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

Doubly robust estimation combines a form of outcome regression with a model for the exposure (i.e., the propensity score) to estimate the causal effect of an exposure on an outcome. When used individually to estimate a causal effect, both outcome regression and propensity score methods are unbiased only if the statistical model is correctly specified. The doubly robust estimator combines these 2 approaches such that only 1 of the 2 models need be correctly specified to obtain an unbiased effect estimator. In this introduction to doubly robust estimators, the authors present a conceptual overview of doubly robust estimation, a simple worked example, results from a simulation study examining performance of estimated and bootstrapped standard errors, and a discussion of the potential advantages and limitations of this method. The supplementary material for this paper, which is posted on the Journal's Web site (http://aje.oupjournals.org/), includes a demonstration of the doubly robust property (Web Appendix 1) and a description of a SAS macro (SAS Institute, Inc., Cary, North Carolina) for doubly robust estimation, available for download at http://www.unc.edu/~mfunk/dr/.

Summary A treatment regime is a rule that assigns a treatment, among a set of possible treatments, to a patient as a function of his/her observed characteristics, hence “personalizing” treatment to the patient. The goal is to identify the optimal treatment regime that, if followed by the entire population of patients, would lead to the best outcome on average. Given data from a clinical trial or observational study, for a single treatment decision, the optimal regime can be found by assuming a regression model for the expected outcome conditional on treatment and covariates, where, for a given set of covariates, the optimal treatment is the one that yields the most favorable expected outcome. However, treatment assignment via such a regime is suspect if the regression model is incorrectly specified. Recognizing that, even if misspecified, such a regression model defines a class of regimes, we instead consider finding the optimal regime within such a class by finding the regime the optimizes an estimator of overall population mean outcome. To take into account possible confounding in an observational study and to increase precision, we use a doubly robust augmented inverse probability weighted estimator for this purpose. Simulations and application to data from a breast cancer clinical trial demonstrate the performance of the method.

SUMMARYThere is considerable debate regarding whether and how covariate adjusted analyses should be used in the comparison of treatments in randomized clinical trials. Substantial baseline covariate information is routinely collected in such trials, and one goal of adjustment is to exploit covariates associated with outcome to increase precision of estimation of the treatment effect. However, concerns are routinely raised over the potential for bias when the covariates used are selected post hoc; and the potential for adjustment based on a model of the relationship between outcome, covariates, and treatment to invite a "fishing expedition" for that leading to the most dramatic effect estimate. By appealing to the theory of semiparametrics, we are led naturally to a characterization of all treatment effect estimators and to principled, practically-feasible methods for covariate adjustment that yield the desired gains in efficiency and that allow covariate relationships to be identified and exploited while circumventing the usual concerns. The methods and strategies for their implementation in practice are presented. Simulation studies and an application to data from an HIV clinical trial demonstrate the performance of the techniques relative to existing methods.

Great Britain, 1995. No. of pages: xv + 359. Price: S32. ISBN: 0-412- 98431-9The central statistical model of this book is the non-linear mixed model for continuous responses, that is, a non-linear random regression model. The data structure can be thought of as one or more groups of individuals who are followed over a period of time over which some response is measured repeatedly.The book consists of 12 chapters, which can be roughly grouped into three categories: prerequisites; model formulation and inference, and applications.The prerequisite part comprises the first three chapters: the introduction; the fundamentals of ordinary non-linear regression, and the theory of hierarchical linear models (classical two-stage random regression models).The introductory chapter serves as a very fine appetizer, offering a series of examples with very clearly stated purposes.Chapter 2 gives an outline of the theory of ordinary non-linear regression (with only one 'individual'), with focus on the modelling of inhomogeneity in the variance as a parametric function of the mean. Useful practical and computational guidance is given, although in no detail.Chapter 3 gives an almost self-contained exposition of the classical normal theory inference in the hierarchical (two-stage) linear model (linear mixed effects model). The computation of ML and REML and best linear unbiased predictors (BLUP) for the random effects are given, and the relation to Bayes estimation is discussed. Numerical algorithms (Newton-Raphson, EM) are discussed, together with implementations in the standard software packages SAS, BMDP and S + . The last page gives an extremely useful bibliography.Chapters 2 and 3 together form the natural building bricks for the two-stage non-linear models, which are introduced in Chapter 4. The first stage of the model describes the mean value behaviour of a single individual as a non-linear function, depending on possible covariates, and with a specified variance and covariance structure which in theory can be quite arbitrary. In the second stage the parameters from the various individuals are assumed to follow some specified distribution, typically a normal distribution with unknown parameters.To be specific, the models considered allow for the following: A key issue in the use of such general models in applied work is the possibility of making adequate diagnostics. At present, few appropriate methods exist, and this book does not fill this gap. There are, however, some suggestions, for instance in the advice of the specification of the distribution of random effects. Whereas the normal distribution is usually assumed without much specific reason, it is here advocated to specify a more flexible class of distributions, allowing for bimodality, so that a possible inhomogeneity in the random effects can be detected and possibly included in the model as an explicit dependence on some covariate.A model this flexible can easily cause identifiability problems, as we approach the limit of information available in the data. Ofte...

Normality of random effects is a routine assumption for the linear mixed model, but it may be unrealistic, obscuring important features of among-individual variation. We relax this assumption by approximating the random effects density by the seminonparameteric (SNP) representation of Gallant and Nychka (1987, Econometrics 55, 363-390), which includes normality as a special case and provides flexibility in capturing a broad range of nonnormal behavior, controlled by a user-chosen tuning parameter. An advantage is that the marginal likelihood may be expressed in closed form, so inference may be carried out using standard optimization techniques. We demonstrate that standard information criteria may be used to choose the tuning parameter and detect departures from normality, and we illustrate the approach via simulation and using longitudinal data from the Framingham study.

SummaryThe primary goal of a randomized clinical trial is to make comparisons among two or more treatments. For example, in a two-arm trial with continuous response, the focus may be on the difference in treatment means; with more than two treatments, the comparison may be based on pairwise differences. With binary outcomes, pairwise odds-ratios or log-odds ratios may be used. In general, comparisons may be based on meaningful parameters in a relevant statistical model. Standard analyses for estimation and testing in this context typically are based on the data collected on response and treatment assignment only. In many trials, auxiliary baseline covariate information may also be available, and it is of interest to exploit these data to improve the efficiency of inferences. Taking a semiparametric theory perspective, we propose a broadly-applicable approach to adjustment for auxiliary covariates to achieve more efficient estimators and tests for treatment parameters in the analysis of randomized clinical trials. Simulations and applications demonstrate the performance of the methods.

Abstract Considerable recent interest has focused on doubly robust estimators for a population mean response in the presence of incomplete data, which involve models for both the propensity score and the regression of outcome on covariates. The usual doubly robust estimator may yield severely biased inferences if neither of these models is correctly specified and can exhibit nonnegligible bias if the estimated propensity score is close to zero for some observations. We propose alternative doubly robust estimators that achieve comparable or improved performance relative to existing methods, even with some estimated propensity scores close to zero.