A tutorial on propensity score estimation for multiple treatments using generalized boosted models

The use of propensity scores to control for pretreatment imbalances on observed variables in non‐randomized or observational studies examining the causal effects of treatments or interventions has become widespread over the past decade. For settings with two conditions of interest such as a treatment and a control, inverse probability of treatment weighted estimation with propensity scores estimated via boosted models has been shown in simulation studies to yield causal effect estimates with desirable properties. There are tools (e.g., the twang package in R) and guidance for implementing this method with two treatments. However, there is not such guidance for analyses of three or more treatments. The goals of this paper are twofold: (1) to provide step‐by‐step guidance for researchers who want to implement propensity score weighting for multiple treatments and (2) to propose the use of generalized boosted models (GBM) for estimation of the necessary propensity score weights. We define the causal quantities that may be of interest to studies of multiple treatments and derive weighted estimators of those quantities. We present a detailed plan for using GBM to estimate propensity scores and using those scores to estimate weights and causal effects. We also provide tools for assessing balance and overlap of pretreatment variables among treatment groups in the context of multiple treatments. A case study examining the effects of three treatment programs for adolescent substance abuse demonstrates the methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Models for the propensity score that contemplate the positivity assumption and their application to missing data and causality

Generalized linear models are often assumed to fit propensity scores, which are used to compute inverse probability weighted (IPW) estimators. To derive the asymptotic properties of IPW estimators, the propensity score is supposed to be bounded away from zero. This condition is known in the literature as strict positivity (or positivity assumption), and, in practice, when it does not hold, IPW estimators are very unstable and have a large variability. Although strict positivity is often assumed, it is not upheld when some of the covariates are unbounded. In real data sets, a data‐generating process that violates the positivity assumption may lead to wrong inference because of the inaccuracy in the estimations. In this work, we attempt to conciliate between the strict positivity condition and the theory of generalized linear models by incorporating an extra parameter, which results in an explicit lower bound for the propensity score. An additional parameter is added to fulfil the overlap assumption in the causal framework.     

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

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.     

Moving towards best practice when using inverse probability of treatment weighting (IPTW) using the propensity score to estimate causal treatment effects in observational studies

The propensity score is defined as a subject's probability of treatment selection, conditional on observed baseline covariates. Weighting subjects by the inverse probability of treatment received creates a synthetic sample in which treatment assignment is independent of measured baseline covariates. Inverse probability of treatment weighting (IPTW) using the propensity score allows one to obtain unbiased estimates of average treatment effects. However, these estimates are only valid if there are no residual systematic differences in observed baseline characteristics between treated and control subjects in the sample weighted by the estimated inverse probability of treatment. We report on a systematic literature review, in which we found that the use of IPTW has increased rapidly in recent years, but that in the most recent year, a majority of studies did not formally examine whether weighting balanced measured covariates between treatment groups. We then proceed to describe a suite of quantitative and qualitative methods that allow one to assess whether measured baseline covariates are balanced between treatment groups in the weighted sample. The quantitative methods use the weighted standardized difference to compare means, prevalences, higher‐order moments, and interactions. The qualitative methods employ graphical methods to compare the distribution of continuous baseline covariates between treated and control subjects in the weighted sample. Finally, we illustrate the application of these methods in an empirical case study. We propose a formal set of balance diagnostics that contribute towards an evolving concept of ‘best practice’ when using IPTW to estimate causal treatment effects using observational data. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

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.     

The use of propensity score methods with survival or time‐to‐event outcomes: reporting measures of effect similar to those used in randomized experiments

