An alternative empirical likelihood method in missing response problems and causal inference

Missing responses are common problems in medical, social, and economic studies. When responses are missing at random, a complete case data analysis may result in biases. A popular debias method is inverse probability weighting proposed by Horvitz and Thompson. To improve efficiency, Robins et al. proposed an augmented inverse probability weighting method. The augmented inverse probability weighting estimator has a double‐robustness property and achieves the semiparametric efficiency lower bound when the regression model and propensity score model are both correctly specified. In this paper, we introduce an empirical likelihood‐based estimator as an alternative to Qin and Zhang (2007). Our proposed estimator is also doubly robust and locally efficient. Simulation results show that the proposed estimator has better performance when the propensity score is correctly modeled. Moreover, the proposed method can be applied in the estimation of average treatment effect in observational causal inferences. Finally, we apply our method to an observational study of smoking, using data from the Cardiovascular Outcomes in Renal Atherosclerotic Lesions clinical trial. Copyright © 2016 John Wiley & Sons, Ltd.     

Should a propensity score model be super? The utility of ensemble procedures for causal adjustment

In investigations of the effect of treatment on outcome, the propensity score is a tool to eliminate imbalance in the distribution of confounding variables between treatment groups. Recent work has suggested that Super Learner, an ensemble method, outperforms logistic regression in nonlinear settings; however, experience with real-data analyses tends to show overfitting of the propensity score model using this approach. We investigated a wide range of simulated settings of varying complexities including simulations based on real data to compare the performances of logistic regression, generalized boosted models, and Super Learner in providing balance and for estimating the average treatment effect via propensity score regression, propensity score matching, and inverse probability of treatment weighting. We found that Super Learner and logistic regression are comparable in terms of covariate balance, bias, and mean squared error (MSE); however, Super Learner is computationally very expensive thus leaving no clear advantage to the more complex approach. Propensity scores estimated by generalized boosted models were inferior to the other two estimation approaches. We also found that propensity score regression adjustment was superior to either matching or inverse weighting when the form of the dependence on the treatment on the outcome is correctly specified.     

The performance of different propensity‐score methods for estimating differences in proportions (risk differences or absolute risk reductions) in observational studies

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.     

On regression adjustment for the propensity score

Propensity scores are widely adopted in observational research because they enable adjustment for high‐dimensional confounders without requiring models for their association with the outcome of interest. The results of statistical analyses based on stratification, matching or inverse weighting by the propensity score are therefore less susceptible to model extrapolation than those based solely on outcome regression models. This is attractive because extrapolation in outcome regression models may be alarming, yet difficult to diagnose, when the exposed and unexposed individuals have very different covariate distributions. Standard regression adjustment for the propensity score forms an alternative to the aforementioned propensity score methods, but the benefits of this are less clear because it still involves modelling the outcome in addition to the propensity score. In this article, we develop novel insights into the properties of this adjustment method. We demonstrate that standard tests of the null hypothesis of no exposure effect (based on robust variance estimators), as well as particular standardised effects obtained from such adjusted regression models, are robust against misspecification of the outcome model when a propensity score model is correctly specified; they are thus not vulnerable to the aforementioned problem of extrapolation. We moreover propose efficient estimators for these standardised effects, which retain a useful causal interpretation even when the propensity score model is misspecified, provided the outcome regression model is correctly specified. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Doubly robust estimation of the weighted average treatment effect for a target population

The weighted average treatment effect is a causal measure for the comparison of interventions in a specific target population, which may be different from the population where data are sampled from. For instance, when the goal is to introduce a new treatment to a target population, the question is what efficacy (or effectiveness) can be gained by switching patients from a standard of care (control) to this new treatment, for which the average treatment effect for the control estimand can be applied. In this paper, we propose two estimators based on augmented inverse probability weighting to estimate the weighted average treatment effect for a well-defined target population (ie, there exists a predefined target function of covariates that characterizes the population of interest, for example, a function of age to focus on elderly diabetic patients using samples from the US population). The first proposed estimator is doubly robust if the target function is known or can be correctly specified. The second proposed estimator is doubly robust if the target function has a linear dependence on the propensity score, which can be used to estimate the average treatment effect for the treated and the average treatment effect for the control. We demonstrate the properties of the proposed estimators through theoretical proof and simulation studies. We also apply our proposed methods in a comparison of glucagon-like peptide-1 receptor agonists therapy and insulin therapy among patients with type 2 diabetes, using the UK Clinical Practice Research Datalink data.     

