methods for estimating average treatment effects, under the assumption of no unmeasured confounders, include regression models; propensity score adjustments using stratification, weighting, or matching; and doubly robust estimators (a combination of both). Researchers continue to debate about the best estimator for outcomes such as health care cost data, as they are usually characterized by an asymmetric distribution and heterogeneous treatment effects,. Challenges in finding the right specifications for regression models are well documented in the literature. Propensity score estimators are proposed as alternatives to overcoming these challenges. Using simulations, we find that in moderate size samples (n= 5000), balancing on propensity scores that are estimated from saturated specifications can balance the covariate means across treatment arms but fails to balance higher-order moments and covariances amongst covariates. Therefore, unlike regression model, even if a formal model for outcomes is not required, propensity score estimators can be inefficient at best and biased at worst for health care cost data. Our simulation study, designed to take a ‘proof by contradiction’ approach, proves that no one estimator can be considered the best under all data generating processes for outcomes such as costs. The inverse-propensity weighted estimator is most likely to be unbiased under alternate data generating processes but is prone to bias under misspecification of the propensity score model and is inefficient compared to an unbiased regression estimator. Our results show that there are no ‘magic bullets’ when it comes to estimating treatment effects in health care costs. Care should be taken before naively applying any one estimator to estimate average treatment effects in these data. We illustrate the performance of alternative methods in a cost dataset on breast cancer treatment.
doi:10.1007/s10742-011-0072-8
PMCID: PMC3244728  PMID: 22199462
Propensity score; non-linear regression; average treatment effect; health care costs
3.  Doubly Robust Estimation of Causal Effects 
American Journal of Epidemiology  2011;173(7):761-767.
Doubly robust estimation combines a form of outcome regression with a model for the exposure (i.e., the propensity score) to estimate the causal effect of an exposure on an outcome. When used individually to estimate a causal effect, both outcome regression and propensity score methods are unbiased only if the statistical model is correctly specified. The doubly robust estimator combines these 2 approaches such that only 1 of the 2 models need be correctly specified to obtain an unbiased effect estimator. In this introduction to doubly robust estimators, the authors present a conceptual overview of doubly robust estimation, a simple worked example, results from a simulation study examining performance of estimated and bootstrapped standard errors, and a discussion of the potential advantages and limitations of this method. The supplementary material for this paper, which is posted on the Journal's Web site (http://aje.oupjournals.org/), includes a demonstration of the doubly robust property (Web Appendix 1) and a description of a SAS macro (SAS Institute, Inc., Cary, North Carolina) for doubly robust estimation, available for download at http://www.unc.edu/∼mfunk/dr/.
doi:10.1093/aje/kwq439
PMCID: PMC3070495  PMID: 21385832
causal inference; epidemiologic methods; propensity score
4.  Evaluating the Effect of Early Versus Late ARV Regimen Change if Failure on an Initial Regimen: Results From the AIDS Clinical Trials Group Study A5095 
The current goal of initial antiretroviral (ARV) therapy is suppression of plasma human immunodeficiency virus (HIV)-1 RNA levels to below 200 copies per milliliter. A proportion of HIV-infected patients who initiate antiretroviral therapy in clinical practice or antiretroviral clinical trials either fail to suppress HIV-1 RNA or have HIV-1 RNA levels rebound on therapy. Frequently, these patients have sustained CD4 cell counts responses and limited or no clinical symptoms and, therefore, have potentially limited indications for altering therapy which they may be tolerating well despite increased viral replication. On the other hand, increased viral replication on therapy leads to selection of resistance mutations to the antiretroviral agents comprising their therapy and potentially cross-resistance to other agents in the same class decreasing the likelihood of response to subsequent antiretroviral therapy. The optimal time to switch antiretroviral therapy to ensure sustained virologic suppression and prevent clinical events in patients who have rebound in their HIV-1 RNA, yet are stable, is not known. Randomized clinical trials to compare early versus delayed switching have been difficult to design and more difficult to enroll. In some clinical trials, such as the AIDS Clinical Trials Group (ACTG) Study A5095, patients randomized to initial antiretroviral treatment combinations, who fail to suppress HIV-1 RNA or have a rebound of HIV-1 RNA on therapy are allowed to switch from the initial ARV regimen to a new regimen, based on clinician and patient decisions. We delineate a statistical framework to estimate the effect of early versus late regimen change using data from ACTG A5095 in the context of two-stage designs.
