Larry Wasserman

Regression discontinuity and kink designs are typically analyzed through mean effects, even when treatment changes the shape of the entire outcome distribution. To address this, we introduce distributional discontinuity designs, a framework for estimating causal effects for a scalar outcome at the boundary of a discontinuity in treatment assignment. Our estimand is the Wasserstein distance between limiting conditional outcome distributions; a single scale-interpretable measure of distribution shift. We show that this weakly bounds the average treatment effect, where equality holds if and only if the treatment effect is purely additive; thus, departure from equality measures effect heterogeneity. To further encode effect heterogeneity we show that the Wasserstein distance admits an orthogonal decomposition into squared differences in $L$-moments, thereby quantifying the contribution from location, scale, skewness, and higher-order shape components to the overall distributional distance. Next, we extend this framework to distributional kink designs by evaluating the Wasserstein derivative at a policy kink; this describes the flow of probability mass through the kink. In the case of fuzzy kink designs, we derive new identification results. Finally, we apply our methods on real data by re-analyzing two natural experiments to compare our distributional effects to traditional causal estimands.

Reconstruction of the ancestral protein repertoire offers valuable insights into the tempo and mode of protein content evolution, but can be highly sensitive to model choice. We used a phylogenetic Birth-Death-Gain model to investigate the evolution of the metazoan protein domain repertoire. Domains, protein modules with a distinct structure and function, represent the basic components of protein repertoire. Given a species tree and a census of protein domain families in present-day species, we estimated the most likely rates of domain family origination, duplication, and loss. Rates were allowed to vary across species lineages and domain families, decoupling these factors. Statistical hierarchical clustering of family-specific rates reveals groups of domains evolving in concert. Moreover, we observe a strong and significant association between family rate and family function. Interestingly, families with functions associated with metazoan innovations tend to have the fastest rates. We further inferred the expected ancestral domain content and the history of domain family gains, losses, expansions, and contractions in each species lineage. Our analysis reveals an ongoing process of domain family replacement and resizing, consistent with extensive remodeling of the protein domain repertoire. This stands in contrast to recent reports of widespread loss during metazoan evolution, which were obtained with more constrained models. The use of a powerful, probabilistic Birth-Death-Gain model reveals an unexpected level of genomic plasticity.

We introduce causal geodesy, a framework for studying the landscape of stochastic interventions that lie between the two extremes of performing no intervention, and performing a sharp intervention that sets an exposure equal to a specific value. We define this framework by constructing paths of distributions that smoothly interpolate between the treatment density and a point mass at the target intervention. Thus, each path starts at a purely observational (or correlational) quantity and moves into a counterfactual world. Of particular interest are paths that correspond to geodesics in some metric, i.e. the shortest path. We then consider the interpretation and estimation of the corresponding causal effects as we move along the path from correlation toward causation.

Domains, sequence fragments that encode protein modules with a distinct structure and function, are the basic building blocks of proteins. The set of domains encoded in the genome serves as the functional toolkit of the species. Here, we use a phylogenetic Birth-Death-Gain model to investigate the evolution of this protein toolkit in metazoa. Given a species tree and the set of protein domain families in each present-day species, this approach estimates the most likely rates of domain origination, duplication and loss. Statistical hierarchical clustering of domain family rates reveals sets of domains with similar rate profiles, consistent with groups of domains evolving in concert. Moreover, we find that domains with similar functions tend to have similar rate profiles. Interestingly, domains with functions associated with metazoan innovations, including immune response, cell adhesion, tissue repair, and signal transduction, tend to have the fastest rates. We further infer the expected ancestral domain content and the history of domain family gains, losses, expansions, and contractions on each branch of the species tree. In contrast to recent reports of widespread loss during metazoan evolution, we observe little evidence of genome streamlining. Rather, our analysis reveals an ongoing process of domain family replacement and resizing, consistent with extensive remodeling of the protein domain repertoire. The use of a powerful, probabilistic Birth-Death-Gain model reveals an unexpected level of genomic plasticity and a striking harmony between the evolution of domain usage in metazoan proteins and organismal innovation.

