Survival analysis using inverse probability of treatment weighted methods based on the generalized propensity score

In survival analysis, treatment effects are commonly evaluated based on survival curves and hazard ratios as causal treatment effects. In observational studies, these estimates may be biased due to confounding factors. The inverse probability of treatment weighted (IPTW) method based on the propensity score is one of the approaches utilized to adjust for confounding factors between binary treatment groups. As a generalization of this methodology, we developed an exact formula for an IPTW log‐rank test based on the generalized propensity score for survival data. This makes it possible to compare the group differences of IPTW Kaplan–Meier estimators of survival curves using an IPTW log‐rank test for multi‐valued treatments. As causal treatment effects, the hazard ratio can be estimated using the IPTW approach. If the treatments correspond to ordered levels of a treatment, the proposed method can be easily extended to the analysis of treatment effect patterns with contrast statistics. In this paper, the proposed method is illustrated with data from the Kyushu Lipid Intervention Study (KLIS), which investigated the primary preventive effects of pravastatin on coronary heart disease (CHD). The results of the proposed method suggested that pravastatin treatment reduces the risk of CHD and that compliance to pravastatin treatment is important for the prevention of CHD. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparison of causal effect estimators under exposure misclassification

Over the past decades, various principles for causal effect estimation have been proposed, all differing in terms of how they adjust for measured confounders: either via traditional regression adjustment, by adjusting for the expected exposure given those confounders (e.g., the propensity score), or by inversely weighting each subject's data by the likelihood of the observed exposure, given those confounders. When the exposure is measured with error, this raises the question whether these different estimation strategies might be differently affected and whether one of them is to be preferred for that reason. In this article, we investigate this by comparing inverse probability of treatment weighted (IPTW) estimators and doubly robust estimators for the exposure effect in linear marginal structural mean models (MSM) with G-estimators, propensity score (PS) adjusted estimators and ordinary least squares (OLS) estimators for the exposure effect in linear regression models. We find analytically that these estimators are equally affected when exposure misclassification is independent of the confounders, but not otherwise. Simulation studies reveal similar results for time-varying exposures and when the model of interest includes a logistic link.     

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.     

Variance estimation of the risk difference when using propensity-score matching and weighting with time-to-event outcomes

Observational studies are increasingly being used in medicine to estimate the effects of treatments or exposures on outcomes. To minimize the potential for confounding when estimating treatment effects, propensity score methods are frequently implemented. Often outcomes are the time to event. While it is common to report the treatment effect as a relative effect, such as the hazard ratio, reporting the effect using an absolute measure of effect is also important. One commonly used absolute measure of effect is the risk difference or difference in probability of the occurrence of an event within a specified duration of follow-up between a treatment and comparison group. We first describe methods for point and variance estimation of the risk difference when using weighting or matching based on the propensity score when outcomes are time-to-event. Next, we conducted Monte Carlo simulations to compare the relative performance of these methods with respect to bias of the point estimate, accuracy of variance estimates, and coverage of estimated confidence intervals. The results of the simulation generally support the use of weighting methods (untrimmed ATT weights and IPTW) or caliper matching when the prevalence of treatment is low for point estimation. For standard error estimation the simulation results support the use of weighted robust standard errors, bootstrap methods, or matching with a naïve standard error (i.e., Greenwood method). The methods considered in the article are illustrated using a real-world example in which we estimate the effect of discharge prescribing of statins on patients hospitalized for acute myocardial infarction.     

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.     

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.     

