Volume 3, Issue 1 -

In the Regression Discontinuity (RD) design, units are assigned a treatment based on whether their value of an observed covariate is above or below a fixed cutoff. Under the assumption that the distribution of potential confounders changes continuously around the cutoff, the discontinuous jump in the probability of treatment assignment can be used to identify the treatment effect. Although a recent strand of the RD literature advocates interpreting this design as a local randomized experiment, the standard approach to estimation and inference is based solely on continuity assumptions that do not justify this interpretation. In this article, we provide precise conditions in a randomization inference context under which this interpretation is directly justified and develop exact finite-sample inference procedures based on them. Our randomization inference framework is motivated by the observation that only a few observations might be available close enough to the threshold where local randomization is plausible, and hence standard large-sample procedures may be suspect. Our proposed methodology is intended as a complement and a robustness check to standard RD inference approaches. We illustrate our framework with a study of two measures of party-level advantage in U.S. Senate elections, where the number of close races is small and our framework is well suited for the empirical analysis.

In this article, we study the causal inference problem with a continuous treatment variable using propensity score-based methods. For a continuous treatment, the generalized propensity score is defined as the conditional density of the treatment-level given covariates (confounders). The dose–response function is then estimated by inverse probability weighting, where the weights are calculated from the estimated propensity scores. When the dimension of the covariates is large, the traditional nonparametric density estimation suffers from the curse of dimensionality. Some researchers have suggested a two-step estimation procedure by first modeling the mean function. In this study, we suggest a boosting algorithm to estimate the mean function of the treatment given covariates. In boosting, an important tuning parameter is the number of trees to be generated, which essentially determines the trade-off between bias and variance of the causal estimator. We propose a criterion called average absolute correlation coefficient (AACC) to determine the optimal number of trees. Simulation results show that the proposed approach performs better than a simple linear approximation or L2 boosting. The proposed methodology is also illustrated through the Early Dieting in Girls study, which examines the influence of mothers’ overall weight concern on daughters’ dieting behavior.

“ M -Bias,” as it is called in the epidemiologic literature, is the bias introduced by conditioning on a pretreatment covariate due to a particular “ M -Structure” between two latent factors, an observed treatment, an outcome, and a “collider.” This potential source of bias, which can occur even when the treatment and the outcome are not confounded, has been a source of considerable controversy. We here present formulae for identifying under which circumstances biases are inflated or reduced. In particular, we show that the magnitude of M -Bias in linear structural equation models tends to be relatively small compared to confounding bias, suggesting that it is generally not a serious concern in many applied settings. These theoretical results are consistent with recent empirical findings from simulation studies. We also generalize the M -Bias setting (1) to allow for the correlation between the latent factors to be nonzero and (2) to allow for the collider to be a confounder between the treatment and the outcome. These results demonstrate that mild deviations from the M -Structure tend to increase confounding bias more rapidly than M -Bias, suggesting that choosing to condition on any given covariate is generally the superior choice. As an application, we re-examine a controversial example between Professors Donald Rubin and Judea Pearl.

We consider estimation of and inference for the mean outcome under the optimal dynamic two time-point treatment rule defined as the rule that maximizes the mean outcome under the dynamic treatment, where the candidate rules are restricted to depend only on a user-supplied subset of the baseline and intermediate covariates. This estimation problem is addressed in a statistical model for the data distribution that is nonparametric beyond possible knowledge about the treatment and censoring mechanism. This contrasts from the current literature that relies on parametric assumptions. We establish that the mean of the counterfactual outcome under the optimal dynamic treatment is a pathwise differentiable parameter under conditions, and develop a targeted minimum loss-based estimator (TMLE) of this target parameter. We establish asymptotic linearity and statistical inference for this estimator under specified conditions. In a sequentially randomized trial the statistical inference relies upon a second-order difference between the estimator of the optimal dynamic treatment and the optimal dynamic treatment to be asymptotically negligible, which may be a problematic condition when the rule is based on multivariate time-dependent covariates. To avoid this condition, we also develop TMLEs and statistical inference for data adaptive target parameters that are defined in terms of the mean outcome under the estimate of the optimal dynamic treatment. In particular, we develop a novel cross-validated TMLE approach that provides asymptotic inference under minimal conditions, avoiding the need for any empirical process conditions. We offer simulation results to support our theoretical findings.

This work investigates the intersection property of conditional independence. It states that for random variables A,B,C$$A,B,C$$ and X we have that X⊥⊥A|B,C$$X \bot \bot A{\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} B,C$$ and X⊥⊥B|A,C$$X\, \bot \bot\, B{\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} A,C$$ implies X⊥⊥(A,B)|C$$X\, \bot \bot\, (A,B){\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} C$$. Here, “⊥⊥$$ \bot \bot $$” stands for statistical independence. Under the assumption that the joint distribution has a density that is continuous in A,B$$A,B$$ and C , we provide necessary and sufficient conditions under which the intersection property holds. The result has direct applications to causal inference: it leads to strictly weaker conditions under which the graphical structure becomes identifiable from the joint distribution of an additive noise model.

Failure (or success) in finding a statistically significant effect of a large-scale intervention may be due to choices made in the evaluation. To highlight the potential limitations and pitfalls of some common identification strategies used for estimating causal effects of community-level interventions, we apply a roadmap for causal inference to a pre-post evaluation of a national nutrition program in Madagascar. Selection into the program was non-random and strongly associated with the pre-treatment (lagged) outcome. Using structural causal models (SCM), directed acyclic graphs (DAGs) and simulated data, we illustrate that an estimand with the outcome defined as the post-treatment outcome controls for confounding by the lagged outcome but not by possible unmeasured confounders. Two separate differencing estimands (of the pre- and post-treatment outcome) have the potential to adjust for a certain type of unmeasured confounding, but introduce bias if the additional identification assumptions they rely on are not met. In order to illustrate the practical impact of choice between three common identification strategies and their corresponding estimands, we used observational data from the community nutrition program in Madagascar to estimate each of these three estimands. Specifically, we estimated the average treatment effect of the program on the community mean nutritional status of children 5 years and under and found that the estimate based on the post-treatment estimand was about a quarter of the magnitude of either of the differencing estimands (0.066 SD vs. 0.26–0.27 SD increase in mean weight-for-age z-score). Choice of estimand clearly has important implications for the interpretation of the success of the program to improve nutritional status of young children. A careful appraisal of the assumptions underlying the causal model is imperative before committing to a statistical model and progressing to estimation. However, knowledge about the data-generating process must be sufficient in order to choose the identification strategy that gets us closest to the truth.

In this issue of the Causal, Casual, and Curious column, I compare several ways of extracting information from post-treatment variables and call attention to some peculiar relationships among them. In particular, I contrast do -calculus conditioning with counterfactual conditioning and discuss their interpretations and scopes of applications. These relationships have come up in conversations with readers, students and curious colleagues, so I will present them in a question–answers format.