In causal inference, the joint law of a set of counterfactual random variables is generally not identified. But many interesting quantities are functions of the joint distribution. For example, the individual treatment effect is a difference of counterfactuals and any functional of this difference such as the variance, the quantiles and density, all depend on this joint distribution. For binary treatments, many researchers have found identifiable bounds on these quantities. We extend this idea to continuous treatments. We show that a conservative version of the joint law – corresponding to the smallest treatment effect – is identified. The notion of “conservative” depends on how we choose to measure the causal effect and we consider a few such measures. Finding this law uses recent results from optimal transport theory. Under this conservative law we can bound causal effects and we may construct inferences for each individual’s counterfactual dose-response curve. Intuitively, this is the flattest counterfactual curve for each subject that is consistent with the distribution of the observables. If the outcome is univariate then, under mild conditions, this curve is simply the quantile function of the counterfactual distribution that passes through the observed point. This curve corresponds to a nonparametric rank preserving structural model.

Recent methodological research in causal inference has focused on effects of stochastic interventions, which assign treatment randomly, often according to subject-specific covariates. In this work, we demonstrate that the usual notion of stochastic interventions have a surprising property: when there is unmeasured confounding, bounds on their effects do not collapse when the policy approaches the observational regime. As an alternative, we propose to study generalized policies, treatment rules that can depend on covariates, the natural value of treatment, and auxiliary randomness. We show that certain generalized policy formulations can resolve the "non-collapsing" bound issue: bounds narrow to a point when the target treatment distribution approaches that in the observed data. Moreover, drawing connections to the theory of optimal transport, we characterize generalized policies that minimize worst-case bound width in various sensitivity analysis models, as well as corresponding sharp bounds on their causal effects. These optimal policies are new, and can have a more parsimonious interpretation compared to their usual stochastic policy analogues. Finally, we develop flexible, efficient, and robust estimators for the sharp nonparametric bounds that emerge from the framework.

Understanding treatment effect heterogeneity is vital for scientific and policy research. However, identifying and evaluating heterogeneous treatment effects pose significant challenges due to the typically unknown subgroup structure. Recently, a novel approach, causal k-means clustering, has emerged to assess heterogeneity of treatment effect by applying the k-means algorithm to unknown counterfactual regression functions. In this paper, we expand upon this framework by integrating hierarchical and density-based clustering algorithms. We propose plug-in estimators that are simple and readily implementable using off-the-shelf algorithms. Unlike k-means clustering, which requires the margin condition, our proposed estimators do not rely on strong structural assumptions on the outcome process. We go on to study their rate of convergence, and show that under the minimal regularity conditions, the additional cost of causal clustering is essentially the estimation error of the outcome regression functions. Our findings significantly extend the capabilities of the causal clustering framework, thereby contributing to the progression of methodologies for identifying homogeneous subgroups in treatment response, consequently facilitating more nuanced and targeted interventions. The proposed methods also open up new avenues for clustering with generic pseudo-outcomes. We explore finite sample properties via simulation, and illustrate the proposed methods in voting and employment projection datasets.

Standard regression adjustment gives inconsistent estimates of causal effects when there are time-varying treatment effects and time-varying covariates. Loosely speaking, the issue is that some covariates are post-treatment variables because they may be affected by prior treatment status, and regressing out post-treatment variables causes bias. More precisely, the bias is due to certain non-confounding latent variables that create colliders in the causal graph. These latent variables, which we call phantoms, do not harm the identifiability of the causal effect, but they render naive regression estimates inconsistent. Motivated by this, we ask: how can we modify regression methods so that they hold up even in the presence of phantoms? We develop an estimator for this setting based on regression modeling (linear, log-linear, probit and Cox regression), proving that it is consistent for a reasonable causal estimand. In particular, the estimator is a regression model fit with a simple adjustment for collinearity, making it easy to understand and implement with standard regression software. The proposed estimators are instances of the parametric g-formula, extending the regression-with-residuals approach to several canonical nonlinear models.

We develop and analyse the HulC, an intuitive and general method for constructing confidence sets using the convex hull of estimates constructed from subsets of the data. Unlike classical methods which are based on estimating the (limiting) distribution of an estimator, the HulC is often simpler to use and effectively bypasses this step. In comparison to the bootstrap, the HulC requires fewer regularity conditions and succeeds in many examples where the bootstrap provably fails. Unlike sub-sampling, the HulC does not require knowledge of the rate of convergence of the estimators on which it is based. The validity of the HulC requires knowledge of the (asymptotic) median bias of the estimators. We further analyse a variant of our basic method, called the Adaptive HulC, which is fully data-driven and estimates the median bias using sub-sampling. We discuss these methods in the context of several challenging inferential problems which arise in parametric, semi-parametric, and non-parametric inference. Although our focus is on validity under weak regularity conditions, we also provide some general results on the width of the HulC confidence sets, showing that in many cases the HulC confidence sets have near-optimal width.