Augmented Inverse Probability Weighted Estimator Research Articles

Augmented inverse probability weighted estimation for conditional treatment effect

The Augmented Inverse Probability Weighted (AIPW) estimation has been extensively studied in various scenarios across diverse research areas. This investigation explores the asymptotic properties of the AIPW estimator for the conditional average treatment effect. Under various combinations of the parametric, semiparametric, and nonparametric structure in the nuisance propensity score and outcome regression models, we discuss the asymptotic bias and compare the asymptotic variances among the corresponding estimators. The study covers the asymptotic properties with no model misspecified; with either propensity score or outcome regressions locally/globally misspecified; and with all models locally/globally misspecified. To provide a deeper insight into the nature of these estimators and out of curiosity, we reveal the phenomenon that the asymptotic variances, with model-misspecification, could sometimes be even smaller than those with all models correctly specified. We also conduct a numerical study to validate the theoretical results.

Subgroup identification and variable selection for treatment decision making

When treatment effect heterogeneity exists, identifying the subgroup of patients who would benefit from an active treatment relative to a control is an important question. This article focuses on subgroup identification in the presence of a large dimensional set of covariates, with the number of covariates possibly greater than the sample size. We approach this problem from the perspective of optimal treatment decision rules and propose methods that can simultaneously estimate the treatment decision rule and select prescriptive variables important for treatment decision making and subgroup identification. The proposed methods are built within a robust classification framework based on doubly robust augmented inverse probability weighted estimators (AIPWE), hence sharing the robustness property. An L1 (lasso-type) penalty is used within the classification framework to target selection of prescriptive variables. We further propose a backward elimination process for fine-tuning selection. The methods can be conveniently implemented by taking advantage of standard software for logistic regression and lasso. The methods are evaluated by extensive simulation studies which demonstrated the superior and robust performance of the proposed methods relative to existing ones. In addition, the estimated decision rules from the proposed methods are considerably simpler than other methods. We applied various methods to identify the subgroup of patients suitable for each of the two commonly used anticoagulants in terms of bleeding risk for patients with acute myocardial infarction undergoing percutaneous coronary intervention.

Robust method for optimal treatment decision making based on survival data.

Identifying the optimal treatment decision rule, where the best treatment for an individual varies according to his/her characteristics, is of great importance when treatment effect heterogeneity exists. We develop methods for estimating the optimal treatment decision rule based on data with survival time as the primary endpoint. Our methods are based on a flexible semiparametric accelerated failure time model, where only the treatment contrast (ie, the difference in means between treatments) is parameterized and all other aspects are unspecified. An individual's treatment contrast is firstly estimated robustly by an augmented inverse probability weighted estimator (AIPWE). Then the optimal decision rule is estimated by minimizing the loss between the treatment contrast and the AIPWE contrast. Two loss functions with different strategies to account for censoring are proposed. The proposed loss functions distinguish from existing ones in that they are based on treatment contrasts, which completely determine the optimal treatment rule. Our methods can further incorporate a penalty term to select variables that are only important for treatment decision making, while taking advantage of all covariates predictive of outcomes to improve performance. Comprehensive simulation studies have been conducted to evaluate performances of the proposed methods relative to existing methods. The proposed methods are illustrated with an application to the ACTG 175 clinical trial on HIV-infected patients.

Multinomial logistic regression with missing outcome data: An application to cancer subtypes

Many diseases such as cancer and heart diseases are heterogeneous and it is of great interest to study the disease risk specific to the subtypes in relation to genetic and environmental risk factors. However, due to logistic and cost reasons, the subtype information for the disease is missing for some subjects. In this article, we investigate methods for multinomial logistic regression with missing outcome data, including a bootstrap hot deck multiple imputation (BHMI), simple inverse probability weighted (SIPW), augmented inverse probability weighted (AIPW), and expected estimating equation (EEE) estimators. These methods are important approaches for missing data regression. The BHMI modifies the standard hot deck multiple imputation method such that it can provide valid confidence interval estimation. Under the situation when the covariates are discrete, the SIPW, AIPW, and EEE estimators are numerically identical. When the covariates are continuous, nonparametric smoothers can be applied to estimate the selection probabilities and the estimating scores. These methods perform similarly. Extensive simulations show that all of these methods yield unbiased estimators while the complete-case (CC) analysis can be biased if the missingness depends on the observed data. Our simulations also demonstrate that these methods can gain substantial efficiency compared with the CC analysis. The methods are applied to a colorectal cancer study in which cancer subtype data are missing among some study individuals.

Assessing the accuracy of predictive models with interval-censored data.

SummaryWe develop methods for assessing the predictive accuracy of a given event time model when the validation sample is comprised of case n}{}K interval-censored data. An imputation-based, an inverse probability weighted (IPW), and an augmented inverse probability weighted (AIPW) estimator are developed and evaluated for the mean prediction error and the area under the receiver operating characteristic curve when the goal is to predict event status at a landmark time. The weights used for the IPW and AIPW estimators are obtained by fitting a multistate model which jointly considers the event process, the recurrent assessment process, and loss to follow-up. We empirically investigate the performance of the proposed methods and illustrate their application in the context of a motivating rheumatology study in which human leukocyte antigen markers are used to predict disease progression status in patients with psoriatic arthritis.

