Social scientists use conjoint analysis, which is based on randomized experiments with a factorial design, to analyze multidimensional preferences in a population. In such experiments, several factors, each with multiple levels, are randomized to form a large number of possible treatment conditions. To explore causal interaction in factorial experiments, we propose a new definition of 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 high-order interaction. We show that the AMIE can be nonparametrically estimated using the ANOVA regression with weighted zero-sum constraints. These two properties enable us to directly regularize the AMIEs by collapsing levels and selecting factors within a penalized ANOVA framework. This 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 method is implemented through the open-source software.