In causal inference, a large class of doubly robust estimators are derived through semiparametric theory with applications to missing data problems. This class of estimators is motivated through geometric arguments and relies on large samples for good performance. By now, several authors have noted that a doubly robust estimator may be suboptimal when the outcome model is misspecified even if it is semiparametric efficient when the outcome regression model is correctly specified. Through auxiliary variables, two-stage designs, and within the contextual backdrop of our scientific problem and clinical study, we propose improved doubly robust, locally efficient estimators of a population mean and average causal effect for early versus delayed switching to second-line ARV treatment regimens. Our analysis of the ACTG A5095 data further demonstrates how methods that use auxiliary variables can improve over methods that ignore them. Using the methods developed here, we conclude that patients who switch within 8 weeks of virologic failure have better clinical outcomes, on average, than patients who delay switching to a new second-line ARV regimen after failing on the initial regimen. Ordinary statistical methods fail to find such differences. This article has online supplementary material.
doi:10.1080/01621459.2011.646932
PMCID: PMC3545451  PMID: 23329858
Causal inference; Double robustness; Longitudinal data analysis; Missing data; Rubin causal model; Semiparametric efficient estimation
5.  Improved Doubly Robust Estimation when Data are Monotonely Coarsened, with Application to Longitudinal Studies with Dropout 
Biometrics  2010;67(2):536-545.
Summary
A routine challenge is that of making inference on parameters in a statistical model of interest from longitudinal data subject to drop out, which are a special case of the more general setting of monotonely coarsened data. Considerable recent attention has focused on doubly robust estimators, which in this context involve positing models for both the missingness (more generally, coarsening) mechanism and aspects of the distribution of the full data, that have the appealing property of yielding consistent inferences if only one of these models is correctly specified. Doubly robust estimators have been criticized for potentially disastrous performance when both of these models are even only mildly misspecified. We propose a doubly robust estimator applicable in general monotone coarsening problems that achieves comparable or improved performance relative to existing doubly robust methods, which we demonstrate via simulation studies and by application to data from an AIDS clinical trial.
doi:10.1111/j.1541-0420.2010.01476.x
PMCID: PMC3061242  PMID: 20731640
Coarsening at random; Discrete hazard; Dropout; Longitudinal data; Missing at random
6.  Test the reliability of doubly robust estimation with missing response data 
Biometrics  2014;70(2):289-298.
Summary
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 is 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 estimates method for handling missing data because of its robustness to the misspecification of either the missing data model or the conditional mean model, i.e. 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 doubly robust estimates method. In this paper, we propose a testing method to assess the reliability of the estimator derived from the doubly robust 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.
doi:10.1111/biom.12150
PMCID: PMC4061274  PMID: 24571677
Auxiliary; doubly robust; estimating equation; goodness of fit; missing data
7.  A new estimation with minimum trace of asymptotic covariance matrix for incomplete longitudinal data with a surrogate process 
Statistics in medicine  2013;32(27):10.1002/sim.5875.
Missing data is a very common problem in medical and social studies, especially when data are collected longitudinally. It is a challenging problem to utilize observed data effectively. Many papers on missing data problems can be found in statistical literature. It is well known that the inverse weighted estimation is neither efficient nor robust. On the other hand, the doubly robust (DR) method can improve the efficiency and robustness. As is known, the DR estimation requires a missing data model (i.e., a model for the probability that data are observed) and a working regression model (i.e., a model for the outcome variable given covariates and surrogate variables). Because the DR estimating function has mean zero for any parameters in the working regression model when the missing data model is correctly specified, in this paper, we derive a formula for the estimator of the parameters of the working regression model that yields the optimally efficient estimator of the marginal mean model (the parameters of interest) when the missing data model is correctly specified. Furthermore, the proposed method also inherits the DR property. Simulation studies demonstrate the greater efficiency of the proposed method compared with the standard DR method. A longitudinal dementia data set is used for illustration.
doi:10.1002/sim.5875
PMCID: PMC3808493  PMID: 23744541
longitudinal data; missing data; optimal; surrogate outcome
8.  Doubly Robust Nonparametric Multiple Imputation for Ignorable Missing Data 
Statistica Sinica  2012;22:149-172.
