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/.

That conditioning on a common effect of exposure and outcome may cause selection, or collider-stratification, bias is not intuitive. We provide two hypothetical examples to convey concepts underlying bias due to conditioning on a collider. In the first example, fever is a common effect of influenza and consumption of a tainted egg-salad sandwich. In the second example, case-status is a common effect of a genotype and an environmental factor. In both examples, conditioning on the common effect imparts an association between two otherwise independent variables; we call this selection bias.

It is common to present multiple adjusted effect estimates from a single model in a single table. For example, a table might show odds ratios for one or more exposures and also for several confounders from a single logistic regression. This can lead to mistaken interpretations of these estimates. We use causal diagrams to display the sources of the problems. Presentation of exposure and confounder effect estimates from a single model may lead to several interpretative difficulties, inviting confusion of direct-effect estimates with total-effect estimates for covariates in the model. These effect estimates may also be confounded even though the effect estimate for the main exposure is not confounded. Interpretation of these effect estimates is further complicated by heterogeneity (variation, modification) of the exposure effect measure across covariate levels. We offer suggestions to limit potential misunderstandings when multiple effect estimates are presented, including precise distinction between total and direct effect measures from a single model, and use of multiple models tailored to yield total-effect estimates for covariates.

SummaryObjective-Propensity scores for the analysis of observational data are typically estimated using logistic regression. Our objective in this Review was to assess machine learning alternatives to logistic regression which may accomplish the same goals but with fewer assumptions or greater accuracy.Study Design and Setting-We identified alternative methods for propensity score estimation and/or classification from the public health, biostatistics, discrete mathematics, and computer science literature, and evaluated these algorithms for applicability to the problem of propensity score estimation, potential advantages over logistic regression, and ease of use.Results-We identified four techniques as alternatives to logistic regression: neural networks, support vector machines, decision trees (CART), and meta-classifiers (in particular, boosting).Conclusion-While the assumptions of logistic regression are well understood, those assumptions are frequently ignored. All four alternatives have advantages and disadvantages compared with logistic regression. Boosting (meta-classifiers) and to a lesser extent decision trees (particularly CART) appear to be most promising for use in the context of propensity score analysis, but extensive simulation studies are needed to establish their utility in practice.

We derive and compare the operating characteristics of hierarchical and square array-based testing algorithms for case identification in the presence of testing error. The operating characteristics investigated include efficiency (i.e., expected number of tests per specimen) and error rates (i.e., sensitivity, specificity, positive and negative predictive values, per-family error rate, and per-comparison error rate). The methodology is illustrated by comparing different pooling algorithms for the detection of individuals recently infected with HIV in North Carolina and Malawi.