Propensity score estimation with missing values using a multiple imputation missingness pattern (MIMP) approach

Propensity scores have been used widely as a bias reduction method to estimate the treatment effect in nonrandomized studies. Since many covariates are generally included in the model for estimating the propensity scores, the proportion of subjects with at least one missing covariate could be large. While many methods have been proposed for propensity score‐based estimation in the presence of missing covariates, little has been published comparing the performance of these methods. In this article we propose a novel method called multiple imputation missingness pattern (MIMP) and compare it with the naive estimator (ignoring propensity score) and three commonly used methods of handling missing covariates in propensity score‐based estimation (separate estimation of propensity scores within each pattern of missing data, multiple imputation and discarding missing data) under different mechanisms of missing data and degree of correlation among covariates. Simulation shows that all adjusted estimators are much less biased than the naive estimator. Under certain conditions MIMP provides benefits (smaller bias and mean‐squared error) compared with existing alternatives. Copyright © 2009 John Wiley & Sons, Ltd.     

On variance estimate for covariate adjustment by propensity score analysis

Propensity score (PS) methods have been used extensively to adjust for confounding factors in the statistical analysis of observational data in comparative effectiveness research. There are four major PS‐based adjustment approaches: PS matching, PS stratification, covariate adjustment by PS, and PS‐based inverse probability weighting. Though covariate adjustment by PS is one of the most frequently used PS‐based methods in clinical research, the conventional variance estimation of the treatment effects estimate under covariate adjustment by PS is biased. As Stampf et al. have shown, this bias in variance estimation is likely to lead to invalid statistical inference and could result in erroneous public health conclusions (e.g., food and drug safety and adverse events surveillance). To address this issue, we propose a two‐stage analytic procedure to develop a valid variance estimator for the covariate adjustment by PS analysis strategy. We also carry out a simple empirical bootstrap resampling scheme. Both proposed procedures are implemented in an R function for public use. Extensive simulation results demonstrate the bias in the conventional variance estimator and show that both proposed variance estimators offer valid estimates for the true variance, and they are robust to complex confounding structures. The proposed methods are illustrated for a post‐surgery pain study. Copyright © 2016 John Wiley & Sons, Ltd.     

Using a monotone single-index model to stabilize the propensity score in missing data problems and causal inference

The augmented inverse weighting method is one of the most popular methods for estimating the mean of the response in causal inference and missing data problems. An important component of this method is the propensity score. Popular parametric models for the propensity score include the logistic, probit, and complementary log-log models. A common feature of these models is that the propensity score is a monotonic function of a linear combination of the explanatory variables. To avoid the need to choose a model, we model the propensity score via a semiparametric single-index model, in which the score is an unknown monotonic nondecreasing function of the given single index. Under this new model, the augmented inverse weighting estimator (AIWE) of the mean of the response is asymptotically linear, semiparametrically efficient, and more robust than existing estimators. Moreover, we have made a surprising observation. The inverse probability weighting and AIWEs based on a correctly specified parametric model may have worse performance than their counterparts based on a nonparametric model. A heuristic explanation of this phenomenon is provided. A real-data example is used to illustrate the proposed methods.     

