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.     

The use of prognostic scores for causal inference with general treatment regimes

In nonrandomised studies, inferring causal effects requires appropriate methods for addressing confounding bias. Although it is common to adopt propensity score analysis to this purpose, prognostic score analysis has recently been proposed as an alternative strategy. While both approaches were originally introduced to estimate causal effects for binary interventions, the theory of propensity score has since been extended to the case of general treatment regimes. Indeed, many treatments are not assigned in a binary fashion and require a certain extent of dosing. Hence, researchers may often be interested in estimating treatment effects across multiple exposures. To the best of our knowledge, the prognostic score analysis has not been yet generalised to this case. In this article, we describe the theory of prognostic scores for causal inference with general treatment regimes. Our methods can be applied to compare multiple treatments using nonrandomised data, a topic of great relevance in contemporary evaluations of clinical interventions. We propose estimators for the average treatment effects in different populations of interest, the validity of which is assessed through a series of simulations. Finally, we present an illustrative case in which we estimate the effect of the delay to Aspirin administration on a composite outcome of death or dependence at 6 months in stroke patients.     

On the aggregation of published prognostic scores for causal inference in observational studies

As real world evidence on drug efficacy involves nonrandomized studies, statistical methods adjusting for confounding are needed. In this context, prognostic score (PGS) analysis has recently been proposed as a method for causal inference. It aims to restore balance across the different treatment groups by identifying subjects with a similar prognosis for a given reference exposure (“control”). This requires the development of a multivariable prognostic model in the control arm of the study sample, which is then extrapolated to the different treatment arms. Unfortunately, large cohorts for developing prognostic models are not always available. Prognostic models are therefore subject to a dilemma between overfitting and parsimony; the latter being prone to a violation of the assumption of no unmeasured confounders when important covariates are ignored. Although it is possible to limit overfitting by using penalization strategies, an alternative approach is to adopt evidence synthesis. Aggregating previously published prognostic models may improve the generalizability of PGS, while taking account of a large set of covariates—even when limited individual participant data are available. In this article, we extend a method for prediction model aggregation to PGS analysis in nonrandomized studies. We conduct extensive simulations to assess the validity of model aggregation, compared with other methods of PGS analysis for estimating marginal treatment effects. We show that aggregating existing PGS into a “meta-score” is robust to misspecification, even when elementary scores wrongfully omit confounders or focus on different outcomes. We illustrate our methods in a setting of treatments for asthma.     

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.     

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.     

A novel approach for propensity score matching and stratification for multiple treatments: Application to an electronic health record–derived study

Currently, methods for conducting multiple treatment propensity scoring in the presence of high-dimensional covariate spaces that result from “big data” are lacking—the most prominent method relies on inverse probability treatment weighting (IPTW). However, IPTW only utilizes one element of the generalized propensity score (GPS) vector, which can lead to a loss of information and inadequate covariate balance in the presence of multiple treatments. This limitation motivates the development of a novel propensity score method that uses the entire GPS vector to establish a scalar balancing score that, when adjusted for, achieves covariate balance in the presence of potentially high-dimensional covariates. Specifically, the generalized propensity score cumulative distribution function (GPS-CDF) method is introduced. A one-parameter power function fits the CDF of the GPS vector and a resulting scalar balancing score is used for matching and/or stratification. Simulation results show superior performance of the new method compared to IPTW both in achieving covariate balance and estimating average treatment effects in the presence of multiple treatments. The proposed approach is applied to a study derived from electronic medical records to determine the causal relationship between three different vasopressors and mortality in patients with non-traumatic aneurysmal subarachnoid hemorrhage. Results suggest that the GPS-CDF method performs well when applied to large observational studies with multiple treatments that have large covariate spaces.     

A pilot design for observational studies: Using abundant data thoughtfully

Observational studies often benefit from an abundance of observational units. This can lead to studies that—while challenged by issues of internal validity—have inferences derived from sample sizes substantially larger than randomized controlled trials. But is the information provided by an observational unit best used in the analysis phase? We propose the use of a “pilot design,” in which observations are expended in the design phase of the study, and the posttreatment information from these observations is used to improve study design. In modern observational studies, which are data rich but control poor, pilot designs can be used to gain information about the structure of posttreatment variation. This information can then be used to improve instrumental variable designs, propensity score matching, doubly robust estimation, and other observational study designs. We illustrate one version of a pilot design, which aims to reduce within-set heterogeneity and improve performance in sensitivity analyses. This version of a pilot design expends observational units during the design phase to fit a prognostic model, avoiding concerns of overfitting. In addition, it enables the construction of “assignment-control plots,” which visualize the relationship between propensity and prognostic scores. We first show some examples of these plots, then we demonstrate in a simulation setting how this alternative use of the observations can lead to gains in terms of both treatment effect estimation and sensitivity analyses of unobserved confounding.     