Missing data are common in medical and social science studies and often pose a serious challenge in data analysis. Multiple imputation methods are popular and natural tools for handling missing data, replacing each missing value with a set of plausible values that represent the uncertainty about the underlying values. We consider a case of missing at random (MAR) and investigate the estimation of the marginal mean of an outcome variable in the presence of missing values when a set of fully observed covariates is available. We propose a new nonparametric multiple imputation (MI) approach that uses two working models to achieve dimension reduction and define the imputing sets for the missing observations. Compared with existing nonparametric imputation procedures, our approach can better handle covariates of high dimension, and is doubly robust in the sense that the resulting estimator remains consistent if either of the working models is correctly specified. Compared with existing doubly robust methods, our nonparametric MI approach is more robust to the misspecification of both working models; it also avoids the use of inverse-weighting and hence is less sensitive to missing probabilities that are close to 1. We propose a sensitivity analysis for evaluating the validity of the working models, allowing investigators to choose the optimal weights so that the resulting estimator relies either completely or more heavily on the working model that is likely to be correctly specified and achieves improved efficiency. We investigate the asymptotic properties of the proposed estimator, and perform simulation studies to show that the proposed method compares favorably with some existing methods in finite samples. The proposed method is further illustrated using data from a colorectal adenoma study.
PMCID: PMC3280694  PMID: 22347786
Doubly robust; Missing at random; Multiple imputation; Nearest neighbor; Nonparametric imputation; Sensitivity analysis
9.  The performance of different propensity-score methods for estimating differences in proportions (risk differences or absolute risk reductions) in observational studies 
Statistics in Medicine  2010;29(20):2137-2148.
Propensity score methods are increasingly being used to estimate the effects of treatments on health outcomes using observational data. There are four methods for using the propensity score to estimate treatment effects: covariate adjustment using the propensity score, stratification on the propensity score, propensity-score matching, and inverse probability of treatment weighting (IPTW) using the propensity score. When outcomes are binary, the effect of treatment on the outcome can be described using odds ratios, relative risks, risk differences, or the number needed to treat. Several clinical commentators suggested that risk differences and numbers needed to treat are more meaningful for clinical decision making than are odds ratios or relative risks. However, there is a paucity of information about the relative performance of the different propensity-score methods for estimating risk differences. We conducted a series of Monte Carlo simulations to examine this issue. We examined bias, variance estimation, coverage of confidence intervals, mean-squared error (MSE), and type I error rates. A doubly robust version of IPTW had superior performance compared with the other propensity-score methods. It resulted in unbiased estimation of risk differences, treatment effects with the lowest standard errors, confidence intervals with the correct coverage rates, and correct type I error rates. Stratification, matching on the propensity score, and covariate adjustment using the propensity score resulted in minor to modest bias in estimating risk differences. Estimators based on IPTW had lower MSE compared with other propensity-score methods. Differences between IPTW and propensity-score matching may reflect that these two methods estimate the average treatment effect and the average treatment effect for the treated, respectively. Copyright © 2010 John Wiley & Sons, Ltd.
doi:10.1002/sim.3854
PMCID: PMC3068290  PMID: 20108233
propensity score; observational study; binary data; risk difference; number needed to treat; matching; IPTW; inverse probability of treatment weighting; propensity-score matching
10.  Bias and variance trade-offs when combining propensity score weighting and regression: with an application to HIV status and homeless men 
The quality of propensity scores is traditionally measured by assessing how well they make the distributions of covariates in the treatment and control groups match, which we refer to as “good balance”. Good balance guarantees less biased estimates of the treatment effect. However, the cost of achieving good balance is that the variance of the estimates increases due to a reduction in effective sample size, either through the introduction of propensity score weights or dropping cases when propensity score matching. In this paper, we investigate whether it is best to optimize the balance or to settle for a less than optimal balance and use double robust estimation to adjust for remaining differences. We compare treatment effect estimates from regression, propensity score weighting, and double robust estimation with varying levels of effort expended to achieve balance using data from a study about the differences in outcomes by HIV status in heterosexually active homeless men residing in Los Angeles. Because of how costly data collection efforts are for this population, it is important to find an alternative estimation method that does not reduce effective sample size as much as methods that aggressively aim to optimize balance. Results from a simulation study suggest that there are instances in which we can obtain more precise treatment effect estimates without increasing bias too much by using a combination of regression and propensity score weights that achieve a less than optimal balance. There is a bias-variance tradeoff at work in propensity score estimation; every step toward better balance usually means an increase in variance and at some point a marginal decrease in bias may not be worth the associated increase in variance.