On kernel machine learning for propensity score estimation under complex confounding structures

Post marketing data offer rich information and cost-effective resources for physicians and policy-makers to address some critical scientific questions in clinical practice. However, the complex confounding structures (e.g., nonlinear and nonadditive interactions) embedded in these observational data often pose major analytical challenges for proper analysis to draw valid conclusions. Furthermore, often made available as electronic health records (EHRs), these data are usually massive with hundreds of thousands observational records, which introduce additional computational challenges. In this paper, for comparative effectiveness analysis, we propose a statistically robust yet computationally efficient propensity score (PS) approach to adjust for the complex confounding structures. Specifically, we propose a kernel-based machine learning method for flexibly and robustly PS modeling to obtain valid PS estimation from observational data with complex confounding structures. The estimated propensity score is then used in the second stage analysis to obtain the consistent average treatment effect estimate. An empirical variance estimator based on the bootstrap is adopted. A split-and-merge algorithm is further developed to reduce the computational workload of the proposed method for big data, and to obtain a valid variance estimator of the average treatment effect estimate as a by-product. As shown by extensive numerical studies and an application to postoperative pain EHR data comparative effectiveness analysis, the proposed approach consistently outperforms other competing methods, demonstrating its practical utility.     

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.     

Propensity‐score‐based priors for Bayesian augmented control design

Drug developers are required to demonstrate substantial evidence of effectiveness through the conduct of adequate and well‐controlled (A&WC) studies to obtain marketing approval of their medicine. What constitutes A&WC is interpreted as the conduct of randomized controlled trials (RCTs). However, these trials are sometimes unfeasible because of their size, duration, and cost. One way to reduce sample size is to leverage information on the control through a prior. One consideration when forming data‐driven prior is the consistency of the external and the current data. It is essential to make this process less susceptible to choosing information that only helps improve the chances toward making an effectiveness claim. For this purpose, propensity score methods are employed for two reasons: (1) it gives the probability of a patient to be in the trial, and (2) it minimizes selection bias by pairing together treatment and control within the trial and control subjects in the external data that are similar in terms of their pretreatment characteristics. Two matching schemes based on propensity scores, estimated through generalized boosted methods, are applied to a real example with the objective of using external data to perform Bayesian augmented control in a trial where the allocation is disproportionate. The simulation results show that the data augmentation process prevents prior and data conflict and improves the precision of the estimator of the average treatment effect.     

Propensity score prediction for electronic healthcare databases using super learner and high-dimensional propensity score methods

ABSTRACT

The optimal learner for prediction modeling varies depending on the underlying data-generating distribution. Super Learner (SL) is a generic ensemble learning algorithm that uses cross-validation to select among a ‘library’ of candidate prediction models. While SL has been widely studied in a number of settings, it has not been thoroughly evaluated in large electronic healthcare databases that are common in pharmacoepidemiology and comparative effectiveness research. In this study, we applied and evaluated the performance of SL in its ability to predict the propensity score (PS), the conditional probability of treatment assignment given baseline covariates, using three electronic healthcare databases. We considered a library of algorithms that consisted of both nonparametric and parametric models. We also proposed a novel strategy for prediction modeling that combines SL with the high-dimensional propensity score (hdPS) variable selection algorithm. Predictive performance was assessed using three metrics: the negative log-likelihood, area under the curve (AUC), and time complexity. Results showed that the best individual algorithm, in terms of predictive performance, varied across datasets. The SL was able to adapt to the given dataset and optimize predictive performance relative to any individual learner. Combining the SL with the hdPS was the most consistent prediction method and may be promising for PS estimation and prediction modeling in electronic healthcare databases.     

Estimation of average treatment effects based on parametric propensity score model

In this paper, the estimation of average treatment effects is examined given that the propensity score is of a parametric form with some unknown parameters. Under the assumption that the treatment is ignorable given some observed characteristics, the MLEs for those unknown parameters in the probability assignment model have been achieved firstly and then three estimators have been defined by the inverse probability weighted, regression and imputation methods, respectively. All the estimators are shown asymptotically normal and more importantly, the substantial efficiency gains of the first two estimates have been obtained theoretically compared with the existing estimators in Hahn (1998) and Hirano et al. (2003), i.e., the inverse weighted probability estimator and the regression estimator have smaller asymptotic variances. Our simulation analysis verifies the theoretical results in terms of biases, SEs and MSEs.     

