Inverse Probability Weighting Estimators Research Articles

Confounding-adjustment methods for the causal difference in medians

BackgroundWith continuous outcomes, the average causal effect is typically defined using a contrast of expected potential outcomes. However, in the presence of skewed outcome data, the expectation (population mean) may no longer be meaningful. In practice the typical approach is to continue defining the estimand this way or transform the outcome to obtain a more symmetric distribution, although neither approach may be entirely satisfactory. Alternatively the causal effect can be redefined as a contrast of median potential outcomes, yet discussion of confounding-adjustment methods to estimate the causal difference in medians is limited. In this study we described and compared confounding-adjustment methods to address this gap.MethodsThe methods considered were multivariable quantile regression, an inverse probability weighted (IPW) estimator, weighted quantile regression (another form of IPW) and two little-known implementations of g-computation for this problem. Methods were evaluated within a simulation study under varying degrees of skewness in the outcome and applied to an empirical study using data from the Longitudinal Study of Australian Children.ResultsSimulation results indicated the IPW estimator, weighted quantile regression and g-computation implementations minimised bias across all settings when the relevant models were correctly specified, with g-computation additionally minimising the variance. Multivariable quantile regression, which relies on a constant-effect assumption, consistently yielded biased results. Application to the empirical study illustrated the practical value of these methods.ConclusionThe presented methods provide appealing avenues for estimating the causal difference in medians.

Estimating average treatment effects for clustered RCTs with recruitment bias.

In clustered randomized controlled trials (RCTs), sample recruitment is often conducted after cluster randomization. This timing can lead to recruitment bias if access to the intervention affects the composition of study-eligible cluster entrants and study consenters. This article develops a potential outcomes framework in such settings that yields a causal estimand that pertains to the always-recruited in either research condition. A consistent inverse probability weighting (IPW) estimator is developed using data on recruits only, and a generalized estimating equations approach is used to obtain robust clustered SE estimators that adjust for estimation error in the IPW weights. A simple data collection strategy is discussed to improve the predictive accuracy of the logit propensity score models. Simulations show that the IPW estimator achieves nominal confidence interval coverage under the assumed identification conditions. An empirical application demonstrates the methods using data from an RCT testing the effects of a behavioral health intervention in schools. An R program for estimation is available for download.

Estimation in optimal treatment regimes based on mean residual lifetimes with right-censored data.

An optimal individualized treatment regime (ITR) is a decision rule in allocating the best treatment to each patient and, hence, maximizing overall benefits. In this paper, we propose a novel framework based on nonparametric inverse probability weighting (IPW) and augmented inverse probability weighting (AIPW) estimators of the value function when the data are subject to right censoring. In contrast to most existing approaches that are designed to maximize the expected survival time under a binary treatment framework, the proposed method targets maximizing the mean residual lifetime of patients. Specifically, the proposed IPW method searches the optimal ITR by maximizing an estimator for the overall population outcome directly, without specifying the regression model for the conditional mean residual lifetime, whereas the AIPW method integrates the model information of the mean residual lifetime to improve the robustness. Furthermore, to overcome the computational difficulty in a nonsmooth value estimator, smoothed IPW and AIPW estimators are constructed. In theory, we establish the asymptotic properties of the proposed method under suitable regularity conditions. The empirical performances of the proposed IPW and AIPW estimators are evaluated using simulation studies and are further illustrated with an application to the real-world data set from the Acquired Immunodeficiency Syndrome Clinical Trial Group Protocol 175 (ACTG175).

Estimating Causal Effects of HIV Prevention Interventions with Interference in Network-based Studies among People Who Inject Drugs.