PMCID: PMC3433039  PMID: 22956891
Propensity score; Double robust estimation; HIV status; Homeless men
11.  Collaborative Double Robust Targeted Maximum Likelihood Estimation* 
Collaborative double robust targeted maximum likelihood estimators represent a fundamental further advance over standard targeted maximum likelihood estimators of a pathwise differentiable parameter of a data generating distribution in a semiparametric model, introduced in van der Laan, Rubin (2006). The targeted maximum likelihood approach involves fluctuating an initial estimate of a relevant factor (Q) of the density of the observed data, in order to make a bias/variance tradeoff targeted towards the parameter of interest. The fluctuation involves estimation of a nuisance parameter portion of the likelihood, g. TMLE has been shown to be consistent and asymptotically normally distributed (CAN) under regularity conditions, when either one of these two factors of the likelihood of the data is correctly specified, and it is semiparametric efficient if both are correctly specified.
In this article we provide a template for applying collaborative targeted maximum likelihood estimation (C-TMLE) to the estimation of pathwise differentiable parameters in semi-parametric models. The procedure creates a sequence of candidate targeted maximum likelihood estimators based on an initial estimate for Q coupled with a succession of increasingly non-parametric estimates for g. In a departure from current state of the art nuisance parameter estimation, C-TMLE estimates of g are constructed based on a loss function for the targeted maximum likelihood estimator of the relevant factor Q that uses the nuisance parameter to carry out the fluctuation, instead of a loss function for the nuisance parameter itself. Likelihood-based cross-validation is used to select the best estimator among all candidate TMLE estimators of Q0 in this sequence. A penalized-likelihood loss function for Q is suggested when the parameter of interest is borderline-identifiable.
We present theoretical results for “collaborative double robustness,” demonstrating that the collaborative targeted maximum likelihood estimator is CAN even when Q and g are both mis-specified, providing that g solves a specified score equation implied by the difference between the Q and the true Q0. This marks an improvement over the current definition of double robustness in the estimating equation literature.
We also establish an asymptotic linearity theorem for the C-DR-TMLE of the target parameter, showing that the C-DR-TMLE is more adaptive to the truth, and, as a consequence, can even be super efficient if the first stage density estimator does an excellent job itself with respect to the target parameter.
This research provides a template for targeted efficient and robust loss-based learning of a particular target feature of the probability distribution of the data within large (infinite dimensional) semi-parametric models, while still providing statistical inference in terms of confidence intervals and p-values. This research also breaks with a taboo (e.g., in the propensity score literature in the field of causal inference) on using the relevant part of likelihood to fine-tune the fitting of the nuisance parameter/censoring mechanism/treatment mechanism.
doi:10.2202/1557-4679.1181
PMCID: PMC2898626  PMID: 20628637
asymptotic linearity; coarsening at random; causal effect; censored data; crossvalidation; collaborative double robust; double robust; efficient influence curve; estimating function; estimator selection; influence curve; G-computation; locally efficient; loss-function; marginal structural model; maximum likelihood estimation; model selection; pathwise derivative; semiparametric model; sieve; super efficiency; super-learning; targeted maximum likelihood estimation; targeted nuisance parameter estimator selection; variable importance
12.  Balance diagnostics for comparing the distribution of baseline covariates between treatment groups in propensity-score matched samples 
Statistics in Medicine  2009;28(25):3083-3107.
The propensity score is a subject's probability of treatment, conditional on observed baseline covariates. Conditional on the true propensity score, treated and untreated subjects have similar distributions of observed baseline covariates. Propensity-score matching is a popular method of using the propensity score in the medical literature. Using this approach, matched sets of treated and untreated subjects with similar values of the propensity score are formed. Inferences about treatment effect made using propensity-score matching are valid only if, in the matched sample, treated and untreated subjects have similar distributions of measured baseline covariates. In this paper we discuss the following methods for assessing whether the propensity score model has been correctly specified: comparing means and prevalences of baseline characteristics using standardized differences; ratios comparing the variance of continuous covariates between treated and untreated subjects; comparison of higher order moments and interactions; five-number summaries; and graphical methods such as quantile–quantile plots, side-by-side boxplots, and non-parametric density plots for comparing the distribution of baseline covariates between treatment groups. We describe methods to determine the sampling distribution of the standardized difference when the true standardized difference is equal to zero, thereby allowing one to determine the range of standardized differences that are plausible with the propensity score model having been correctly specified. We highlight the limitations of some previously used methods for assessing the adequacy of the specification of the propensity-score model. In particular, methods based on comparing the distribution of the estimated propensity score between treated and untreated subjects are uninformative. Copyright © 2009 John Wiley & Sons, Ltd.