Model averaging procedure for varying-coefficient partially linear models with missing responses

This paper is concerned with model averaging procedure for varying-coefficient partially linear models with missing responses. The profile least-squares estimation process and inverse probability weighted method are employed to estimate regression coefficients of the partially restricted models, in which the propensity score is estimated by the covariate balancing propensity score method. The estimators of the linear parameters are shown to be asymptotically normal. Then we develop the focused information criterion, formulate the frequentist model averaging estimators and construct the corresponding confidence intervals. Some simulation studies are conducted to examine the finite sample performance of the proposed methods. We find that the covariate balancing propensity score improves the performance of the inverse probability weighted estimator. We also demonstrate the superiority of the proposed model averaging estimators over those of existing strategies in terms of mean squared error and coverage probability. Finally, our approach is further applied to a real data example.     

The challenges of evaluating dose response in flexible‐dose trials using marginal structural models

Assessing dose response from flexible‐dose clinical trials is problematic. The true dose effect may be obscured and even reversed in observed data because dose is related to both previous and subsequent outcomes. To remove selection bias, we propose marginal structural models, inverse probability of treatment‐weighting (IPTW) methodology. Potential clinical outcomes are compared across dose groups using a marginal structural model (MSM) based on a weighted pooled repeated measures analysis (generalized estimating equations with robust estimates of standard errors), with dose effect represented by current dose and recent dose history, and weights estimated from the data (via logistic regression) and determined as products of (i) inverse probability of receiving dose assignments that were actually received and (ii) inverse probability of remaining on treatment by this time. In simulations, this method led to almost unbiased estimates of true dose effect under various scenarios. Results were compared with those obtained by unweighted analyses and by weighted analyses under various model specifications. The simulation showed that the IPTW MSM methodology is highly sensitive to model misspecification even when weights are known. Practitioners applying MSM should be cautious about the challenges of implementing MSM with real clinical data. Clinical trial data are used to illustrate the methodology. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

网络访问固定样本调查的统计推断研究

如何解决网络访问固定样本调查的统计推断问题,是大数据背景下网络调查面临的严重挑战。针对此问题,提出将网络访问固定样本的调查样本与概率样本结合,利用倾向得分逆加权和加权组调整构造伪权数来估计目标总体,进一步采用基于有放回概率抽样的Vwr方法、基于广义回归估计的Vgreg方法与Jackknife方法来估计方差,并比较不同方法估计的效果。研究表明:无论概率样本的样本量较大还是较小,本研究所提出的总体均值估计方法效果较好,并且在方差估计中Jackknife方法的估计效果最好。     

Optimal caliper widths for propensity-score matching when estimating differences in means and differences in proportions in observational studies

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.     

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.     

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。     

Treatment Evaluation in the Presence of Sample Selection

Sample selection and attrition are inherent in a range of treatment evaluation problems such as the estimation of the returns to schooling or training. Conventional estimators tackling selection bias typically rely on restrictive functional form assumptions that are unlikely to hold in reality. This paper shows identification of average and quantile treatment effects in the presence of the double selection problem into (i) a selective subpopulation (e.g., working—selection on unobservables) and (ii) a binary treatment (e.g., training—selection on observables) based on weighting observations by the inverse of a nested propensity score that characterizes either selection probability. Weighting estimators based on parametric propensity score models are applied to female labor market data to estimate the returns to education.     

Comparison of clustering algorithms on generalized propensity score in observational studies: a simulation study

In observational studies, unbalanced observed covariates between treatment groups often cause biased inferences on the estimation of treatment effects. Recently, generalized propensity score (GPS) has been proposed to overcome this problem; however, a practical technique to apply the GPS is lacking. This study demonstrates how clustering algorithms can be used to group similar subjects based on transformed GPS. We compare four popular clustering algorithms: k-means clustering (KMC), model-based clustering, fuzzy c-means clustering and partitioning around medoids based on the following three criteria: average dissimilarity between subjects within clusters, average Dunn index and average silhouette width under four various covariate scenarios. Simulation studies show that the KMC algorithm has overall better performance compared with the other three clustering algorithms. Therefore, we recommend using the KMC algorithm to group similar subjects based on the transformed GPS.