Flexible regression approach to propensity score analysis and its relationship with matching and weighting

In propensity score analysis, the frequently used regression adjustment involves regressing the outcome on the estimated propensity score and treatment indicator. This approach can be highly efficient when model assumptions are valid, but can lead to biased results when the assumptions are violated. We extend the simple regression adjustment to a varying coefficient regression model that allows for nonlinear association between outcome and propensity score. We discuss its connection with some propensity score matching and weighting methods, and show that the proposed analytical framework can shed light on the intrinsic connection among some mainstream propensity score approaches (stratification, regression, kernel matching, and inverse probability weighting) and handle commonly used causal estimands. We derive analytic point and variance estimators that properly take into account the sampling variability in the estimated propensity score. Extensive simulations show that the proposed approach possesses desired finite sample properties and demonstrates competitive performance in comparison with other methods estimating the same causal estimand. The proposed methodology is illustrated with a study on right heart catheterization.     

The performance of different propensity score methods for estimating marginal odds ratios

The propensity score which is the probability of exposure to a specific treatment conditional on observed variables. Conditioning on the propensity score results in unbiased estimation of the expected difference in observed responses to two treatments. In the medical literature, propensity score methods are frequently used for estimating odds ratios. The performance of propensity score methods for estimating marginal odds ratios has not been studied. We performed a series of Monte Carlo simulations to assess the performance of propensity score matching, stratifying on the propensity score, and covariate adjustment using the propensity score to estimate marginal odds ratios. We assessed bias, precision, and mean-squared error (MSE) of the propensity score estimators, in addition to the proportion of bias eliminated due to conditioning on the propensity score. When the true marginal odds ratio was one, then matching on the propensity score and covariate adjustment using the propensity score resulted in unbiased estimation of the true treatment effect, whereas stratification on the propensity score resulted in minor bias in estimating the true marginal odds ratio. When the true marginal odds ratio ranged from 2 to 10, then matching on the propensity score resulted in the least bias, with a relative biases ranging from 2.3 to 13.3 per cent. Stratifying on the propensity score resulted in moderate bias, with relative biases ranging from 15.8 to 59.2 per cent. For both methods, relative bias was proportional to the true odds ratio. Finally, matching on the propensity score tended to result in estimators with the lowest MSE.     

Estimating propensity scores with missing covariate data using general location mixture models

In many observational studies, analysts estimate causal effects using propensity scores, e.g. by matching, sub-classifying, or inverse probability weighting based on the scores. Estimation of propensity scores is complicated when some values of the covariates are missing. Analysts can use multiple imputation to create completed data sets from which propensity scores can be estimated. We propose a general location mixture model for imputations that assumes that the control units are a latent mixture of (i) units whose covariates are drawn from the same distributions as the treated units' covariates and (ii) units whose covariates are drawn from different distributions. This formulation reduces the influence of control units outside the treated units' region of the covariate space on the estimation of parameters in the imputation model, which can result in more plausible imputations. In turn, this can result in more reliable estimates of propensity scores and better balance in the true covariate distributions when matching or sub-classifying. We illustrate the benefits of the latent class modeling approach with simulations and with an observational study of the effect of breast feeding on children's cognitive abilities.     

Variance reduction in randomised trials by inverse probability weighting using the propensity score

In individually randomised controlled trials, adjustment for baseline characteristics is often undertaken to increase precision of the treatment effect estimate. This is usually performed using covariate adjustment in outcome regression models. An alternative method of adjustment is to use inverse probability‐of‐treatment weighting (IPTW), on the basis of estimated propensity scores. We calculate the large‐sample marginal variance of IPTW estimators of the mean difference for continuous outcomes, and risk difference, risk ratio or odds ratio for binary outcomes. We show that IPTW adjustment always increases the precision of the treatment effect estimate. For continuous outcomes, we demonstrate that the IPTW estimator has the same large‐sample marginal variance as the standard analysis of covariance estimator. However, ignoring the estimation of the propensity score in the calculation of the variance leads to the erroneous conclusion that the IPTW treatment effect estimator has the same variance as an unadjusted estimator; thus, it is important to use a variance estimator that correctly takes into account the estimation of the propensity score. The IPTW approach has particular advantages when estimating risk differences or risk ratios. In this case, non‐convergence of covariate‐adjusted outcome regression models frequently occurs. Such problems can be circumvented by using the IPTW adjustment approach. © 2013 The authors. Statistics in Medicine published by John Wiley & Sons, Ltd.     