doi:10.1002/sim.3697
PMCID: PMC3472075  PMID: 19757444
balance; goodness-of-fit; observational study; propensity score; matching; propensity-score matching; standardized difference; bias
13.  A robust method using propensity score stratification for correcting verification bias for binary tests 
Sensitivity and specificity are common measures of the accuracy of a diagnostic test. The usual estimators of these quantities are unbiased if data on the diagnostic test result and the true disease status are obtained from all subjects in an appropriately selected sample. In some studies, verification of the true disease status is performed only for a subset of subjects, possibly depending on the result of the diagnostic test and other characteristics of the subjects. Estimators of sensitivity and specificity based on this subset of subjects are typically biased; this is known as verification bias. Methods have been proposed to correct verification bias under the assumption that the missing data on disease status are missing at random (MAR), that is, the probability of missingness depends on the true (missing) disease status only through the test result and observed covariate information. When some of the covariates are continuous, or the number of covariates is relatively large, the existing methods require parametric models for the probability of disease or the probability of verification (given the test result and covariates), and hence are subject to model misspecification. We propose a new method for correcting verification bias based on the propensity score, defined as the predicted probability of verification given the test result and observed covariates. This is estimated separately for those with positive and negative test results. The new method classifies the verified sample into several subsamples that have homogeneous propensity scores and allows correction for verification bias. Simulation studies demonstrate that the new estimators are more robust to model misspecification than existing methods, but still perform well when the models for the probability of disease and probability of verification are correctly specified.
doi:10.1093/biostatistics/kxr020
PMCID: PMC3276270  PMID: 21856650
Diagnostic test; Model misspecification; Propensity score; Sensitivity; Specificity
14.  The performance of different propensity score methods for estimating marginal hazard ratios 
Statistics in Medicine  2012;32(16):2837-2849.
Propensity score methods are increasingly being used to reduce or minimize the effects of confounding when estimating the effects of treatments, exposures, or interventions when using observational or non-randomized data. Under the assumption of no unmeasured confounders, previous research has shown that propensity score methods allow for unbiased estimation of linear treatment effects (e.g., differences in means or proportions). However, in biomedical research, time-to-event outcomes occur frequently. There is a paucity of research into the performance of different propensity score methods for estimating the effect of treatment on time-to-event outcomes. Furthermore, propensity score methods allow for the estimation of marginal or population-average treatment effects. We conducted an extensive series of Monte Carlo simulations to examine the performance of propensity score matching (1:1 greedy nearest-neighbor matching within propensity score calipers), stratification on the propensity score, inverse probability of treatment weighting (IPTW) using the propensity score, and covariate adjustment using the propensity score to estimate marginal hazard ratios. We found that both propensity score matching and IPTW using the propensity score allow for the estimation of marginal hazard ratios with minimal bias. Of these two approaches, IPTW using the propensity score resulted in estimates with lower mean squared error when estimating the effect of treatment in the treated. Stratification on the propensity score and covariate adjustment using the propensity score result in biased estimation of both marginal and conditional hazard ratios. Applied researchers are encouraged to use propensity score matching and IPTW using the propensity score when estimating the relative effect of treatment on time-to-event outcomes. Copyright © 2012 John Wiley & Sons, Ltd.
doi:10.1002/sim.5705
PMCID: PMC3747460  PMID: 23239115
propensity score; survival analysis; inverse probability of treatment weighting (IPTW); Monte Carlo simulations; observational study; time-to-event outcomes
15.  Propensity Score Weighting with Multilevel Data 
Statistics in medicine  2013;32(19):3373-3387.
Propensity score methods are being increasingly used as a less parametric alternative to traditional regression to balance observed differences across groups in both descriptive and causal comparisons. Data collected in many disciplines often have analytically relevant multilevel or clustered structure. The propensity score, however, was developed and has been used primarily with unstructured data. We present and compare several propensity-score-weighted estimators for clustered data, including marginal, cluster-weighted and doubly-robust estimators. Using both analytical derivations and Monte Carlo simulations, we illustrate bias arising when the usual assumptions of propensity score analysis do not hold for multilevel data. We show that exploiting the multilevel structure, either parametrically or nonparametrically, in at least one stage of the propensity score analysis can greatly reduce these biases. These methods are applied to a study of racial disparities in breast cancer screening among beneficiaries in Medicare health plans.
