Estimation of the average causal effect via multiple propensity score stratification

Suppose we are interested in estimating the average causal effect (ACE) for the population mean from observational study. Because of simplicity and ease of interpretation, stratification by a propensity score (PS) is widely used to adjust for influence of confounding factors in estimation of the ACE. Appropriateness of the estimation by the PS stratification relies on correct specification of the PS. We propose an estimator based on stratification with multiple PS models by clustering techniques instead of model selection. If one of them correctly specifies, the proposed estimator removes bias and thus is more robust than the standard PS stratification.     

Inference for proportional hazard model with propensity score

Since the publication of the seminal paper by Cox (1972), proportional hazard model has become very popular in regression analysis for right censored data. In observational studies, treatment assignment may depend on observed covariates. If these confounding variables are not accounted for properly, the inference based on the Cox proportional hazard model may perform poorly. As shown in Rosenbaum and Rubin (1983), under the strongly ignorable treatment assignment assumption, conditioning on the propensity score yields valid causal effect estimates. Therefore we incorporate the propensity score into the Cox model for causal inference with survival data. We derive the asymptotic property of the maximum partial likelihood estimator when the model is correctly specified. Simulation results show that our method performs quite well for observational data. The approach is applied to a real dataset on the time of readmission of trauma patients. We also derive the asymptotic property of the maximum partial likelihood estimator with a robust variance estimator, when the model is incorrectly specified.     

Using generalized doubly robust estimator to estimate average treatment effects of multiple treatments in observational studies

The generalized doubly robust estimator is proposed for estimating the average treatment effect (ATE) of multiple treatments based on the generalized propensity score (GPS). In medical researches where observational studies are conducted, estimations of ATEs are usually biased since the covariate distributions could be unbalanced among treatments. To overcome this problem, Imbens [The role of the propensity score in estimating dose-response functions, Biometrika 87 (2000), pp. 706–710] and Feng et al. [Generalized propensity score for estimating the average treatment effect of multiple treatments, Stat. Med. (2011), in press. Available at: http://onlinelibrary.wiley.com/doi/10.1002/sim.4168/abstract] proposed weighted estimators that are extensions of a ratio estimator based on GPS to estimate ATEs with multiple treatments. However, the ratio estimator always produces a larger empirical sample variance than the doubly robust estimator, which estimates an ATE between two treatments based on the estimated propensity score (PS). We conduct a simulation study to compare the performance of our proposed estimator with Imbens’ and Feng et al.’s estimators, and simulation results show that our proposed estimator outperforms their estimators in terms of bias, empirical sample variance and mean-squared error of the estimated ATEs.     

Propensity score applied to survival data analysis through proportional hazards models: a Monte Carlo study

Propensity score methods are increasingly used in medical literature to estimate treatment effect using data from observational studies. Despite many papers on propensity score analysis, few have focused on the analysis of survival data. Even within the framework of the popular proportional hazard model, the choice among marginal, stratified or adjusted models remains unclear. A Monte Carlo simulation study was used to compare the performance of several survival models to estimate both marginal and conditional treatment effects. The impact of accounting or not for pairing when analysing propensity‐score‐matched survival data was assessed. In addition, the influence of unmeasured confounders was investigated. After matching on the propensity score, both marginal and conditional treatment effects could be reliably estimated. Ignoring the paired structure of the data led to an increased test size due to an overestimated variance of the treatment effect. Among the various survival models considered, stratified models systematically showed poorer performance. Omitting a covariate in the propensity score model led to a biased estimation of treatment effect, but replacement of the unmeasured confounder by a correlated one allowed a marked decrease in this bias. Our study showed that propensity scores applied to survival data can lead to unbiased estimation of both marginal and conditional treatment effect, when marginal and adjusted Cox models are used. In all cases, it is necessary to account for pairing when analysing propensity‐score‐matched data, using a robust estimator of the variance. Copyright © 2012 John Wiley & Sons, Ltd.     

