On the use of propensity scores in principal causal effect estimation

We examine the practicality of propensity score methods for estimating causal treatment effects conditional on intermediate posttreatment outcomes (principal effects) in the context of randomized experiments. In particular, we focus on the sensitivity of principal causal effect estimates to violation of principal ignorability, which is the primary assumption that underlies the use of propensity score methods to estimate principal effects. Under principal ignorability (PI), principal strata membership is conditionally independent of the potential outcome under control given the pre‐treatment covariates; i.e. there are no differences in the potential outcomes under control across principal strata given the observed pretreatment covariates. Under this assumption, principal scores modeling principal strata membership can be estimated based solely on the observed covariates and used to predict strata membership and estimate principal effects. While this assumption underlies the use of propensity scores in this setting, sensitivity to violations of it has not been studied rigorously. In this paper, we explicitly define PI using the outcome model (although we do not actually use this outcome model in estimating principal scores) and systematically examine how deviations from the assumption affect estimates, including how the strength of association between principal stratum membership and covariates modifies the performance. We find that when PI is violated, very strong covariate predictors of stratum membership are needed to yield accurate estimates of principal effects. Copyright © 2009 John Wiley & Sons, Ltd.     

Conditioning on the propensity score can result in biased estimation of common measures of treatment effect: a Monte Carlo study

Propensity score methods are increasingly being used to estimate causal treatment effects in the medical literature. Conditioning on the propensity score results in unbiased estimation of the expected difference in observed responses to two treatments. The degree to which conditioning on the propensity score introduces bias into the estimation of the conditional odds ratio or conditional hazard ratio, which are frequently used as measures of treatment effect in observational studies, has not been extensively studied. We conducted Monte Carlo simulations to determine the degree to which propensity score matching, stratification on the quintiles of the propensity score, and covariate adjustment using the propensity score result in biased estimation of conditional odds ratios, hazard ratios, and rate ratios. We found that conditioning on the propensity score resulted in biased estimation of the true conditional odds ratio and the true conditional hazard ratio. In all scenarios examined, treatment effects were biased towards the null treatment effect. However, conditioning on the propensity score did not result in biased estimation of the true conditional rate ratio. In contrast, conventional regression methods allowed unbiased estimation of the true conditional treatment effect when all variables associated with the outcome were included in the regression model. The observed bias in propensity score methods is due to the fact that regression models allow one to estimate conditional treatment effects, whereas propensity score methods allow one to estimate marginal treatment effects. In several settings with non-linear treatment effects, marginal and conditional treatment effects do not coincide.     

Causal models for randomized trials with two active treatments and continuous compliance

In many clinical trials, compliance with assigned treatment could be measured on a continuous scale (e.g., the proportion of assigned treatment actually taken). In general, inference about principal causal effects can be challenging, particularly when there are two active treatments; the problem is exacerbated for continuous measures of compliance. We address this issue by first proposing a structural model for the principal effects. We then specify compliance models within each arm of the study. These marginal models are identifiable. The joint distribution of the observed and counterfactual compliance variables is assumed to follow a Gaussian copula model, which links the two marginal models and includes a dependence parameter that cannot be identified. This dependence parameter can be varied as part of a sensitivity analysis. We illustrate the methodology with an analysis of data from a smoking cessation trial. As part of the analysis, we estimate causal effects at particular levels of the compliance variables and within subpopulations that have similar compliance behavior.     

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.     

Two-stage instrumental variable methods for estimating the causal odds ratio: analysis of bias