doi:10.1002/sim.5786
PMCID: PMC3710526  PMID: 23526267
balance; multilevel; propensity score; racial disparity; treatment effect; unmeasured confounders; weighting
16.  A Bayesian MCMC approach to survival analysis with doubly-censored data 
Doubly-censored data refers to time to event data for which both the originating and failure times are censored. In studies involving AIDS incubation time or survival after dementia onset, for example, data are frequently doubly-censored because the date of the originating event is interval-censored and the date of the failure event usually is right-censored. The primary interest is in the distribution of elapsed times between the originating and failure events and its relationship to exposures and risk factors. The estimating equation approach [Sun, et al. 1999. Regression analysis of doubly censored failure time data with applications to AIDS studies. Biometrics 55, 909-914] and its extensions assume the same distribution of originating event times for all subjects. This paper demonstrates the importance of utilizing additional covariates to impute originating event times, i.e., more accurate estimation of originating event times may lead to less biased parameter estimates for elapsed time. The Bayesian MCMC method is shown to be a suitable approach for analyzing doubly-censored data and allows a rich class of survival models. The performance of the proposed estimation method is compared to that of other conventional methods through simulations. Two examples, an AIDS cohort study and a population-based dementia study, are used for illustration. Sample code is shown in the appendix.
doi:10.1016/j.csda.2010.02.025
PMCID: PMC2877214  PMID: 20514348
AIDS; dementia; doubly censored data; incubation period; MCMC; midpoint imputation
17.  Doubly Robust Estimates for Binary Longitudinal Data Analysis with Missing Response and Missing Covariates 
Biometrics  2011;67(3):10.1111/j.1541-0420.2010.01541.x.
Summary
Longitudinal studies often feature incomplete response and covariate data. Likelihood-based methods such as the expectation–maximization algorithm give consistent estimators for model parameters when data are missing at random (MAR) provided that the response model and the missing covariate model are correctly specified; however, we do not need to specify the missing data mechanism. An alternative method is the weighted estimating equation, which gives consistent estimators if the missing data and response models are correctly specified; however, we do not need to specify the distribution of the covariates that have missing values. In this article, we develop a doubly robust estimation method for longitudinal data with missing response and missing covariate when data are MAR. This method is appealing in that it can provide consistent estimators if either the missing data model or the missing covariate model is correctly specified. Simulation studies demonstrate that this method performs well in a variety of situations.
doi:10.1111/j.1541-0420.2010.01541.x
PMCID: PMC3652597  PMID: 21281272
Doubly robust; Estimating equation; Missing at random; Missing covariate; Missing response
18.  Optimal caliper widths for propensity-score matching when estimating differences in means and differences in proportions in observational studies 
Pharmaceutical Statistics  2010;10(2):150-161.
In a study comparing the effects of two treatments, the propensity score is the probability of assignment to one treatment conditional on a subject's measured baseline covariates. Propensity-score matching is increasingly being used to estimate the effects of exposures using observational data. In the most common implementation of propensity-score matching, pairs of treated and untreated subjects are formed whose propensity scores differ by at most a pre-specified amount (the caliper width). There has been a little research into the optimal caliper width. We conducted an extensive series of Monte Carlo simulations to determine the optimal caliper width for estimating differences in means (for continuous outcomes) and risk differences (for binary outcomes). When estimating differences in means or risk differences, we recommend that researchers match on the logit of the propensity score using calipers of width equal to 0.2 of the standard deviation of the logit of the propensity score. When at least some of the covariates were continuous, then either this value, or one close to it, minimized the mean square error of the resultant estimated treatment effect. It also eliminated at least 98% of the bias in the crude estimator, and it resulted in confidence intervals with approximately the correct coverage rates. Furthermore, the empirical type I error rate was approximately correct. When all of the covariates were binary, then the choice of caliper width had a much smaller impact on the performance of estimation of risk differences and differences in means. Copyright © 2010 John Wiley & Sons, Ltd.
doi:10.1002/pst.433
PMCID: PMC3120982  PMID: 20925139
propensity score; observational study; binary data; risk difference; propensity-score matching; Monte Carlo simulations; bias; matching
19.  On a Closed-Form Doubly Robust Estimator of the Adjusted Odds Ratio for a Binary Exposure 
American Journal of Epidemiology  2013;177(11):1314-1316.