Comparison of various machine learning algorithms for estimating generalized propensity score

In this paper, we conducted a simulation study to evaluate the performance of four algorithms: multinomial logistic regression (MLR), bagging (BAG), random forest (RF), and gradient boosting (GB), for estimating generalized propensity score (GPS). Similar to the propensity score (PS), the ultimate goal of using GPS is to estimate unbiased average treatment effects (ATEs) in observational studies. We used the GPS estimates computed from these four algorithms with the generalized doubly robust (GDR) estimator to estimate ATEs in observational studies. We evaluated these ATE estimates in terms of bias and mean squared error (MSE). Simulation results show that overall, the GB algorithm produced the best ATE estimates based on these evaluation criteria. Thus, we recommend using the GB algorithm for estimating GPS in practice.     

Correlation and efficiency of propensity score-based estimators for average causal effects

Propensity score-based estimators are commonly used to estimate causal effects in evaluation research. To reduce bias in observational studies, researchers might be tempted to include many, perhaps correlated, covariates when estimating the propensity score model. Taking into account that the propensity score is estimated, this study investigates how the efficiency of matching, inverse probability weighting, and doubly robust estimators change under the case of correlated covariates. Propositions regarding the large sample variances under certain assumptions on the data-generating process are given. The propositions are supplemented by several numerical large sample and finite sample results from a wide range of models. The results show that the covariate correlations may increase or decrease the variances of the estimators. There are several factors that influence how correlation affects the variance of the estimators, including the choice of estimator, the strength of the confounding toward outcome and treatment, and whether a constant or non-constant causal effect is present.     

Multiply robust matching estimators of average and quantile treatment effects

Propensity score matching has been a long-standing tradition for handling confounding in causal inference, however, requiring stringent model assumptions. In this article, we propose novel double score matching (DSM) utilizing both the propensity score and prognostic score. To gain the protection of possible model misspecification, we posit multiple candidate models for each score. We show that the debiasing DSM estimator achieves the multiple robustness property in that it is consistent if any one of the score models is correctly specified. We characterize the asymptotic distribution for the DSM estimator requiring only one correct model specification based on the martingale representations of the matching estimators and theory for local normal experiments. We also provide a two-stage replication method for variance estimation and extend DSM for quantile estimation. Simulation demonstrates DSM outperforms single-score matching and prevailing multiply robust weighting estimators in the presence of extreme propensity scores.     

Efficient estimation in a regression model with missing responses

This article examines methods to efficiently estimate the mean response in a linear model with an unknown error distribution under the assumption that the responses are missing at random. We show how the asymptotic variance is affected by the estimator of the regression parameter, and by the imputation method. To estimate the regression parameter, the ordinary least squares is efficient only if the error distribution happens to be normal. If the errors are not normal, then we propose a one step improvement estimator or a maximum empirical likelihood estimator to efficiently estimate the parameter.To investigate the imputation’s impact on the estimation of the mean response, we compare the listwise deletion method and the propensity score method (which do not use imputation at all), and two imputation methods. We demonstrate that listwise deletion and the propensity score method are inefficient. Partial imputation, where only the missing responses are imputed, is compared to full imputation, where both missing and non-missing responses are imputed. Our results reveal that, in general, full imputation is better than partial imputation. However, when the regression parameter is estimated very poorly, the partial imputation will outperform full imputation. The efficient estimator for the mean response is the full imputation estimator that utilizes an efficient estimator of the parameter.     