Assessing the performance of the generalized propensity score for estimating the effect of quantitative or continuous exposures on binary outcomes

Propensity score methods are increasingly being used to estimate the effects of treatments and exposures when using observational data. The propensity score was initially developed for use with binary exposures. The generalized propensity score (GPS) is an extension of the propensity score for use with quantitative or continuous exposures (eg, dose or quantity of medication, income, or years of education). We used Monte Carlo simulations to examine the performance of different methods of using the GPS to estimate the effect of continuous exposures on binary outcomes. We examined covariate adjustment using the GPS and weighting using weights based on the inverse of the GPS. We examined both the use of ordinary least squares to estimate the propensity function and the use of the covariate balancing propensity score algorithm. The use of methods based on the GPS was compared with the use of G‐computation. All methods resulted in essentially unbiased estimation of the population dose‐response function. However, GPS‐based weighting tended to result in estimates that displayed greater variability and had higher mean squared error when the magnitude of confounding was strong. Of the methods based on the GPS, covariate adjustment using the GPS tended to result in estimates with lower variability and mean squared error when the magnitude of confounding was strong. We illustrate the application of these methods by estimating the effect of average neighborhood income on the probability of death within 1 year of hospitalization for an acute myocardial infarction.     

Variance estimation when using inverse probability of treatment weighting (IPTW) with survival analysis

Propensity score methods are used to reduce the effects of observed confounding when using observational data to estimate the effects of treatments or exposures. A popular method of using the propensity score is inverse probability of treatment weighting (IPTW). When using this method, a weight is calculated for each subject that is equal to the inverse of the probability of receiving the treatment that was actually received. These weights are then incorporated into the analyses to minimize the effects of observed confounding. Previous research has found that these methods result in unbiased estimation when estimating the effect of treatment on survival outcomes. However, conventional methods of variance estimation were shown to result in biased estimates of standard error. In this study, we conducted an extensive set of Monte Carlo simulations to examine different methods of variance estimation when using a weighted Cox proportional hazards model to estimate the effect of treatment. We considered three variance estimation methods: (i) a naïve model‐based variance estimator; (ii) a robust sandwich‐type variance estimator; and (iii) a bootstrap variance estimator. We considered estimation of both the average treatment effect and the average treatment effect in the treated. We found that the use of a bootstrap estimator resulted in approximately correct estimates of standard errors and confidence intervals with the correct coverage rates. The other estimators resulted in biased estimates of standard errors and confidence intervals with incorrect coverage rates. Our simulations were informed by a case study examining the effect of statin prescribing on mortality. © 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

Regression-adjusted matching and double-robust methods for estimating average treatment effects in health economic evaluation

Regression, propensity score (PS) and double-robust (DR) methods can reduce selection bias when estimating average treatment effects (ATEs). Economic evaluations of health care interventions exemplify complex data structures, in that the covariate–endpoint relationships tend to be highly non-linear, with highly skewed cost and health outcome endpoints. When either the regression or PS model is correct, DR methods can provide unbiased, efficient estimates of ATEs, but generally the specification of both models is unknown. Regression-adjusted matching can also protect against bias from model misspecification, but has not been compared to DR methods. This paper compares regression-adjusted matching to selected DR methods (weighted regression and augmented inverse probability of treatment weighting) as well as to regression and PS methods for addressing selection bias in cost-effectiveness analyses (CEA). We contrast the methods in a CEA of a pharmaceutical intervention, where there are extreme estimated PSs, hence unstable inverse probability of treatment (IPT) weights. The case study motivates a simulation which considers settings with functional form misspecification in the PS and endpoint regression models (e.g. cost model with log instead of identity link), stable and unstable PS weights. We find that in the realistic setting of unstable IPT weights and misspecifications to the PS and regression models, regression-adjusted matching reports less bias than DR methods. We conclude that regression-adjusted matching is a relatively robust method for estimating ATEs in applications with complex data structures exemplified by CEA.     