We present closed-form expressions of asymptotic bias for the causal odds ratio from two estimation approaches of instrumental variable logistic regression: (i) the two-stage predictor substitution (2SPS) method and (ii) the two-stage residual inclusion (2SRI) approach. Under the 2SPS approach, the first stage model yields the predicted value of treatment as a function of an instrument and covariates, and in the second stage model for the outcome, this predicted value replaces the observed value of treatment as a covariate. Under the 2SRI approach, the first stage is the same, but the residual term of the first stage regression is included in the second stage regression, retaining the observed treatment as a covariate. Our bias assessment is for a different context from that of Terza (J. Health Econ. 2008; 27(3):531-543), who focused on the causal odds ratio conditional on the unmeasured confounder, whereas we focus on the causal odds ratio among compliers under the principal stratification framework. Our closed-form bias results show that the 2SPS logistic regression generates asymptotically biased estimates of this causal odds ratio when there is no unmeasured confounding and that this bias increases with increasing unmeasured confounding. The 2SRI logistic regression is asymptotically unbiased when there is no unmeasured confounding, but when there is unmeasured confounding, there is bias and it increases with increasing unmeasured confounding. The closed-form bias results provide guidance for using these IV logistic regression methods. Our simulation results are consistent with our closed-form analytic results under different combinations of parameter settings.     

Testing concordance of instrumental variable effects in generalized linear models with application to Mendelian randomization

Instrumental variable regression is one way to overcome unmeasured confounding and estimate causal effect in observational studies. Built on structural mean models, there has been considerable work recently developed for consistent estimation of causal relative risk and causal odds ratio. Such models can sometimes suffer from identification issues for weak instruments. This hampered the applicability of Mendelian randomization analysis in genetic epidemiology. When there are multiple genetic variants available as instrumental variables, and causal effect is defined in a generalized linear model in the presence of unmeasured confounders, we propose to test concordance between instrumental variable effects on the intermediate exposure and instrumental variable effects on the disease outcome, as a means to test the causal effect. We show that a class of generalized least squares estimators provide valid and consistent tests of causality. For causal effect of a continuous exposure on a dichotomous outcome in logistic models, the proposed estimators are shown to be asymptotically conservative. When the disease outcome is rare, such estimators are consistent because of the log‐linear approximation of the logistic function. Optimality of such estimators relative to the well‐known two‐stage least squares estimator and the double‐logistic structural mean model is further discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

A comparison of different matching designs in case-control studies: An empirical example using continuous exposures,continuous confounders and incidence of myocardial infarction

The paper presents a case-control study involving a disease, exposures and several continuous confounders. The relative efficiency and validity of a fully matched design is compared with random sampling of controls. We test a viable option of a partially matched design when inability to match all study subjects on all confounders occurs. The degree of bias in the odds ratios introduced by the different designs and by the different analytic models is assessed in comparison with the estimates obtained from a total cohort, from which both cases and controls were selected. Matched designs and analytic strategies are also evaluated in terms of the variances of the odds ratios. The results indicate that matching on continuous variables may lead to a more precise estimate of odds ratio than statistical control of confounding in unmatched designs. Partial selection of controls by matching may be a useful strategy when complete matching cannot be achieved; in practice, partial matching achieves most of the benefits of full matching.     

Factors Associated with Adherence to the Mediterranean Diet in the Adult Population

Our aim was to analyze the variables associated with adherence to the Mediterranean diet in the adult population. We conducted a cross-sectional study in an established cohort of 1,553 healthy study participants (mean age=55±14 years; 60.3% women). Mediterranean diet adherence was evaluated based on a 14-item questionnaire and the Mediterranean diet adherence screener, which defines adequate adherence as a score of ≥9. Physical activity was evaluated using the 7-day physical activity record. Sociodemographic, biological, and anthropometric variables were also evaluated. The differences between Mediterranean diet compliers and noncompliers are defined by the consumption of fruit, red meats, carbonated beverages, wine, fish/shellfish, legumes, pasta, and rice (P<0.01). Adherence was lower among individuals younger than 49 years of age. In the first age tertile, adherence was greater in women and in nonobese individuals, and the triglyceride levels were lower among compliers. In the second age tertile, the compliers exercised more and had a lower body fat percentage. In the third age tertile, the compliers also possessed less body fat. The logistic regression analysis revealed the following factors associated with improved Mediterranean diet adherence: more physical exercise (odds ratio=1.588), older age (odds ratio=2.162), and moderate alcohol consumption (odds ratio=1.342). The factors associated with improved Mediterranean diet adherence included female sex, age older than 62 years, moderate alcohol consumption, and more than 17 metabolic equivalents (METs)/h/wk of physical exercise. Poorer adherence was associated with males and obesity.     