Propensity score matching and stratification using multiparty data without pooling

Matching and stratification based on confounding factors or propensity scores (PS) are powerful approaches for reducing confounding bias in indirect treatment comparisons. However, implementing these approaches requires pooled individual patient data (IPD). The research presented here was motivated by an indirect comparison between a single-armed trial in acute myeloid leukemia (AML), and two external AML registries with current treatments for a control. For confidentiality reasons, IPD cannot be pooled. Common approaches to adjusting confounding bias, such as PS matching or stratification, cannot be applied as 1) a model for PS, for example, a logistic model, cannot be fitted without pooling covariate data; 2) pooling response data may be necessary for some statistical inference (e.g., estimating the SE of mean difference of matched pairs) after PS matching. We propose a set of approaches that do not require pooling IPD, using a combination of methods including a linear discriminant for matching and stratification, and secure multiparty computation for estimation of within-pair sample variance and for calculations involving multiple control sources. The approaches only need to share aggregated data offline, rather than real-time secure data transfer, as required by typical secure multiparty computation for model fitting. For survival analysis, we propose an approach using restricted mean survival time. A simulation study was conducted to evaluate this approach in several scenarios, in particular, with a mixture of continuous and binary covariates. The results confirmed the robustness and efficiency of the proposed approach. A real data example is also provided for illustration.     

Correcting for Population Stratification in Genomewide Association Studies

Genomewide association studies have become the primary tool for discovering the genetic basis of complex human diseases. Such studies are susceptible to the confounding effects of population stratification, in that the combination of allele-frequency heterogeneity with disease-risk heterogeneity among different ancestral subpopulations can induce spurious associations between genetic variants and disease. This article provides a statistically rigorous and computationally feasible solution to this challenging problem of unmeasured confounders. We show that the odds ratio of disease with a genetic variant is identifiable if and only if the genotype is independent of the unknown population substructure conditional on a set of observed ancestry-informative markers in the disease-free population. Under this condition, the odds ratio of interest can be estimated by fitting a semiparametric logistic regression model with an arbitrary function of a propensity score relating the genotype probability to ancestry-informative markers. Approximating the unknown function of the propensity score by B-splines, we derive a consistent and asymptotically normal estimator for the odds ratio of interest with a consistent variance estimator. Simulation studies demonstrate that the proposed inference procedures perform well in realistic settings. An application to the well-known Wellcome Trust Case-Control Study is presented. Supplemental materials are available online.     

Asymptotic theory and inference of predictive mean matching imputation using a superpopulation model framework

Predictive mean matching imputation is popular for handling item nonresponse in survey sampling. In this article, we study the asymptotic properties of the predictive mean matching estimator for finite-population inference using a superpopulation model framework. We also clarify conditions for its robustness. For variance estimation, the conventional bootstrap inference is invalid for matching estimators with a fixed number of matches due to the nonsmoothness nature of the matching estimator. We propose a new replication variance estimator, which is asymptotically valid. The key strategy is to construct replicates directly based on the linear terms of the martingale representation for the matching estimator, instead of individual records of variables. Simulation studies confirm that the proposed method provides valid inference.     

LASSO-type estimators for semiparametric nonlinear mixed-effects models estimation

Parametric nonlinear mixed effects models (NLMEs) are now widely used in biometrical studies, especially in pharmacokinetics research and HIV dynamics models, due to, among other aspects, the computational advances achieved during the last years. However, this kind of models may not be flexible enough for complex longitudinal data analysis. Semiparametric NLMEs (SNMMs) have been proposed as an extension of NLMEs. These models are a good compromise and retain nice features of both parametric and nonparametric models resulting in more flexible models than standard parametric NLMEs. However, SNMMs are complex models for which estimation still remains a challenge. Previous estimation procedures are based on a combination of log-likelihood approximation methods for parametric estimation and smoothing splines techniques for nonparametric estimation. In this work, we propose new estimation strategies in SNMMs. On the one hand, we use the Stochastic Approximation version of EM algorithm (SAEM) to obtain exact ML and REML estimates of the fixed effects and variance components. On the other hand, we propose a LASSO-type method to estimate the unknown nonlinear function. We derive oracle inequalities for this nonparametric estimator. We combine the two approaches in a general estimation procedure that we illustrate with simulations and through the analysis of a real data set of price evolution in on-line auctions.     