Interval estimation for treatment effects using propensity score matching

In causal studies without random assignment of treatment, causal effects can be estimated using matched treated and control samples, where matches are obtained using estimated propensity scores. Propensity score matching can reduce bias in treatment effect estimators in cases where the matched samples have overlapping covariate distributions. Despite its application in many applied problems, there is no universally employed approach to interval estimation when using propensity score matching. In this article, we present and evaluate approaches to interval estimation when using propensity score matching.     

Uncertainty in the design stage of two-stage Bayesian propensity score analysis

The two-stage process of propensity score analysis (PSA) includes a design stage where propensity scores (PSs) are estimated and implemented to approximate a randomized experiment and an analysis stage where treatment effects are estimated conditional on the design. This article considers how uncertainty associated with the design stage impacts estimation of causal effects in the analysis stage. Such design uncertainty can derive from the fact that the PS itself is an estimated quantity, but also from other features of the design stage tied to choice of PS implementation. This article offers a procedure for obtaining the posterior distribution of causal effects after marginalizing over a distribution of design-stage outputs, lending a degree of formality to Bayesian methods for PSA that have gained attention in recent literature. Formulation of a probability distribution for the design-stage output depends on how the PS is implemented in the design stage, and propagation of uncertainty into causal estimates depends on how the treatment effect is estimated in the analysis stage. We explore these differences within a sample of commonly used PS implementations (quantile stratification, nearest-neighbor matching, caliper matching, inverse probability of treatment weighting, and doubly robust estimation) and investigate in a simulation study the impact of statistician choice in PS model and implementation on the degree of between- and within-design variability in the estimated treatment effect. The methods are then deployed in an investigation of the association between levels of fine particulate air pollution and elevated exposure to emissions from coal-fired power plants.     

Comparing the performance of propensity score methods in healthcare database studies with rare outcomes

Nonrandomized studies of treatments from electronic healthcare databases are critical for producing the evidence necessary to making informed treatment decisions, but often rely on comparing rates of events observed in a small number of patients. In addition, studies constructed from electronic healthcare databases, for example, administrative claims data, often adjust for many, possibly hundreds, of potential confounders. Despite the importance of maximizing efficiency when there are many confounders and few observed outcome events, there has been relatively little research on the relative performance of different propensity score methods in this context. In this paper, we compare a wide variety of propensity‐based estimators of the marginal relative risk. In contrast to prior research that has focused on specific statistical methods in isolation of other analytic choices, we instead consider a method to be defined by the complete multistep process from propensity score modeling to final treatment effect estimation. Propensity score model estimation methods considered include ordinary logistic regression, Bayesian logistic regression, lasso, and boosted regression trees. Methods for utilizing the propensity score include pair matching, full matching, decile strata, fine strata, regression adjustment using one or two nonlinear splines, inverse propensity weighting, and matching weights. We evaluate methods via a ‘plasmode’ simulation study, which creates simulated datasets on the basis of a real cohort study of two treatments constructed from administrative claims data. Our results suggest that regression adjustment and matching weights, regardless of the propensity score model estimation method, provide lower bias and mean squared error in the context of rare binary outcomes. Copyright © 2017 John Wiley & Sons, Ltd.     

不同混杂结构下广义倾向性评分法的模拟研究及应用

目的 通过构建存在不同混杂结构的广义倾向性评分(generalized propensity score,GPS)模型和结局模型,探索比较三种GPS估计法:广义倾向性评分-最小二乘法(generalized propensity score-ordinary least squares,GPS-OLS),广义倾向性评分...