Propensity score methods are increasingly being used to estimate causal treatment effects in observational studies. In medical and epidemiological studies, outcomes are frequently time‐to‐event in nature. Propensity‐score methods are often applied incorrectly when estimating the effect of treatment on time‐to‐event outcomes. This article describes how two different propensity score methods (matching and inverse probability of treatment weighting) can be used to estimate the measures of effect that are frequently reported in randomized controlled trials: (i) marginal survival curves, which describe survival in the population if all subjects were treated or if all subjects were untreated; and (ii) marginal hazard ratios. The use of these propensity score methods allows one to replicate the measures of effect that are commonly reported in randomized controlled trials with time‐to‐event outcomes: both absolute and relative reductions in the probability of an event occurring can be determined. We also provide guidance on variable selection for the propensity score model, highlight methods for assessing the balance of baseline covariates between treated and untreated subjects, and describe the implementation of a sensitivity analysis to assess the effect of unmeasured confounding variables on the estimated treatment effect when outcomes are time‐to‐event in nature. The methods in the paper are illustrated by estimating the effect of discharge statin prescribing on the risk of death in a sample of patients hospitalized with acute myocardial infarction. In this tutorial article, we describe and illustrate all the steps necessary to conduct a comprehensive analysis of the effect of treatment on time‐to‐event outcomes. © 2013 The authors. Statistics in Medicine published by John Wiley & Sons, Ltd.     

Bayesian propensity score analysis for observational data

In the analysis of observational data, stratifying patients on the estimated propensity scores reduces confounding from measured variables. Confidence intervals for the treatment effect are typically calculated without acknowledging uncertainty in the estimated propensity scores, and intuitively this may yield inferences, which are falsely precise. In this paper, we describe a Bayesian method that models the propensity score as a latent variable. We consider observational studies with a dichotomous treatment, dichotomous outcome, and measured confounders where the log odds ratio is the measure of effect. Markov chain Monte Carlo is used for posterior simulation. We study the impact of modelling uncertainty in the propensity scores in a case study investigating the effect of statin therapy on mortality in Ontario patients discharged from hospital following acute myocardial infarction. Our analysis reveals that the Bayesian credible interval for the treatment effect is 10 per cent wider compared with a conventional propensity score analysis. Using simulations, we show that when the association between treatment and confounders is weak, then this increases uncertainty in the estimated propensity scores. Bayesian interval estimates for the treatment effect are longer on average, though there is little improvement in coverage probability. A novel feature of the proposed method is that it fits models for the treatment and outcome simultaneously rather than one at a time. The method uses the outcome variable to inform the fit of the propensity model. We explore the performance of the estimated propensity scores using cross‐validation. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Assessing exposure effects on gene expression

In observational genomics data sets, there is often confounding of the effect of an exposure on gene expression. To adjust for confounding when estimating the exposure effect, a common approach involves including potential confounders as covariates with the exposure in a regression model of gene expression. However, when the exposure and confounders interact to influence gene expression, the fitted regression model does not necessarily estimate the overall effect of the exposure. Using inverse probability weighting (IPW) or the parametric g-formula in these instances is straightforward to apply and yields consistent effect estimates. IPW can readily be integrated into a genomics data analysis pipeline with upstream data processing and normalization, while the g-formula can be implemented by making simple alterations to the regression model. The regression, IPW, and g-formula approaches to exposure effect estimation are compared herein using simulations; advantages and disadvantages of each approach are explored. The methods are applied to a case study estimating the effect of current smoking on gene expression in adipose tissue.     

Bayesian estimation of the average treatment effect on the treated using inverse weighting

We develop a Bayesian approach to estimate the average treatment effect on the treated in the presence of confounding. The approach builds on developments proposed by Saarela et al in the context of marginal structural models, using importance sampling weights to adjust for confounding and estimate a causal effect. The Bayesian bootstrap is adopted to approximate posterior distributions of interest and avoid the issue of feedback that arises in Bayesian causal estimation relying on a joint likelihood. We present results from simulation studies to estimate the average treatment effect on the treated, evaluating the impact of sample size and the strength of confounding on estimation. We illustrate our approach using the classic Right Heart Catheterization data set and find a negative causal effect of the exposure on 30-day survival, in accordance with previous analyses of these data. We also apply our approach to the data set of the National Center for Health Statistics Birth Data and obtain a negative effect of maternal smoking during pregnancy on birth weight.     

Stratification and weighting via the propensity score in estimation of causal treatment effects: a comparative study

Estimation of treatment effects with causal interpretation from observational data is complicated because exposure to treatment may be confounded with subject characteristics. The propensity score, the probability of treatment exposure conditional on covariates, is the basis for two approaches to adjusting for confounding: methods based on stratification of observations by quantiles of estimated propensity scores and methods based on weighting observations by the inverse of estimated propensity scores. We review popular versions of these approaches and related methods offering improved precision, describe theoretical properties and highlight their implications for practice, and present extensive comparisons of performance that provide guidance for practical use.     