Improved Doubly Robust Estimation in Marginal Mean Models for Dynamic Regimes

Abstract Doubly robust (DR) estimators are an important class of statistics derived from a theory of semiparametric efficiency. They have become a popular tool in causal inference, including applications to dynamic treatment regimes. The doubly robust estimators for the mean response to a dynamic treatment regime may be conceived through the augmented inverse probability weighted (AIPW) estimating function, defined as the sum of the inverse probability weighted (IPW) estimating function and an augmentation term. The IPW estimating function of the causal estimand via marginal structural model is defined as the complete-case score function for those subjects whose treatment sequence is consistent with the dynamic regime in question divided by the probability of observing the treatment sequence given the subject's treatment and covariate histories. The augmentation term is derived by projecting the IPW estimating function onto the nuisance tangent space and has mean-zero under the truth. The IPW estimator of the causal estimand is consistent if (i) the treatment assignment mechanism is correctly modeled and the AIPW estimator is consistent if either (i) is true or (ii) nested functions of intermediate and final outcomes are correctly modeled. Hence, the AIPW estimator is doubly robust and, moreover, the AIPW is semiparametric efficient if both (i) and (ii) are true simultaneously. Unfortunately, DR estimators can be inferior when either (i) or (ii) is true and the other false. In this case, the misspecified parts of the model can have a detrimental effect on the variance of the DR estimator. We propose an improved DR estimator of causal estimand in dynamic treatment regimes through a technique originally developed by [4] which aims to mitigate the ill-effects of model misspecification through a constrained optimization. In addition to solving a doubly robust system of equations, the improved DR estimator simultaneously minimizes the asymptotic variance of the estimator under a correctly specified treatment assignment mechanism but misspecification of intermediate and final outcome models. We illustrate the desirable operating characteristics of the estimator through Monte Carlo studies and apply the methods to data from a randomized study of integrilin therapy for patients undergoing coronary stent implantation. The methods proposed here are new and may be used to further improve personalized medicine, in general.

Improved methods for moment restriction models with data combination and an application to two‐sample instrumental variable estimation

AbstractCombining‐100 information from multiple samples is often needed in biomedical and economic studies, but differences between these samples must be appropriately taken into account in the analysis of the combined data. We study the estimation for moment restriction models with data combined from two samples under an ignorability‐type assumption while allowing for different marginal distributions of variables common to both samples. Suppose that an outcome regression (OR) model and a propensity score (PS) model are specified. By leveraging semi‐parametric efficiency theory, we derive an augmented inverse probability‐weighted (AIPW) estimator that is locally efficient and doubly robust with respect to these models. Furthermore, we develop calibrated regression and likelihood estimators that are not only locally efficient and doubly robust but also intrinsically efficient in achieving smaller variances than the AIPW estimator when the PS model is correctly specified but the OR model may be mispecified. As an important application, we study the two‐sample instrumental variable problem and derive the corresponding estimators while allowing for incompatible distributions of variables common to the two samples. Finally, we provide a simulation study and an econometric application on public housing projects to demonstrate the superior performance of our improved estimators. The Canadian Journal of Statistics 48: 259–284; 2020 © 2019 Statistical Society of Canada

Achieving semiparametric efficiency bound in longitudinal data analysis with dropouts

In longitudinal data analysis with dropouts, despite its local efficiency in theory, the augmented inverse probability weighted (AIPW) estimator hardly achieves the semiparametric efficiency bound in practice, even if the variance–covariance of the longitudinal outcomes is correctly modeled. In this paper, we propose a method based on conditional empirical likelihood. Assuming missing at random (MAR) mechanism, our estimator is doubly robust and locally efficient. Unlike the AIPW estimator, our estimator does not require to model any second moments, including the variance–covariance of the longitudinal outcomes, in order to achieve the semiparametric efficiency bound.

Missing data approaches for probability regression models with missing outcomes with applications.

In this paper, we investigate several well known approaches for missing data and their relationships for the parametric probability regression model Pβ(Y|X) when outcome of interest Y is subject to missingness. We explore the relationships between the mean score method, the inverse probability weighting (IPW) method and the augmented inverse probability weighted (AIPW) method with some interesting findings. The asymptotic distributions of the IPW and AIPW estimators are derived and their efficiencies are compared. Our analysis details how efficiency may be gained from the AIPW estimator over the IPW estimator through estimation of validation probability and augmentation. We show that the AIPW estimator that is based on augmentation using the full set of observed variables is more efficient than the AIPW estimator that is based on augmentation using a subset of observed variables. The developed approaches are applied to Poisson regression model with missing outcomes based on auxiliary outcomes and a validated sample for true outcomes. We show that, by stratifying based on a set of discrete variables, the proposed statistical procedure can be formulated to analyze automated records that only contain summarized information at categorical levels. The proposed methods are applied to analyze influenza vaccine efficacy for an influenza vaccine study conducted in Temple-Belton, Texas during the 2000-2001 influenza season.