Effects of correlated covariates on the asymptotic efficiency of matching and inverse probability weighting estimators for causal inference

In observational studies, the overall aim when fitting a model for the propensity score is to reduce bias for an estimator of the causal effect. To make the assumption of an unconfounded treatment plausible researchers might include many, possibly correlated, covariates in the propensity score model. In this paper, we study how the asymptotic efficiency of matching and inverse probability weighting estimators for average causal effects change when the covariates are correlated. We investigate the case with multivariate normal covariates, a logistic model for the propensity score and linear models for the potential outcomes and show results under different model assumptions. We show that the correlation can both increase and decrease the large sample variances of the estimators, and that the correlation affects the asymptotic efficiency of the estimators differently, both with regard to direction and magnitude. Moreover, the strength of the confounding towards the outcome and the treatment plays an important role.     

Controlling for confounding via propensity score methods can result in biased estimation of the conditional AUC: A simulation study

In the medical literature, there has been an increased interest in evaluating association between exposure and outcomes using nonrandomized observational studies. However, because assignments to exposure are not random in observational studies, comparisons of outcomes between exposed and nonexposed subjects must account for the effect of confounders. Propensity score methods have been widely used to control for confounding, when estimating exposure effect. Previous studies have shown that conditioning on the propensity score results in biased estimation of conditional odds ratio and hazard ratio. However, research is lacking on the performance of propensity score methods for covariate adjustment when estimating the area under the ROC curve (AUC). In this paper, AUC is proposed as measure of effect when outcomes are continuous. The AUC is interpreted as the probability that a randomly selected nonexposed subject has a better response than a randomly selected exposed subject. A series of simulations has been conducted to examine the performance of propensity score methods when association between exposure and outcomes is quantified by AUC; this includes determining the optimal choice of variables for the propensity score models. Additionally, the propensity score approach is compared with that of the conventional regression approach to adjust for covariates with the AUC. The choice of the best estimator depends on bias, relative bias, and root mean squared error. Finally, an example looking at the relationship of depression/anxiety and pain intensity in people with sickle cell disease is used to illustrate the estimation of the adjusted AUC using the proposed approaches.     

基于协变量平衡加权的平均处理效应的稳健有效估计

倾向性得分是估计平均处理效应的重要工具。但在观察性研究中,通常会由于协变量在处理组与对照组分布的不平衡性而导致极端倾向性得分的出现,即存在十分接近于0或1的倾向性得分,这使得因果推断的强可忽略假设接近于违背,进而导致平均处理效应的估计出现较大的偏差与方差。Li等(2018a)提出了协变量平衡加权法,在无混杂性假设下通过实现协变量分布的加权平衡,解决了极端倾向性得分带来的影响。本文在此基础上,提出了基于协变量平衡加权法的稳健且有效的估计方法,并通过引入超级学习算法提升了模型在实证应用中的稳健性;更进一步,将前一方法推广至理论上不依赖于结果回归模型和倾向性得分模型假设的基于协变量平衡加权的稳健有效估计。蒙特卡洛模拟表明,本文提出的两种方法在结果回归模型和倾向性得分模型均存在误设时仍具有极小的偏差和方差。实证部分将两种方法应用于右心导管插入术数据,发现右心导管插入术大约会增加患者6. 3%死亡率。     

Variance reduction for quantile estimates in simulations via nonlinear controls

Linear controls are a well known simple technique for achieving variance reduction in computer simulation. Unfortunately the effectiveness of a linear control depends upon the correlation between the statistic of interest and the control, which is often low. Since statistics often have a nonlinear relation-ship with the potential control variables, nonlinear controls offer a means for improvement over linear controls. This paper focuses on the use of nonlinear controls for reducing the variance of quantile estimates in simulation. It is shown that one can substantially reduce the analytic effort required to develop a nonlinear control from a quantile estimator by using a strictly monotone transformation to create the nonlinear control. It is also shown that as one increases the sample size for the quantile estimator, the asymptotic multivariate normal distribution of the quantile of interest and the control reduces the effectiveness of the nonlinear control to that of the linear control. However, the data has to be sectioned to obtain an estimate of the variance of the controlled quantile estimate. Graphical methods are suggested for selecting the section size that maximizes the effectiveness of the nonlinear control     

