In many observational studies, analysts estimate treatment effects using propensity scores, e.g. by matching or sub-classifying on the scores. When some values of the covariates are missing, analysts can use multiple imputation to fill in the missing data, estimate propensity scores based on the m completed datasets, and use the propensity scores to estimate treatment effects. We compare two approaches to implement this process. In the first, the analyst estimates the treatment effect using propensity score matching within each completed data set, and averages the m treatment effect estimates. In the second approach, the analyst averages the m propensity scores for each record across the completed datasets, and performs propensity score matching with these averaged scores to estimate the treatment effect. We compare properties of both methods via simulation studies using artificial and real data. The simulations suggest that the second method has greater potential to produce substantial bias reductions than the first, particularly when the missing values are predictive of treatment assignment.

In logistic regression, separation occurs when a linear combination of the predictors can perfectly classify part or all of the observations in the sample, and as a result, finite maximum likelihood estimates of the regression coefficients do not exist. Gelman et al. (2008) recommended independent Cauchy distributions as default priors for the regression coefficients in logistic regression, even in the case of separation, and reported posterior modes in their analyses. As the mean does not exist for the Cauchy prior, a natural question is whether the posterior means of the regression coefficients exist under separation. We prove theorems that provide necessary and sufficient conditions for the existence of posterior means under independent Cauchy priors for the logit link and a general family of link functions, including the probit link. We also study the existence of posterior means under multivariate Cauchy priors. For full Bayesian inference, we develop a Gibbs sampler based on Pólya-Gamma data augmentation to sample from the posterior distribution under independent Student-t priors including Cauchy priors, and provide a companion R package in the supplement. We demonstrate empirically that even when the posterior means of the regression coefficients exist under separation, the magnitude of the posterior samples for Cauchy priors may be unusually large, and the corresponding Gibbs sampler shows extremely slow mixing. While alternative algorithms such as the No-U-Turn Sampler in Stan can greatly improve mixing, in order to resolve the issue of extremely heavy tailed posteriors for Cauchy priors under separation, one would need to consider lighter tailed priors such as normal priors or Student-t priors with degrees of freedom larger than one.

Abstract. To limit disclosures, statistical agencies and other data disseminators can release partially synthetic, public use microdata sets. These comprise the units originally surveyed; but some collected values, for example, sensitive values at high risk of disclosure or values of key identifiers, are replaced with multiple draws from statistical models. Because the original records are on the file, there remain risks of identifications. In this paper, we describe how to evaluate identification disclosure risks in partially synthetic data, accounting for released information from the multiple datasets, the model used to generate synthetic values, and the approach used to select values to synthesize. We illustrate the computations using the Survey of Youths in Custody.

Abstract[In many observational studies, researchers estimate causal effects using propensity scores, e.g., by matching or sub-classifying on the scores. Estimation of propensity scores is complicated when some values of the covariates are missing. We propose to use multiple imputation to create completed datasets, from which propensity scores can be estimated, with a general location mixture model. The model assumes that the control units are a latent mixture of (i) units whose covariates are drawn from the same distributions as the treated AbstractIn many observational studies, researchers estimate causal effects using propensity scores, e.g., by matching or sub-classifying on the scores. Estimation of propensity scores is complicated when some values of the covariates are missing. We propose to use multiple imputation to create completed datasets, from which propensity scores can be estimated, with a general location mixture model. The model assumes that the control units are a latent mixture of (i) units whose covariates are drawn from the same distributions as the treated units' covariates and (ii) units whose covariates are drawn from different distributions. This formulation reduces the influence of control units outside the treated units' region of the covariate space on the estimation of parameters in the imputation model, which can result in more plausible imputations and better balance in the true covariate distributions. We illustrate the benefits of 1 the latent class modeling approach with simulations and with an observational study of the effect of breast feeding on children's cognitive abilities.

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