Rejoinder: Combining biomarkers to optimize patient treatment recommendations

Rejoinder: Combining biomarkers to optimize patient treatment recommendations

Semiparametric proportional mean residual life model with covariates missing at random

In this paper, we consider statistical inference for the proportional mean residual life model when some covariates are missing at random. Simple and augmented inverse probability-weighted estimating equations are used to obtain the estimators of the regression coefficients and baseline mean residual life function. The unknown non-missingness probability and some unknown conditional expectations are estimated by the kernel smoothing technique. We show that the simple inverse probability-weighted estimator with estimated non-missingness probability is more efficient than that with the true non-missingness probability, while the augmented inverse probability-weighted estimator with estimated non-missingness probability and that with the true non-missingness probability have the same efficiency. The asymptotic properties of all the proposed estimators are established. Extensive simulation studies are conducted to examine the finite sample performance of the proposed estimator. At last, the proposed method is applied to the mouse leukaemia data.

Doubly robust estimators of causal exposure effects with missing data in the outcome, exposure or a confounder

We consider the estimation of the causal effect of a binary exposure on a continuous outcome. Confounding and missing data are both likely to occur in practice when observational data are used to estimate this causal effect. In dealing with each of these problems, model misspecification is likely to introduce bias. We present augmented inverse probability weighted (AIPW) estimators that account for both confounding and missing data, with the latter occurring in a single variable only. These estimators have an element of robustness to misspecification of the models used. Our estimators require two models to be specified to deal with confounding and two to deal with missing data. Only one of each of these models needs to be correctly specified. When either the outcome or the exposure of interest is missing, we derive explicit expressions for the AIPW estimator. When a confounder is missing, explicit derivation is complex, so we use a simple algorithm, which can be applied using standard statistical software, to obtain an approximation to the AIPW estimator.

A note on improving the efficiency of inverse probability weighted estimator using the augmentation term

The augmented inverse probability weighted (AIPW) estimator employing the optimal augmentation term is more efficient than the inverse probability weighted (IPW) estimator. However, the AIPW estimator could lose substantial efficiency compared to the IPW estimator when the optimal augmentation term is incorrectly modeled. We propose a modified AIPW (MAIPW) estimator by adapting Tan’s (2010b) “tilde” estimator, which was proposed for structural models, for regression models with missing data. When the missing mechanism is correctly modeled, the proposed MAIPW estimator is more efficient than the IPW estimator, and is more efficient than the AIPW estimator using the same augmentation term, except when the augmentation term is a correct model for the optimal one, in which case both MAIPW and AIPW estimators attain the semiparametric efficiency bound, thus are equally efficient. In addition, like the AIPW estimator, the MAIPW estimator is doubly robust. Through simulation experiments, we compare numerical performances of the MAIPW estimator and some other estimators that attempt to improve efficiency upon the IPW estimator.

Doubly-Robust Parametric Estimation in Moment Conditions Models with Missing-at-Random Data

We consider estimation of finite dimensional parameters defined by a set of moment restrictions when observations on key variables are missing-at-random. We consider the verify-in-sample and verify-out-of-sample cases of Chen, Hong and Tarozzi (2008) and present parametric alternatives to their semiparametric estimators. These alternatives are extension of the Augmented Inverse Probability Weighted (AIPW) estimator of Robins, Rotnitzky and Zhao (1994). We also consider a modification proposed by Cao, Tsiatis and Davidian (2009) and present it as a one-step update of the AIPW estimators. Compared to other parametric estimators, the AIPW estimator and its modification provide additional protection against inconsistency due to parametric misspecification. They are also locally efficient in the absence of misspecification. Simulation results in the context of missing instrumental variables suggest that both estimators are likely to be useful when the researcher is reasonably certain about key components of the parametric specifications.

Joint modeling of missing data due to non-participation and death in longitudinal aging studies.

Specific age-related hypotheses are tested in population-based longitudinal studies. At specific time intervals, both the outcomes of interest and the time-varying covariates are measured. When participants are approached for follow-up, some participants do not provide data. Investigations may show that many have died before the time of follow-up whereas others refused to participate. Some of these non-participants do not provide data at later follow-ups. Few statistical methods for missing data distinguish between 'non-participation' and 'death' among study participants. The augmented inverse probability-weighted estimators are most commonly used in marginal structure models when data are missing at random. Treating non-participation and death as the same, however, may lead to biased estimates and invalid inferences. To overcome this limitation, a multiple inverse probability-weighted approach is presented to account for two types of missing data, non-participation and death, when using a marginal mean model. Under certain conditions, the multiple weighted estimators are consistent and asymptotically normal. Simulation studies will be used to study the finite sample efficiency of the multiple weighted estimators. The proposed method will be applied to study the risk factors associated with the cognitive decline among the aging adults, using data from the Chicago Health and Aging Project (CHAP).