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The regression discontinuity (RD) design (Thistlewaite & Campbell, 1960; Cook, 2008) provides a framework to identify and estimate causal effects from a non-randomized design. Each subject of a RD design is assigned to the treatment (versus assignment to a non-treatment) whenever her/his observed value of the assignment variable equals or exceeds a cutoff value. The RD design provides a "locally-randomized experiment" under remarkably mild conditions, so that the causal effect of treatment outcomes versus non-treatment outcomes can be identified and estimated at the cutoff (Lee, 2008). Such effect estimates are similar to those of a randomized study (Goldberger, 2008/1972). As a result, since 1997, at least 74 RD-based empirical studies have emerged in the ?fields of education, political science, psychology, economics, statistics, criminology, and health science (see van der Klaauw, 2008; Lee & Lemieux, 2010; Bloom, 2012; Wong et al. 2013; Li et al., 2013). Polynomial and local linear models are standard for RD designs (Bloom, 2012; Imbens & Lemieux, 2008). However, these models can produce biased causal effect estimates, due to the presence of outliers of treatment outcomes; and/or due to incorrect choices of the bandwidth parameter for the local linear model. Currently, the correct choice of bandwidth has only been justified by large-sample theory (Imbens & Kalyanaraman, 2012), and the local linear model for quantile regression (Frandsen et al., 2012) suffers from the "quantile crossing" problem. The authors introduce a novel formulation of their Bayesian nonparametric regression model (BLIND, 2012), which provides causal inference for RD designs. It is an infi?nite-mixture model, that allows the entire probability density of the outcome variable to change ?flexibly as a function of the assignment variable. Moreover, the Bayesian model can provide inferences of causal effects, in terms of how the treatment variable impacts the mean, variance, a quantile, distribution function, probability density, hazard function, and/or any other chosen functional of the outcome variable. Moreover, the accurate causal effect estimation relies on a predictively-accurate model for the data. The Bayesian nonparametric regression model attained best overall predictive performance, over many real data sets, compared to many other regression models (BLIND, 2012). Finally, the authors illustrate their Bayesian model through the causal analysis of two real educational data sets. Figures are appended.

Karabatsos and Walker (2011) introduced a new Bayesian nonparametric (BNP) regression model. Through analyses of real and simulated data, they showed that the BNP regression model outperforms other parametric and nonparametric regression models of common use, in terms of predictive accuracy of the outcome (dependent) variable. The other, outperformed, regression models include random-effects/hierarchical linear and generalized linear models, when the random effects were assumed to be normally-distributed (Laird & Ware, 1982; Breslow & Clayton 1993), and when the random effects were more generally modeled by a nonparametric, Dirichlet process (DP) mixture prior (Kleinman & Ibrahim, 1998a,1998b). The authors argue that the new Bayesian nonparametric (BNP) regression model provides a novel, richer, and more valid approach to causal inference, which allows the researcher to investigate how treatments causally change the entire distribution (density) of (potential) outcomes, including not only the mean, but also other features of the outcome variable, such as quantiles (e.g., median, 10th percentile), and the variance. They illustrate the BNP model through the analysis of observational data, to estimate the causal effect of exposure to excellent high school math education (versus non-exposure, the control), on ACT math achievement. In the data analysis, they also compare the predictive accuracy of the new BNP model against other regression models. These other models assume symmetric distributions for the outcomes, and for the inverse-link function of the propensity score model (when specified), and have been recommended for causal inference from observational data. The other models include the normal linear regression model, having one interaction between (1) subject (pre-treatment) covariates, (2) treatment indicators, and (3) indicators of (greater than or equal) 5 matched groups of subjects, formed either by subclassification (Rosenbaum & Rubin, 1984) or optimal full matching on the estimated propensity score. They also compare with the BART model, which provides a very flexible regression of observed outcomes on the treatment variable and the covariates. Extensive data-based simulation studies have shown that, in terms of bias and mean square error in causal effect estimation, these linear regression models and BART outperform normal linear regression of outcomes using (1) propensity-score-based pair-matching or subclassification alone, (2) treatment indicators and estimated propensity scores as covariates, and (3) observation weights defined by inverse of propensity score estimates, when the only covariate is a treatment indicator (Robins, et al. 2000), and when the linear model also includes subject covariates (Kang & Schafer, 2007; Schafer & Kang, 2008; Hill, 2011). These results seemed to hold true, especially when both the outcome and propensity score models were misspecified for the data, which, arguably, almost always occurs in practice. Through the analysis of an observational data set on math achievement, the authors showed that the new BNP regression model can provide richer causal inferences with higher predictive accuracy, compared to typical causal models which focus inference on the mean outcome, and which make restrictive parametric assumptions about the outcome variable and about the propensity score model. The new BNP model allows one to investigate how treatments causally change any interesting aspect of the distribution (density) of (potential) outcomes, in a flexible manner. (Contains 3 tables and 1 figure.) [This research is supported by the Chicago Teacher Partnership Project.]