Epidemiologic studies often aim to estimate the odds ratio for the association between a binary exposure and a binary disease outcome. Because confounding bias is of serious concern in observational studies, investigators typically estimate the adjusted odds ratio in a multivariate logistic regression which conditions on a large number of potential confounders. It is well known that modeling error in specification of the confounders can lead to substantial bias in the adjusted odds ratio for exposure. As a remedy, Tchetgen Tchetgen et al. (Biometrika. 2010;97(1):171–180) recently developed so-called doubly robust estimators of an adjusted odds ratio by carefully combining standard logistic regression with reverse regression analysis, in which exposure is the dependent variable and both the outcome and the confounders are the independent variables. Double robustness implies that only one of the 2 modeling strategies needs to be correct in order to make valid inferences about the odds ratio parameter. In this paper, I aim to introduce this recent methodology into the epidemiologic literature by presenting a simple closed-form doubly robust estimator of the adjusted odds ratio for a binary exposure. A SAS macro (SAS Institute Inc., Cary, North Carolina) is given in an online appendix to facilitate use of the approach in routine epidemiologic practice, and a simulated data example is also provided for the purpose of illustration.
doi:10.1093/aje/kws377
PMCID: PMC3664333  PMID: 23558352
case-control sampling; doubly robust estimator; logistic regression; odds ratio; SAS macro
20.  Estimating Propensity Scores and Causal Survival Functions Using Prevalent Survival Data 
Biometrics  2012;68(3):707-716.
Summary
This article develops semiparametric approaches for estimation of propensity scores and causal survival functions from prevalent survival data. The analytical problem arises when the prevalent sampling is adopted for collecting failure times and, as a result, the covariates are incompletely observed due to their association with failure time. The proposed procedure for estimating propensity scores shares interesting features similar to the likelihood formulation in case-control study, but in our case it requires additional consideration in the intercept term. The result shows that the corrected propensity scores in logistic regression setting can be obtained through standard estimation procedure with specific adjustments on the intercept term. For causal estimation, two different types of missing sources are encountered in our model: one can be explained by potential outcome framework; the other is caused by the prevalent sampling scheme. Statistical analysis without adjusting bias from both sources of missingness will lead to biased results in causal inference. The proposed methods were partly motivated by and applied to the Surveillance, Epidemiology, and End Results (SEER)-Medicare linked data for women diagnosed with breast cancer.
doi:10.1111/j.1541-0420.2012.01754.x
PMCID: PMC3508756  PMID: 22834993
Case-control study; Prevalent sampling; Propensity scores
21.  A Comparison of Multiple Imputation and Fully Augmented Weighted Estimators for Cox Regression with Missing Covariates 
Statistics in medicine  2010;29(25):2592-2604.
Several approaches exist for handling missing covariates in the Cox proportional hazards model. The multiple imputation (MI) is relatively easy to implement with various software available and results in consistent estimates if the imputation model is correct. On the other hand, the fully augmented weighted estimators (FAWEs) recover a substantial proportion of the efficiency and have the doubly robust property. In this paper, we compare the FAWEs and the MI through a comprehensive simulation study. For the MI, we consider the multiple imputation by chained equation (MICE) and focus on two imputation methods: Bayesian linear regression imputation and predictive mean matching. Simulation results show that the imputation methods can be rather sensitive to model misspecification and may have large bias when the censoring time depends on the missing covariates. In contrast, the FAWEs allow the censoring time to depend on the missing covariates and are remarkably robust as long as getting either the conditional expectations or the selection probability correct due to the doubly robust property. The comparison suggests that the FAWEs show the potential for being a competitive and attractive tool for tackling the analysis of survival data with missing covariates.
doi:10.1002/sim.4016
PMCID: PMC4022355  PMID: 20806403
accelerated failure time model; augmented inverse probability weighted estimators; doubly robust property; missing data; proportional hazards model; survival analysis
22.  Evaluating uses of data mining techniques in propensity score estimation: a simulation study† 
SUMMARY
Background
In propensity score modeling, it is a standard practice to optimize the prediction of exposure status based on the covariate information. In a simulation study, we examined in what situations analyses based on various types of exposure propensity score (EPS) models using data mining techniques such as recursive partitioning (RP) and neural networks (NN) produce unbiased and/or efficient results.
Method
We simulated data for a hypothetical cohort study (n=2000) with a binary exposure/outcome and 10 binary/ continuous covariates with seven scenarios differing by non-linear and/or non-additive associations between exposure and covariates. EPS models used logistic regression (LR) (all possible main effects), RP1 (without pruning), RP2 (with pruning), and NN. We calculated c-statistics (C), standard errors (SE), and bias of exposure-effect estimates from outcome models for the PS-matched dataset.
Results
Data mining techniques yielded higher C than LR (mean: NN, 0.86; RPI, 0.79; RP2, 0.72; and LR, 0.76). SE tended to be greater in models with higher C. Overall bias was small for each strategy, although NN estimates tended to be the least biased. C was not correlated with the magnitude of bias (correlation coefficient [COR]=−0.3, p=0.1) but increased SE (COR=0.7, p<0.001).
