Causal Interaction in Factorial Experiments: Application to Conjoint Analysis

ABSTRACT

We study causal interaction in factorial experiments, in which several factors, each with multiple levels, are randomized to form a large number of possible treatment combinations. Examples of such experiments include conjoint analysis, which is often used by social scientists to analyze multidimensional preferences in a population. To characterize the structure of causal interaction in factorial experiments, we propose a new causal interaction effect, called the average marginal interaction effect (AMIE). Unlike the conventional interaction effect, the relative magnitude of the AMIE does not depend on the choice of baseline conditions, making its interpretation intuitive even for higher-order interactions. We show that the AMIE can be nonparametrically estimated using ANOVA regression with weighted zero-sum constraints. Because the AMIEs are invariant to the choice of baseline conditions, we directly regularize them by collapsing levels and selecting factors within a penalized ANOVA framework. This regularized estimation procedure reduces false discovery rate and further facilitates interpretation. Finally, we apply the proposed methodology to the conjoint analysis of ethnic voting behavior in Africa and find clear patterns of causal interaction between politicians’ ethnicity and their prior records. The proposed methodology is implemented in an open source software package. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

KEYWORDS:

• ANOVA
• Causal inference
• Heterogenous treatment effects
• Interaction effects
• Randomized experiments
• Regularization

Acknowledgments

The proposed methods are implemented through open-source software FindIt: Finding Heterogeneous Treatment Effects (Egami, Ratkovic, and Imai 2017), which is freely available as an R package at the Comprehensive R Archive Network (CRAN; https://cran.r-project.org/package=FindIt). The authors thank Elizabeth Carlson for providing them with data and answering their questions. The authors are also grateful for Jens Hainmueller, Walter Mebane, Dustin Tingley, Teppei Yamamoto, Tyler VanderWeele, and seminar participants at Carnegie Mellon University (Statistics), Georgetown University (School of Public Policy), Stanford (Political Science), Umea University (Statistics), University of Bristol (Mathematics), and UCLA (Political Science) for helpful comments on an earlier version of the article.