基于倾向指数匹配法的肝癌患者生存分析

  目的  基于倾向性评分逆概率加权法（IPTW）评价手术、放疗和联合治疗3种方式治疗胃癌患者的疗效, 为胃癌的治疗提供参考依据。  方法  收集2004年1月 — 2013年12月美国国家癌症研究所监测、流行病学与预后项目（SEER）数据库中经胃镜病理诊断确诊的7 005例胃癌患者数据,其中接受手术治疗者3 983例、接受放疗者795例、接受联合治疗者2 227例;采用倾向性评分IPTW法以生存时间和结局为效应指标,分析不同的治疗方法对胃癌患者生存率的影响。  结果  手术组、放疗组和联合治疗组胃癌患者倾向性评分IPTW法加权前中位生存期分别为30、9和38个月,加权后分别为25、11和38个月,加权前、后3组胃癌患者生存曲线间差异均有统计学意义（均P < 0.001）,联合治疗组患者加权前、后的预后均优于手术组和放疗组患者;加权后大部分基线特征绝对标准化平均差异（ASMD）减小,且均 < 0.2,3组患者均衡效果较好;在控制了性别、年龄、种族、婚姻状况、肿瘤大小、病理分化、肿瘤分期、T分期、N分期和远处转移情况等混杂因素后,多因素Cox回归分析结果显示,与手术组胃癌患者相比,放疗组胃癌患者的预后较差（HR = 2.044,95 % CI = 1.770～2.361）,联合治疗组胃癌患者的预后较好（HR = 0.630,95 % CI = 0.573～0.694）。  结论  联合治疗方法疗效优于手术和放疗治疗方法。     

Bias associated with using the estimated propensity score as a regression covariate

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. Copyright © 2013 John Wiley & Sons, Ltd.     

Testing causal effects in observational survival data using propensity score matching design

Time‐to‐event data are very common in observational studies. Unlike randomized experiments, observational studies suffer from both observed and unobserved confounding biases. To adjust for observed confounding in survival analysis, the commonly used methods are the Cox proportional hazards (PH) model, the weighted logrank test, and the inverse probability of treatment weighted Cox PH model. These methods do not rely on fully parametric models, but their practical performances are highly influenced by the validity of the PH assumption. Also, there are few methods addressing the hidden bias in causal survival analysis. We propose a strategy to test for survival function differences based on the matching design and explore sensitivity of the P‐values to assumptions about unmeasured confounding. Specifically, we apply the paired Prentice‐Wilcoxon (PPW) test or the modified PPW test to the propensity score matched data. Simulation studies show that the PPW‐type test has higher power in situations when the PH assumption fails. For potential hidden bias, we develop a sensitivity analysis based on the matched pairs to assess the robustness of our finding, following Rosenbaum's idea for nonsurvival data. For a real data illustration, we apply our method to an observational cohort of chronic liver disease patients from a Mayo Clinic study. The PPW test based on observed data initially shows evidence of a significant treatment effect. But this finding is not robust, as the sensitivity analysis reveals that the P‐value becomes nonsignificant if there exists an unmeasured confounder with a small impact.     

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.     

Avoiding pitfalls when combining multiple imputation and propensity scores

Overcoming bias due to confounding and missing data is challenging when analyzing observational data. Propensity scores are commonly used to account for the first problem and multiple imputation for the latter. Unfortunately, it is not known how best to proceed when both techniques are required. We investigate whether two different approaches to combining propensity scores and multiple imputation (Across and Within) lead to differences in the accuracy or precision of exposure effect estimates. Both approaches start by imputing missing values multiple times. Propensity scores are then estimated for each resulting dataset. Using the Across approach, the mean propensity score across imputations for each subject is used in a single subsequent analysis. Alternatively, the Within approach uses propensity scores individually to obtain exposure effect estimates in each imputation, which are combined to produce an overall estimate. These approaches were compared in a series of Monte Carlo simulations and applied to data from the British Society for Rheumatology Biologics Register. Results indicated that the Within approach produced unbiased estimates with appropriate confidence intervals, whereas the Across approach produced biased results and unrealistic confidence intervals. Researchers are encouraged to implement the Within approach when conducting propensity score analyses with incomplete data.     