Conclusions
Effect estimates from EPS models by simple LR were generally robust. NN models generally provided the least numerically biased estimates. C was not associated with the magnitude of bias but was with the increased SE.
doi:10.1002/pds.1555
PMCID: PMC2905676  PMID: 18311848
propensity score; logistic regression; neural networks; recursive partitioning
23.  Adjustment for Missing Data in Complex Surveys Using Doubly Robust Estimation 
Epidemiology (Cambridge, Mass.)  2010;21(6):863-871.
Background
The Demographic and Health Survey program routinely collects nationally representative information on HIV-related risk behaviors in many countries, using face-to-face interviews and a complex sampling scheme. If respondents skip questions about behaviors perceived as socially undesirable, such interviews may introduce bias. We sought to implement a doubly robust estimator to correct for dependent missing data in this context.
Methods
We applied 3 methods of adjustment for nonresponse on self-reported commercial sexual contact data from the 2005–2006 India Demographic Health Survey to estimate the prevalence of sexual contact between sexually active men and female sex workers. These methods were inverse-probability weighted regression, outcome regression, and doubly robust estimation—a recently-described approach that is more robust to model misspecification.
Results
Compared with an unadjusted prevalence of 0.9% for commercial sexual contact prevalence (95% confidence interval = 0.8%–1.0%), adjustment for nonresponse using doubly robust estimation yielded a prevalence of 1.1% (1.0%–1.2%). We found similar estimates with adjustment by outcome regression and inverse-probability weighting. Marital status was strongly associated with item nonresponse, and correction for nonresponse led to a nearly 80% increase in the prevalence of commercial sexual contact among unmarried men (from 6.9% to 12.1%–12.4%).
Conclusions
Failure to correct for nonresponse produced a bias in self-reported commercial sexual contact. To facilitate the application of these methods (including the doubly robust estimator) to complex survey data settings, we provide analytical variance estimators and the corresponding SAS and MATLAB code. These variance estimators remain valid regardless of whether the modeling assumptions are correct.
doi:10.1097/EDE.0b013e3181f57571
PMCID: PMC4122332  PMID: 20881599
24.  Bias associated with using the estimated propensity score as a regression covariate 
Statistics in medicine  2013;33(1):74-87.
The use of propensity score methods to adjust for selection bias in observational studies has become increasingly popular in public health and medical research. A substantial portion of studies using propensity score adjustment treat the propensity score as a conventional regression predictor. Through a Monte Carlo simulation study, Austin and colleagues. investigated the bias associated with treatment effect estimation when the propensity score is used as a covariate in nonlinear regression models, such as logistic regression and Cox proportional hazards models. We show that the bias exists even in a linear regression model when the estimated propensity score is used and derive the explicit form of the bias. We also conduct an extensive simulation study to compare the performance of such covariate adjustment with propensity score stratification, propensity score matching, inverse probability of treatment weighted method, and nonparametric functional estimation using splines. The simulation scenarios are designed to reflect real data analysis practice. Instead of specifying a known parametric propensity score model, we generate the data by considering various degrees of overlap of the covariate distributions between treated and control groups. Propensity score matching excels when the treated group is contained within a larger control pool, while the model-based adjustment may have an edge when treated and control groups do not have too much overlap. Overall, adjusting for the propensity score through stratification or matching followed by regression or using splines, appears to be a good practical strategy.
doi:10.1002/sim.5884
PMCID: PMC4004383  PMID: 23787715
observational studies; matching; stratification; weighting
25.  The Use of Propensity Scores and Observational Data to Estimate Randomized Controlled Trial Generalizability Bias 
Statistics in medicine  2013;32(20):3552-3568.
While randomized controlled trials (RCT) are considered the “gold standard” for clinical studies, the use of exclusion criteria may impact the external validity of the results. It is unknown whether estimators of effect size are biased by excluding a portion of the target population from enrollment. We propose to use observational data to estimate the bias due to enrollment restrictions, which we term generalizability bias. In this paper we introduce a class of estimators for the generalizability bias and use simulation to study its properties in the presence of non-constant treatment effects. We find the surprising result that our estimators can be unbiased for the true generalizability bias even when all potentially confounding variables are not measured. In addition, our proposed doubly robust estimator performs well even for mis-specified models.
doi:10.1002/sim.5802
PMCID: PMC3967586  PMID: 23553373
Observational studies; Randomized controlled trials; Sample selection error; Propensity score; Causal effect

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