Evaluating causal effects in the presence of interference is challenging in network-based studies of hard-to-reach populations. Like many such populations, people who inject drugs (PWID) are embedded in social networks and often exert influence on others in their network. In our setting, the study design is observational with a non-randomized network-based HIV prevention intervention. Information is available on each participant and their connections that confer possible HIV risk through injection and sexual behaviors. We considered two inverse probability weighted (IPW) estimators to quantify the population-level spillover effects of non-randomized interventions on subsequent health outcomes. We demonstrated that these two IPW estimators are consistent, asymptotically normal, and derived a closed-form estimator for the asymptotic variance, while allowing for overlapping interference sets (groups of individuals in which the interference is assumed possible). A simulation study was conducted to evaluate the finite-sample performance of the estimators. We analyzed data from the Transmission Reduction Intervention Project, which ascertained a network of PWID and their contacts in Athens, Greece, from 2013 to 2015. We evaluated the effects of community alerts on subsequent HIV risk behavior in this observed network, where the connections or links between participants were defined by using substances or having unprotected sex together. In the study, community alerts were distributed to inform people of recent HIV infections among individuals in close proximity in the observed network. The estimates of the risk differences for spillover using either IPW estimator demonstrated a protective effect. The results suggest that HIV risk behavior could be mitigated by exposure to a community alert when an increased risk of HIV is detected in the network.

On variance estimation of target population created by inverse probability weighting

ABSTRACT Inverse probability weighting (IPW) is frequently used to reduce or minimize the observed confounding in observational studies. IPW creates a pseudo-sample by weighting each individual by the inverse of the conditional probability of receiving the treatment level that he/she has actually received. In the pseudo-sample there is no variation among the multiple individuals generated by weighting the same individual in the original sample. This would reduce the variability of the data and therefore bias the variance estimate in the target population. Conventional variance estimation methods for IPW estimators generally ignore this underestimation and tend to produce biased estimates of variance. We here propose a more reasonable method that incorporates this source of variability by using parametric bootstrapping based on intra-stratum variability estimates. This approach firstly uses propensity score stratification and intra-stratum standard deviation to approximate the variability among multiple individuals generated based on a single individual whose propensity score falls within the corresponding stratum. The parametric bootstrapping is then used to incorporate the target variability by re-generating outcomes after adding a random error term to the original data. The performance of the proposed method is compared with three existing methods including the naïve model-based variance estimator, the nonparametric bootstrap variance estimator, and the robust variance estimator in the simulation section. An example of patients with sarcopenia is used to illustrate the implementation of the proposed approach. According to the results, the proposed approach has desirable statistical properties and can be easily implemented using the provided R code.

Inference for covariate-adjusted time-dependent prognosticaccuracy measures.

Evaluating the prognostic performance of candidate markers for future disease onset or progression is one of the major goals in medical research. A marker's prognostic performance refers to how well it separates patients at the high or low risk of a future disease state. Often the discriminative performance of a marker is affected by the patient characteristics (covariates). Time-dependent receiver operating characteristic (ROC) curves that ignore the informativeness of the covariates will lead to biased estimates of the accuracy parameters. We propose a time-dependent ROC curve that accounts for the informativeness of the covariates in the case of censored data. We propose inverse probability weighted (IPW) estimators for estimating the proposed accuracy parameters. We investigate the performance of the IPW estimators through simulation studies and real-life data analysis.

Inverse Probability Weighting Estimation in Completely Randomized Experiments

In addition to treatment assignments and observed outcomes, covariate information is often available prior to randomization in completely randomized experiments that compare an active treatment versus control. The analysis of covariance (ANCOVA) method is commonly applied to adjust for baseline covariates in order to improve precision. We focus on making propensity score-based adjustment to covariates under the completely randomized design in a finite population of experimental units with two treatment groups. We study inverse probability weighting (IPW) estimation of the finite-population average treatment effect for a general class of working propensity score models, which includes generalized linear models for binary data. We provide randomization-based asymptotic analysis of the propensity score approach and explore the finite-population asymptotic behaviors of two IPW estimators of the average treatment effect. We identify a condition under which propensity score-based covariate adjustment is asymptotically equivalent to an ANCOVA-based covariate adjustment and improves precision compared with a simple unadjusted comparison between treatment and control arms. In particular, when the working propensity score is fitted by a generalized linear model for binary data with an intercept term, the asymptotic variance of the IPW estimators is the same for any link function, including identity link, logit link, probit link, and complementary log-log link. We demonstrate these methods using an HIV clinical trial and a post-traumatic stress disorder study. Finally, we present a simulation study comparing the finite-sample performance of IPW and other methods for both continuous and binary outcomes. Supplementary materials for this article are available online.