Bivariate categorical data analysis using normal linear conditional multinomial probability model

Bivariate multinomial data such as the left and right eyes retinopathy status data are analyzed either by using a joint bivariate probability model or by exploiting certain odds ratio‐based association models. However, the joint bivariate probability model yields marginal probabilities, which are complicated functions of marginal and association parameters for both variables, and the odds ratio‐based association model treats the odds ratios involved in the joint probabilities as ‘working’ parameters, which are consequently estimated through certain arbitrary ‘working’ regression models. Also, this later odds ratio‐based model does not provide any easy interpretations of the correlations between two categorical variables. On the basis of pre‐specified marginal probabilities, in this paper, we develop a bivariate normal type linear conditional multinomial probability model to understand the correlations between two categorical variables. The parameters involved in the model are consistently estimated using the optimal likelihood and generalized quasi‐likelihood approaches. The proposed model and the inferences are illustrated through an intensive simulation study as well as an analysis of the well‐known Wisconsin Diabetic Retinopathy status data. Copyright © 2014 John Wiley & Sons, Ltd.     

On the conditional logistic estimator in two‐arm experimental studies with non‐compliance and before–after binary outcomes

The behavior of the conditional logistic estimator is analyzed under a causal model for two‐arm experimental studies with possible non‐compliance in which the effect of the treatment is measured by a binary response variable. We show that, when non‐compliance may only be observed in the treatment arm, the effect (measured on the logit scale) of the treatment on compliers and that of the control on non‐compliers can be identified and consistently estimated under mild conditions. The same does not happen for the effect of the control on compliers. A simple correction of the conditional logistic estimator is then proposed, which allows us to considerably reduce the bias in estimating this quantity and the causal effect of the treatment over control on compliers. A two‐step estimator results on the basis of which we can also set up a Wald test for the hypothesis of absence of a causal effect of the treatment. The asymptotic properties of the estimator are studied by exploiting the general theory on maximum likelihood estimation of misspecified models. Finite‐sample properties of the estimator and of the related Wald test are studied by simulation. The extension of the approach to the case of missing responses is also outlined. The approach is illustrated by an application to a dataset deriving from a study on the efficacy of a training course on the breast self examination practice. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Severity of bias of a simple estimator of the causal odds ratio in Mendelian randomization studies

Mendelian randomization studies estimate causal effects using genetic variants as instruments. Instrumental variable methods are straightforward for linear models, but epidemiologists often use odds ratios to quantify effects. Also, odds ratios are often the quantities reported in meta‐analyses. Many applications of Mendelian randomization dichotomize genotype and estimate the population causal log odds ratio for unit increase in exposure by dividing the genotype‐disease log odds ratio by the difference in mean exposure between genotypes. This ‘Wald‐type’ estimator is biased even in large samples, but whether the magnitude of bias is of practical importance is unclear. We study the large‐sample bias of this estimator in a simple model with a continuous normally distributed exposure, a single unobserved confounder that is not an effect modifier, and interpretable parameters. We focus on parameter values that reflect scenarios in which we apply Mendelian randomization, including realistic values for the degree of confounding and strength of the causal effect. We evaluate this estimator and the causal odds ratio using numerical integration and obtain approximate analytic expressions to check results and gain insight. A small simulation study examines finite sample bias and mild violations of the normality assumption. For our simple data‐generating model, we find that the Wald estimator is asymptotically biased with a bias of around 10% in fairly typical Mendelian randomization scenarios but which can be larger in more extreme situations. Recently developed methods such as structural mean models require fewer untestable assumptions and we recommend their use when the individual‐level data they require are available. The Wald‐type estimator may retain a role as an approximate method for meta‐analysis based on summary data. Copyright © 2012 John Wiley & Sons, Ltd.     

