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Uses of the propensity score to obtain estimates of causal effect have been investigated thoroughly under assumptions of linearity and additivity of exposure effect. When the outcome variable is binary relationships such as collapsibility, valid for the linear model, do not always hold. This article examines uses of the propensity score when both exposure and outcome are binary variables and the parameter of interest is the marginal odds ratio. We review stratification and matching by the propensity score when calculating the Mantel-Haenszel estimator and show that it is consistent for neither the marginal nor conditional odds ratio. We also investigate a marginal odds ratio estimator based on doubly robust estimators and summarize its performance relative to other recently proposed estimators under various conditions, including low exposure prevalence and model misspecification. Finally, we apply all estimators to a case study estimating the effect of Medicare plan type on the quality of care received by African-American breast cancer patients.

We present a method for allocating treatment when subjects arrive in sequence. Based on the theory of propensity scores more commonly used in observational studies, the method balances both discrete and continuous covariates without assuming a model for the outcome. Although we allow for a number of possible specifications, we explore some specific instances in depth. The proposed method is compared with previously suggested sequential randomization and allocation procedures with relationships to some well-known methods highlighted. Through simulations, the deterministic version is shown to achieve both covariate balance and near optimum efficiency with minimal assumptions. We also investigate the properties of selected randomized versions with respect to both optimality and selection bias. We conclude with an application to a pilot study on weight loss. The proposed method is shown to be robust to the number of covariates balanced and the marginal and joint distributions of those covariates., (Copyright © 2013 John Wiley & Sons, Ltd.)