Model misspecification and bias for inverse probability weighting estimators of average causal effects.

Commonly used semiparametric estimators of causal effects specify parametric models for the propensity score (PS) and the conditional outcome. An example is an augmented inverse probability weighting (IPW) estimator, frequently referred to as a doubly robust estimator, because it is consistent if at least one of the two models is correctly specified. However, in many observational studies, the role of the parametric models is often not to provide a representation of the data-generating process but rather to facilitate the adjustment for confounding, making the assumption of at least one true model unlikely to hold. In this paper, we propose a crude analytical approach to study the large-sample bias of estimators when the models are assumed to be approximations of the data-generating process, namely, when all models are misspecified. We apply our approach to three prototypical estimators of the average causal effect, two IPW estimators, using a misspecified PS model, and an augmented IPW (AIPW) estimator, using misspecified models for the outcome regression (OR) and the PS. For the two IPW estimators, we show that normalization, in addition to having a smaller variance, also offers some protection against bias due to model misspecification. To analyze the question of when the use of two misspecified models is better than one we derive necessary and sufficient conditions for when the AIPW estimator has a smaller bias than a simple IPW estimator and when it has a smaller bias than an IPW estimator with normalized weights. If the misspecification of the outcome model is moderate, the comparisons of the biases of the IPW and AIPW estimators show that the AIPW estimator has a smaller bias than the IPW estimators. However, all biases include a scaling with the PS-model error and we suggest caution in modeling the PS whenever such a model is involved. For numerical and finite sample illustrations, we include three simulation studies and corresponding approximations of the large-sample biases. In a dataset from the National Health and Nutrition Examination Survey, we estimate the effect of smoking on blood leadlevels.

Leveraging auxiliary data to improve precision in inverse probability-weighted analyses

PurposeTo demonstrate improvements in the precision of inverse probability-weighted estimators by use of auxiliary variables, i.e., determinants of the outcome that are independent of treatment, missingness or selection. MethodsFirst with simulated data, and then with public data from the National Health and Nutrition Examination Survey (NHANES), we estimated the mean of a continuous outcome using inverse probability weights to account for informative missingness. We assessed gains in precision resulting from the inclusion of auxiliary variables in the model for the weights. We compared the performance of robust and nonparametric bootstrap variance estimators in this setting. ResultsWe found that the inclusion of auxiliary variables reduced the empirical variance of inverse probability-weighted estimators. However, that reduction was not captured in standard errors computed using the robust variance estimator, which is widely used in weighted analyses due to the non-independence of weighted observations. In contrast, a nonparametric bootstrap estimator properly captured the precision gain. ConclusionsEpidemiologists can leverage auxiliary data to improve the precision of weighted estimators by using bootstrap variance estimation, or a closed-form variance estimator that properly accounts for the estimation of the weights, in place of the standard robust variance estimator.