Marginal structural models to estimate the causal effect of zidovudine on the survival of HIV-positive men

Standard methods for survival analysis, such as the time-dependent Cox model, may produce biased effect estimates when there exist time-dependent confounders that are themselves affected by previous treatment or exposure. Marginal structural models are a new class of causal models the parameters of which are estimated through inverse-probability-of-treatment weighting; these models allow for appropriate adjustment for confounding. We describe the marginal structural Cox proportional hazards model and use it to estimate the causal effect of zidovudine on the survival of human immunodeficiency virus-positive men participating in the Multicenter AIDS Cohort Study. In this study, CD4 lymphocyte count is both a time-dependent confounder of the causal effect of zidovudine on survival and is affected by past zidovudine treatment. The crude mortality rate ratio (95% confidence interval) for zidovudine was 3.6 (3.0-4.3), which reflects the presence of confounding. After controlling for baseline CD4 count and other baseline covariates using standard methods, the mortality rate ratio decreased to 2.3 (1.9-2.8). Using a marginal structural Cox model to control further for time-dependent confounding due to CD4 count and other time-dependent covariates, the mortality rate ratio was 0.7 (95% conservative confidence interval = 0.6-1.0). We compare marginal structural models with previously proposed causal methods.     

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.     

Estimating drug effects in the presence of placebo response: Causal inference using growth mixture modeling

Placebo‐controlled randomized trials for antidepressants and other drugs often show a response for a sizeable percentage of the subjects in the placebo group. Potential placebo responders can be assumed to exist also in the drug treatment group, making it difficult to assess the drug effect. A key drug research focus should be to estimate the percentage of individuals among those who responded to the drug who would not have responded to the placebo (‘Drug Only Responders’). This paper investigates a finite mixture model approach to uncover percentages of up to four potential mixture components: Never Responders, Drug Only Responders, Placebo Only Responders, and Always Responders. Two examples are used to illustrate the modeling, a 12‐week antidepressant trial with a continuous outcome (Hamilton D score) and a 7‐week schizophrenia trial with a binary outcome (illness level). The approach is formulated in causal modeling terms using potential outcomes and principal stratification. Growth mixture modeling (GMM) with maximum‐likelihood estimation is used to uncover the different mixture components. The results point to the limitations of the conventional approach of comparing marginal response rates for drug and placebo groups. It is useful to augment such reporting with the GMM‐estimated prevalences for the four classes of subjects and the Drug Only Responder drug effect estimate. Copyright © 2009 John Wiley & Sons, Ltd.     

A comparison of the ability of different propensity score models to balance measured variables between treated and untreated subjects: a Monte Carlo study

The propensity score--the probability of exposure to a specific treatment conditional on observed variables--is increasingly being used in observational studies. Creating strata in which subjects are matched on the propensity score allows one to balance measured variables between treated and untreated subjects. There is an ongoing controversy in the literature as to which variables to include in the propensity score model. Some advocate including those variables that predict treatment assignment, while others suggest including all variables potentially related to the outcome, and still others advocate including only variables that are associated with both treatment and outcome. We provide a case study of the association between drug exposure and mortality to show that including a variable that is related to treatment, but not outcome, does not improve balance and reduces the number of matched pairs available for analysis. In order to investigate this issue more comprehensively, we conducted a series of Monte Carlo simulations of the performance of propensity score models that contained variables related to treatment allocation, or variables that were confounders for the treatment-outcome pair, or variables related to outcome or all variables related to either outcome or treatment or neither. We compared the use of these different propensity scores models in matching and stratification in terms of the extent to which they balanced variables. We demonstrated that all propensity scores models balanced measured confounders between treated and untreated subjects in a propensity-score matched sample. However, including only the true confounders or the variables predictive of the outcome in the propensity score model resulted in a substantially larger number of matched pairs than did using the treatment-allocation model. Stratifying on the quintiles of any propensity score model resulted in residual imbalance between treated and untreated subjects in the upper and lower quintiles. Greater balance between treated and untreated subjects was obtained after matching on the propensity score than after stratifying on the quintiles of the propensity score. When a confounding variable was omitted from any of the propensity score models, then matching or stratifying on the propensity score resulted in residual imbalance in prognostically important variables between treated and untreated subjects. We considered four propensity score models for estimating treatment effects: the model that included only true confounders; the model that included all variables associated with the outcome; the model that included all measured variables; and the model that included all variables associated with treatment selection. Reduction in bias when estimating a null treatment effect was equivalent for all four propensity score models when propensity score matching was used. Reduction in bias was marginally greater for the first two propensity score models than for the last two propensity score models when stratification on the quintiles of the propensity score model was employed. Furthermore, omitting a confounding variable from the propensity score model resulted in biased estimation of the treatment effect. Finally, the mean squared error for estimating a null treatment effect was lower when either of the first two propensity scores was used compared to when either of the last two propensity score models was used.     