A Bootstrap Confidence Interval Procedure for the Treatment Effect Using Propensity Score Subclassification

In the analysis of observational studies, propensity score subclassification has been shown to be a powerful method for adjusting unbalanced covariates for the purpose of causal inferences. One practical difficulty in carrying out such an analysis is to obtain a correct variance estimate for inference, while reducing bias in the estimate of the treatment effect due to an imbalance in the measured covariates. In this paper, we propose a bootstrap procedure for the inferences concerning the average treatment effect; our bootstrap method is based on an extension of Efron's bias-corrected accelerated (BCa) bootstrap confidence interval to a two-sample problem. Unlike the currently available inference procedures based on propensity score subclassifications, the validity of the proposed method does not rely on a particular form of variance estimation. A brief simulation study is included to evaluate the operating characteristics of the proposed procedure. We conclude the paper by illustrating the new procedure through a clinical application comparing the renal effects of two non-steroidal anti-inammatory drugs (NSAIDs).     

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.     

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.     

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.     

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.     

Estimation of treatment effect adjusting for treatment changes using the intensity score method: application to a large primary prevention study for coronary events (MEGA study)

The MEGA study was a prospective, randomized, open-labeled, blinded-endpoints study conducted in Japan to evaluate the primary preventive effect of pravastatin against coronary heart disease (CHD), in which 8214 subjects were randomized to diet or diet plus pravastatin. The intention-to-treat (ITT) analysis showed that pravastatin reduced the incidence of CHD (hazard ratio=0.67; 95 per cent confidence interval (CI): 0.49-0.91) and of stroke events, which was the secondary endpoint in the MEGA study (hazard ratio=0.83; 95 per cent CI: 0.57-1.21). Owing to considerable treatment changes, it is also of interest to estimate the causal effect of treatment that would have been observed had all patients complied with the treatment to which they were assigned. In this paper, we present an intensity score method developed for clinical trials with time-to-event outcomes that correct for treatment changes during follow-up. The proposed method can be easily extended to the estimation of time-dependent treatment effects, where the technique of g-estimation has been difficult to apply in practice. We compared the performances of the proposed method with other methods (as-treated, ITT, and g-estimation analysis) through simulation studies, which showed that the intensity score estimator was unbiased and more efficient. Applying the proposed method to the MEGA study data, several prognostic factors were associated with the process of treatment changes, and after adjusting for these treatment changes, larger treatment effects for pravastatin were observed for both CHD and stroke events. The proposed method provides a valuable and flexible approach for estimating treatment effect adjusting for non-random non-compliance.     

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.     

Generalizability of causal inference in observational studies under retrospective convenience sampling

Many observational studies adopt what we call retrospective convenience sampling (RCS). With the sample size in each arm prespecified, RCS randomly selects subjects from the treatment‐inclined subpopulation into the treatment arm and those from the control‐inclined into the control arm. Samples in each arm are representative of the respective subpopulation, but the proportion of the 2 subpopulations is usually not preserved in the sample data. We show in this work that, under RCS, existing causal effect estimators actually estimate the treatment effect over the sample population instead of the underlying study population. We investigate how to correct existing methods for consistent estimation of the treatment effect over the underlying population. Although RCS is adopted in medical studies for ethical and cost‐effective purposes, it also has a big advantage for statistical inference: When the tendency to receive treatment is low in a study population, treatment effect estimators under RCS, with proper correction, are more efficient than their parallels under random sampling. These properties are investigated both theoretically and through numerical demonstration.     

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

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

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

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

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

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

Matching algorithms for causal inference with multiple treatments

Randomized clinical trials are ideal for estimating causal effects, because the distributions of background covariates are similar in expectation across treatment groups. When estimating causal effects using observational data, matching is a commonly used method to replicate the covariate balance achieved in a randomized clinical trial. Matching algorithms have a rich history dating back to the mid-1900s but have been used mostly to estimate causal effects between two treatment groups. When there are more than two treatments, estimating causal effects requires additional assumptions and techniques. We propose several novel matching algorithms that address the drawbacks of the current methods, and we use simulations to compare current and new methods. All of the methods display improved covariate balance in the matched sets relative to the prematched cohorts. In addition, we provide advice to investigators on which matching algorithms are preferred for different covariate distributions.     

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.