Analyzing propensity matched zero-inflated count outcomes in observational studies

Determining the effectiveness of different treatments from observational data, which are characterized by imbalance between groups due to lack of randomization, is challenging. Propensity matching is often used to rectify imbalances among prognostic variables. However, there are no guidelines on how appropriately to analyze group matched data when the outcome is a zero-inflated count. In addition, there is debate over whether to account for correlation of responses induced by matching and/or whether to adjust for variables used in generating the propensity score in the final analysis. The aim of this research is to compare covariate unadjusted and adjusted zero-inflated Poisson models that do and do not account for the correlation. A simulation study is conducted, demonstrating that it is necessary to adjust for potential residual confounding, but that accounting for correlation is less important. The methods are applied to a biomedical research data set.     

Nonparametric likelihood and doubly robust estimating equations for marginal and nested structural models

This article considers Robins's marginal and nested structural models in the cross‐sectional setting and develops likelihood and regression estimators. First, a nonparametric likelihood method is proposed by retaining a finite subset of all inherent and modelling constraints on the joint distributions of potential outcomes and covariates under a correctly specified propensity score model. A profile likelihood is derived by maximizing the nonparametric likelihood over these joint distributions subject to the retained constraints. The maximum likelihood estimator is intrinsically efficient based on the retained constraints and weakly locally efficient. Second, two regression estimators, named hat and tilde, are derived as first‐order approximations to the likelihood estimator under the propensity score model. The tilde regression estimator is intrinsically and weakly locally efficient and doubly robust. The methods are illustrated by data analysis for an observational study on right heart catheterization. The Canadian Journal of Statistics 38: 609–632; 2010 © 2010 Statistical Society of Canada     

Fast accelerated failure time modeling for case-cohort data

Semiparametric accelerated failure time (AFT) models directly relate the expected failure times to covariates and are a useful alternative to models that work on the hazard function or the survival function. For case-cohort data, much less development has been done with AFT models. In addition to the missing covariates outside of the sub-cohort in controls, challenges from AFT model inferences with full cohort are retained. The regression parameter estimator is hard to compute because the most widely used rank-based estimating equations are not smooth. Further, its variance depends on the unspecified error distribution, and most methods rely on computationally intensive bootstrap to estimate it. We propose fast rank-based inference procedures for AFT models, applying recent methodological advances to the context of case-cohort data. Parameters are estimated with an induced smoothing approach that smooths the estimating functions and facilitates the numerical solution. Variance estimators are obtained through efficient resampling methods for nonsmooth estimating functions that avoids full blown bootstrap. Simulation studies suggest that the recommended procedure provides fast and valid inferences among several competing procedures. Application to a tumor study demonstrates the utility of the proposed method in routine data analysis.     

Using Inverse Probability Weighting Estimators to Evaluate Various Propensity Scores When Treatment Switching Exists

In this paper, we conduct a Monte Carlo simulation study to evaluate three propensity score (PS) scenarios for estimating an average treatment effect (ATE) in observational studies when treatment switching exists: (a) ignoring treatment switching in subjects (UPS), (b) removing subjects with treatment switching (RPS), and (c) adjusting for treatment switching effect (APS) with two inverse probability weighting estimators, IPW1 and IPW2. We evaluate these six estimators in terms of bias, mean squared error (MSE), empirical standard error (ESE), and coverage probability (CP) under various simulation scenarios. Simulation results show that the IPW2 estimator with RPS has relatively good performance.