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.     

Regression standardization with the <Emphasis FontCategory="NonProportional">R</Emphasis> package <Emphasis FontCategory="NonProportional">stdReg</Emphasis>

When studying the association between an exposure and an outcome, it is common to use regression models to adjust for measured confounders. The most common models in epidemiologic research are logistic regression and Cox regression, which estimate conditional (on the confounders) odds ratios and hazard ratios. When the model has been fitted, one can use regression standardization to estimate marginal measures of association. If the measured confounders are sufficient for confounding control, then the marginal association measures can be interpreted as poulation causal effects. In this paper we describe a new R package, stdReg, that carries out regression standardization with generalized linear models (e.g. logistic regression) and Cox regression models. We illustrate the package with several examples, using real data that are publicly available.     

Assessing the ratio of means as a causal estimand in clinical endpoint bioequivalence studies in the presence of intercurrent events

In clinical endpoint bioequivalence studies, the observed per-protocol (PP) population (compliers and completers in general) is usually used in the primary analysis for equivalence assessment. However, intercurrent events, ie, missingness and noncompliance, are not properly handled. The resulting estimand is not causal. Previously, we proposed the first causal framework to assess equivalence in the presence of missing data and noncompliance. We proposed a causal survivor average causal effect (SACE) estimand for the difference of means (DOM). In equivalence assessment, DOM is not as widely used as the ratio of means (ROM). However, no existing formula links the observed PP estimand to the SACE estimand for ROM as exists for DOM. Herein, we propose a similar causal framework for ROM using the principal stratification approach, one of the strategies recommended by the International Conference on Harmonisation (ICH) E9 R1 addendum. We quantify the bias of the observed ROM PP estimand for the SACE estimand, which provides a basis to identify three conditions under which the two estimands are equal. We propose a sensitivity analysis method to evaluate the robustness of the current PP estimator to estimate the SACE estimand. We extend Fieller's confidence interval for the SACE estimand using ROM, which can be applied to many settings. Simulation demonstrates that the PP estimator is biased in either directions and may inflate type 1 error and/or change power when the three identified conditions are violated. Our work can be applied to comparative clinical biosimilar studies.     

Causal inference in longitudinal comparative effectiveness studies with repeated measures of a continuous intermediate variable

We propose a principal stratification approach to assess causal effects in nonrandomized longitudinal comparative effectiveness studies with a binary endpoint outcome and repeated measures of a continuous intermediate variable. Our method is an extension of the principal stratification approach originally proposed for the longitudinal randomized study “Prevention of Suicide in Primary Care Elderly: Collaborative Trial” to assess the treatment effect on the continuous Hamilton depression score adjusting for the heterogeneity of repeatedly measured binary compliance status. Our motivation for this work comes from a comparison of the effect of two glucose‐lowering medications on a clinical cohort of patients with type 2 diabetes. Here, we consider a causal inference problem assessing how well the two medications work relative to one another on two binary endpoint outcomes: cardiovascular disease‐related hospitalization and all‐cause mortality. Clinically, these glucose‐lowering medications can have differential effects on the intermediate outcome, glucose level over time. Ultimately, we want to compare medication effects on the endpoint outcomes among individuals in the same glucose trajectory stratum while accounting for the heterogeneity in baseline covariates (i.e., to obtain ‘principal effects’ on the endpoint outcomes). The proposed method involves a three‐step model estimation procedure. Step 1 identifies principal strata associated with the intermediate variable using hybrid growth mixture modeling analyses. Step 2 obtains the stratum membership using the pseudoclass technique and derives propensity scores for treatment assignment. Step 3 obtains the stratum‐specific treatment effect on the endpoint outcome weighted by inverse propensity probabilities derived from Step 2. Copyright © 2014 John Wiley & Sons, Ltd.