Risk of Cardiovascular Events After COVID-19

We aimed to determine absolute and relative risks of either symptomatic or asymptomatic SARS-CoV-2 infection for late cardiovascular (CV) events and all-cause mortality. We conducted a retrospective double cohort study of patients with either symptomatic or asymptomatic SARS-CoV-2 infection (COVID-19+ cohort) and its documented absence (COVID-19− cohort). The study investigators drew a simple random sample of records from all patients under the Oregon Health & Science University Healthcare (n = 65,585), with available COVID-19 test results, performed March 1, 2020 to September 13, 2020. Exclusion criteria were age <18 years and no established Oregon Health & Science University care. The primary outcome was a composite of CV morbidity and mortality. All-cause mortality was the secondary outcome. The study population included 1,355 patients (mean age 48.7 ± 20.5 years; 770 women [57%], 977 White non-Hispanic [72%]; 1,072 ensured [79%]; 563 with CV disease history [42%]). During a median 6 months at risk, the primary composite outcome was observed in 38 of 319 patients who were COVID-19+ (12%) and 65 of 1,036 patients who were COVID-19− (6%). In the Cox regression, adjusted for demographics, health insurance, and reason for COVID-19 testing, SARS-CoV-2 infection was associated with the risk for primary composite outcome (hazard ratio 1.71, 95% confidence interval 1.06 to 2.78, p = 0.029). Inverse probability-weighted estimation, conditioned for 31 covariates, showed that for every patient who was COVID-19+, the average time to all-cause death was 65.5 days less than when all these patients were COVID-19−: average treatment effect on the treated −65.5 (95% confidence interval −125.4 to −5.61) days, p = 0.032. In conclusion, either symptomatic or asymptomatic SARS-CoV-2 infection is associated with an increased risk for late CV outcomes and has a causal effect on all-cause mortality in a late post-COVID-19 period.

Concordance indices with left-truncated and right-censored data.

In the context of time-to-event analysis, a primary objective is to model the risk of experiencing a particular event in relation to a set of observed predictors. The Concordance Index (C-Index) is a statistic frequently used in practice to assess how well such models discriminate between various risk levels in a population. However, the properties of conventional C-Index estimators when applied to left-truncated time-to-event data have not been well studied, despite the fact that left-truncation is commonly encountered in observational studies. We show that the limiting values of the conventional C-Index estimators depend on the underlying distribution of truncation times, which is similar to the situation with right-censoring as discussed in Uno etal. (2011) [On the C-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data. Statistics in Medicine 30(10), 1105-1117]. We develop a new C-Index estimator based on inverse probability weighting (IPW) that corrects for this limitation, and we generalize this estimator to settings with left-truncated and right-censored data. The proposed IPW estimators are highly robust to the underlying truncation distribution and often outperform the conventional methods in terms of bias, mean squared error, and coverage probability. We apply these estimators to evaluate a predictive survival model for mortality among patients with end-stage renaldisease.

New weighting methods when cases are only a subset of events in a nested case-control study.

Nested case control (NCC) is a sampling method widely used for developing and evaluating risk models with expensive biomarkers on large prospective cohort studies. In a typical NCC design, biomarker values are obtained on a subcohort, where cases consist of all the events (subjects who experience the event during the follow-up). However, when the number of events is not small, due to the cost and limited availability of biospecimen, one may select only a subset of events as cases. We refer to such a variation as the untypical NCC. Unfortunately, existing inverse probability weighted (IPW) estimators for the untypical NCC are biased, and they only focus on relative risk parameters under the proportional hazards (PH) model. In this manuscript, we propose new weighting methods that produce consistent IPW estimators for not only relative risk parameters but also several metrics that evaluate a risk model's predictive performance. We also provide the inference procedure via perturbation resampling, which captures all the variance and between-subject covariance induced by the sampling processes for both case and control selections. In addition, our methods are not limited to the PH model, and they can be applied to the time-specific generalized linear model. Under the typical NCC design, our new weights are equivalent to the weight proposed by Samuelsen; under the untypical NCC, the IPW estimators using our weights have smaller bias and variance than the existing methods. We will demonstrate this improved performance via both analytical and numerical investigations.

Goodness-of-fit tests for a logistic regression model with missing covariates

Model checking for logistic regression with covariates missing at random is considered. Based on the ideas of Copas (1989) and Osius and Rojek (1992) and studies of Homser et al. (1997), proposed are the two-type goodness-of-fit tests, Pearson chi-squared and unweighted residual sum-of-squares tests, in which their test statistics are centralized by subtracting their estimated mean to be mean-zero-form test statistics via the inverse probability weighting (IPW) and nonparametric multiple imputation (MI) methods to solve the missing value problem. The asymptotic properties of these test statistics are established under the null hypothesis and some regularity conditions. The test statistics conducted by using the IPW and MI estimators are asymptotically equivalent. Proposed are the IPW method and two bootstrap re-sampling approaches for estimation of the variances of the proposed test statistics to solve the issue of underestimating their variances by the MI method of Rubin (1987). Simulation studies are carried out to assess the finite-sample power performances of these proposed tests. Two real data examples are used to illustrate the applicability of the proposed tests.