Optimally combining propensity score subclasses

Propensity score methods, such as subclassification, are a common approach to control for confounding when estimating causal effects in non‐randomized studies. Propensity score subclassification groups individuals into subclasses based on their propensity score values. Effect estimates are obtained within each subclass and then combined by weighting by the proportion of observations in each subclass. Combining subclass‐specific estimates by weighting by the inverse variance is a promising alternative approach; a similar strategy is used in meta‐analysis for its efficiency. We use simulation to compare performance of each of the two methods while varying (i) the number of subclasses, (ii) extent of propensity score overlap between the treatment and control groups (i.e., positivity), (iii) incorporation of survey weighting, and (iv) presence of heterogeneous treatment effects across subclasses. Both methods perform well in the absence of positivity violations and with a constant treatment effect with weighting by the inverse variance performing slightly better. Weighting by the proportion in subclass performs better in the presence of heterogeneous treatment effects across subclasses. We apply these methods to an illustrative example estimating the effect of living in a disadvantaged neighborhood on risk of past‐year anxiety and depressive disorders among U.S. urban adolescents. This example entails practical positivity violations but no evidence of treatment effect heterogeneity. In this case, weighting by the inverse variance when combining across propensity score subclasses results in more efficient estimates that ultimately change inference. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimation of conditional and marginal odds ratios using the prognostic score

Introduced by Hansen in 2008, the prognostic score (PGS) has been presented as ‘the prognostic analogue of the propensity score’ (PPS). PPS‐based methods are intended to estimate marginal effects. Most previous studies evaluated the performance of existing PGS‐based methods (adjustment, stratification and matching using the PGS) in situations in which the theoretical conditional and marginal effects are equal (i.e., collapsible situations). To support the use of PGS framework as an alternative to the PPS framework, applied researchers must have reliable information about the type of treatment effect estimated by each method. We propose four new PGS‐based methods, each developed to estimate a specific type of treatment effect. We evaluated the ability of existing and new PGS‐based methods to estimate the conditional treatment effect (CTE), the (marginal) average treatment effect on the whole population (ATE), and the (marginal) average treatment effect on the treated population (ATT), when the odds ratio (a non‐collapsible estimator) is the measure of interest. The performance of PGS‐based methods was assessed by Monte Carlo simulations and compared with PPS‐based methods and multivariate regression analysis. Existing PGS‐based methods did not allow for estimating the ATE and showed unacceptable performance when the proportion of exposed subjects was large. When estimating marginal effects, PPS‐based methods were too conservative, whereas the new PGS‐based methods performed better with low prevalence of exposure, and had coverages closer to the nominal value. When estimating CTE, the new PGS‐based methods performed as well as traditional multivariate regression. Copyright © 2016 John Wiley & Sons, Ltd.     

High‐dimensional propensity score algorithm in comparative effectiveness research with time‐varying interventions

The high‐dimensional propensity score (hdPS) algorithm was proposed for automation of confounding adjustment in problems involving large healthcare databases. It has been evaluated in comparative effectiveness research (CER) with point treatments to handle baseline confounding through matching or covariance adjustment on the hdPS. In observational studies with time‐varying interventions, such hdPS approaches are often inadequate to handle time‐dependent confounding and selection bias. Inverse probability weighting (IPW) estimation to fit marginal structural models can adequately handle these biases under the fundamental assumption of no unmeasured confounders. Upholding of this assumption relies on the selection of an adequate set of covariates for bias adjustment. We describe the application and performance of the hdPS algorithm to improve covariate selection in CER with time‐varying interventions based on IPW estimation and explore stabilization of the resulting estimates using Super Learning. The evaluation is based on both the analysis of electronic health records data in a real‐world CER study of adults with type 2 diabetes and a simulation study. This report (i) establishes the feasibility of IPW estimation with the hdPS algorithm based on large electronic health records databases, (ii) demonstrates little impact on inferences when supplementing the set of expert‐selected covariates using the hdPS algorithm in a setting with extensive background knowledge, (iii) supports the application of the hdPS algorithm in discovery settings with little background knowledge or limited data availability, and (iv) motivates the application of Super Learning to stabilize effect estimates based on the hdPS algorithm. Copyright © 2014 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.