Conjoint experiments provide an attractive way to assess the role of multiple attributes simultaneously on decision-making. However, the randomization of multiple attributes prevents understanding the causal mechanisms that, critically, depend on the relationship between attributes—e.g., how one attribute affects the respondent's belief as to another attribute. This is because conjoint experiments recover controlled effects whereas a substantively important estimand may be the total or indirect effect of one attribute. Unfortunately, existing experimental designs for conjoint experiments cannot estimate these effects. We provide an alternative framework that requires one additional, simple experiment to learn the relationship between attributes among respondents alongside the standard assumptions for causal mediation. Estimation of the relevant effects can be done in a doubly robust fashion using machine learning methods. We illustrate this by conducting a pre-registered experiment on candidate choice and disentangle the effect of different attributes by understanding their mediation through the candidate's party.

Measuring public opinion at subnational geographies is critical to many theories in political science. Multilevel regression and post-stratification (MRP) is a popular tool for doing so, although existing work is limited to measuring opinion on a single survey question. We provide a framework for estimating the joint distribution of opinion on multiple questions ("Multivariate MRP"). To do so, we derive a novel method for variational inference in multinomial logistic regression with many random effects. This requires performing variational inference with high-dimensional fixed effects, but we show that this can be done at a low computational cost. We validate this procedure by estimating public opinion by party in the United States and show that existing methods can be improved considerably by adding contextual covariates on the prior levels of party identification. Substantively, we show how the output of multivariate MRP can be used to study representation across multiple policy issues simultaneously.

How to estimate heterogeneity, e.g. the effect of some variable differing across observations, is a key question in political science. Methods for doing so make simplifying assumptions about the underlying nature of the heterogeneity to draw reliable inferences. This paper allows a common way of simplifying complex phenomenon (placing observations with similar effects into discrete groups) to be integrated into regression analysis. The framework allows researchers to (i) use their prior knowledge to guide which groups are permissible and (ii) appropriately quantify uncertainty. The paper does this by extending work on "structured sparsity" from a traditional penalized likelihood approach to a Bayesian one by deriving new theoretical results and inferential techniques. It shows that this method outperforms state-of-the-art methods for estimating heterogeneous effects when the underlying heterogeneity is grouped and more effectively identifies groups of observations with different effects in observational data.

Multilevel models have long played an important role in a variety of social sciences. We extend this framework by bring to bear recent developments in the machine learning literature to allow for considerable flexibility. We introduce a sparse regression framework that covers both the linear case as well as a logit model for binary outcome data. We leverage recent computational tricks based on data-augmentation to dramatically speed up estimation times with equal or better performance compared to existing approaches. We apply our model in the context of multilevel modelling with post-stratification which has become a common tool for survey researchers.