Optimal sampling for design-based estimators of regression models.

Two-phase designs measure variables of interest on a subcohort where the outcome and covariates are readily available or cheap to collect on all individuals in the cohort. Given limited resource availability, it is of interest to find an optimal design that includes more informative individuals in the final sample. We explore the optimal designs and efficiencies for analyses by design-based estimators. Generalized raking is an efficient class of design-based estimators, and they improve on the inverse-probability weighted (IPW) estimator by adjusting weights based on the auxiliary information. We derive a closed-form solution of the optimal design for estimating regression coefficients from generalized raking estimators. We compare it with the optimal design for analysis via the IPW estimator and other two-phase designs in measurement-error settings. We consider general two-phase designs where the outcome variable and variables of interest can be continuous or discrete. Our results show that the optimal designs for analyses by the two classes of design-based estimators can be very different. The optimal design for analysis via the IPW estimator is optimal for IPW estimation and typically gives near-optimal efficiency for generalized raking estimation, though we show there is potential improvement in some settings.

Early Coronary Angiography Is Associated with Improved 30-Day Outcomes among Patients with Out-of-Hospital Cardiac Arrest.

We evaluated the association between early coronary angiography (CAG) and outcomes in resuscitated out-of-hospital cardiac arrest (OHCA) patients, by linking data from the Singapore Pan-Asian Resuscitation Outcomes Study, with a national registry of cardiac procedures. The 30-day survival and neurological outcome were compared between patients undergoing early CAG (within 1-calender day), versus patients not undergoing early CAG. Inverse probability weighted estimates (IPWE) adjusted for non-randomized CAG. Of 976 resuscitated OHCA patients of cardiac etiology between 2011–2015 (mean(SD) age 64(13) years, 73.7% males), 337 (34.5%) underwent early CAG, of whom, 230 (68.2%) underwent PCI. Those who underwent early CAG were significantly younger (60(12) vs. 66(14) years old), healthier (42% vs. 59% with heart disease; 29% vs. 44% with diabetes), more likely males (86% vs. 67%), and presented with shockable rhythms (69% vs. 36%), compared with those who did not. Early CAG with PCI was associated with better survival and neurological outcome (adjusted odds ratio 1.91 and 1.82 respectively), findings robust to IPWE adjustment. The rates of bleeding and stroke were similar. CAG with PCI within 24 h was associated with improved clinical outcomes after OHCA, without increasing complications. Further studies are required to identify the characteristics of patients who would benefit most from this invasive strategy.

Assess predictive values of a binary diagnostic test under a nested case–control design

ABSTRACT Predictive values of a binary diagnostic test are often evaluated under a random sample design. When the disease is rare, however, such a design might not be as efficient as a nested case–control design where the cases are oversampled from a large existing cohort. Under a nested case–control design, direct proportion estimators of predictive values are biased because cases are oversampled. Consistent estimates of predictive values can be easily obtained by inverse probability weighting (IPW) method. The only difficulty with these IPW estimators has been the absence of expressions for their variances. To fill this gap, in the current paper, we obtain the asymptotic variance formulas for the IPW estimators of predictive values. Unlike their counterparts from weighted logistic regression, our variance formulas take into account the variance of the estimated weights in the IPW estimators of predictive values. We further use the proposed variance formulas to examine the gain in efficiency under a nested case–control design compared with a simple random sampling design. Our results clearly show that when the disease is rare, a nested case–control design can achieve a substantial amount of variance reduction by oversampling cases, compared with a random sample design. Finally, we compare via simulation the accuracy of the proposed variance formulas with the existing methods and illustrate the proposed method by a real data example evaluating the accuracy of D-dimer test.

AIPW: An R Package for Augmented Inverse Probability-Weighted Estimation of Average Causal Effects.

An increasing number of recent studies have suggested that doubly robust estimators with cross-fitting should be used when estimating causal effects with machine learning methods. However, not all existing programs that implement doubly robust estimators support machine learning methods and cross-fitting, or provide estimates on multiplicative scales. To address these needs, we developed AIPW, a software package implementing augmented inverse probability weighting (AIPW) estimation of average causal effects in R (R Foundation for Statistical Computing, Vienna, Austria). Key features of the AIPW package include cross-fitting and flexible covariate adjustment for observational studies and randomized controlled trials (RCTs). In this paper, we use a simulated RCT to illustrate implementation of the AIPW estimator. We also perform a simulation study to evaluate the performance of the AIPW package compared with other doubly robust implementations, including CausalGAM, npcausal, tmle, and tmle3. Our simulation showed that the AIPW package yields performance comparable to that of other programs. Furthermore, we also found that cross-fitting substantively decreases the bias and improves the confidence interval coverage for doubly robust estimators fitted with machine learning algorithms. Our findings suggest that the AIPW package can be a useful tool for estimating average causal effects with machine learning methods in RCTs and observational studies.

Stable inverse probability weighting estimation for longitudinal studies

AbstractWe consider estimation of the average effect of time‐varying dichotomous exposure on outcome using inverse probability weighting (IPW) under the assumption that there is no unmeasured confounding of the exposure–outcome association at each time point. Despite the popularity of IPW, its performance is often poor due to instability of the estimated weights. We develop an estimating equation‐based strategy for the nuisance parameters indexing the weights at each time point, aimed at preventing highly volatile weights and ensuring the stability of IPW estimation. Our proposed approach targets the estimation of the counterfactual mean under a chosen treatment regime and requires fitting a separate propensity score model at each time point. We discuss and examine extensions to enable the fitting of marginal structural models using one propensity score model across all time points. Extensive simulation studies demonstrate adequate performance of our approach compared with the maximum likelihood propensity score estimator and the covariate balancing propensity score estimator.

Inverse probability weighted estimators of vaccine effects accommodating partial interference and censoring.

Estimating population-level effects of a vaccine is challenging because there may be interference, that is, the outcome of one individual may depend on the vaccination status of another individual. Partial interference occurs when individuals can be partitioned into groups such that interference occurs only within groups. In the absence of interference, inverse probability weighted (IPW) estimators are commonly used to draw inference about causal effects of an exposure or treatment. Tchetgen Tchetgen and VanderWeele proposed a modified IPW estimator for causal effects in the presence of partial interference. Motivated by a cholera vaccine study in Bangladesh, this paper considers an extension of the Tchetgen Tchetgen and VanderWeele IPW estimator to the setting where the outcome is subject to right censoring using inverse probability of censoring weights (IPCW). Censoring weights are estimated using proportional hazards frailty models. The large sample properties of the IPCW estimators are derived, and simulation studies are presented demonstrating the estimators' performance in finite samples. The methods are then used to analyze data from the cholera vaccine study.

Variance reduction in the inverse probability weighted estimators for the average treatment effect using the propensity score.

The propensity methodology is widely used in medical research to compare different treatments in designs with a nonrandomized treatment allocation. The inverse probability weighted (IPW) estimators are a primary tool for estimating the average treatment effect but the large variance of these estimators is often a significant concern for their reliable use in practice. Inspired by Rao-Blackwellization, this paper proposes a method to smooth an IPW estimator by replacing the weights in the original estimator by their mean over a distribution of the potential treatment assignment. In our simulation study, the smoothed IPW estimator achieves a substantial variance reduction over its original version with only a small increased bias, for example two-to-sevenfold variance reduction for the three IPW estimators in Lunceford and Davidian [Statistics in Medicine, 23(19), 2937-2960]. In addition, our proposed smoothing can also be applied to the locally efficient and doubly robust estimator for added protection against model misspecification. An implementation in R is provided.