## Causal Interaction in High Dimension

**Authors**: Kosuke Imai and Naoki Egami

While researchers often seek to identify combinations of treatments that induce large causal effects in addition to the separate effects from each individual treatment, the standard approach to causal interaction suffers from the lack of invariance to the choice of baseline condition and the difficulty of interpretation beyond two-way interaction. As an alternative definition of causal interaction effect, we propose the **marginal treatment interaction effect**, whose relative magnitude does not depend on the baseline condition choice and which maintains an intuitive interpretation even for higher-order interactions, allowing researchers to better summarize the structure of casual interactions in high dimension by decomposing the total effects of any treatment combinations into marginal and interaction effects. We establish the identification condition and develop an estimation strategy for the proposed marginal treatment interaction effects. We use as our motivating example conjoint analysis, which largely assumes the absence of causal interactions, and applying our methods reveal insights that standard conjoint analysis overlooks.

**Part 1: **

The focus here is on causal interaction effects. Causal interactions often involve heterogeneous treatment effects, raising questions such as how treatments vary across individuals and the relationship between treatment and pre-treatment covariates, which aspects of a treatment are responsible for the causal effects, and what combination of treatments is optimal for a given individual. There are two different ways of interpreting causal interaction: first, the conditional effect interpretation, in which we evaluate whether the effect of a treatment changes when varying the value of another treatment; and second, the interactive effect interpretation, in which we evaluate whether a combination of treatments induces an*additional effect* beyond the sum of each individual treatment effect. The motivating example is conjoint analysis with a survey experiment involving over 1 million possible treatment combinations.

**Part 2**:

The focus here is on interactive interpretation in high dimension. The key difficulty of the conventional approach to causal interaction is the lack of invariance to the baseline condition. That is, answers about interaction effects will be affected by the choice of baseline condition. To address this shortcoming, this paper instead proposes an invariant marginal treatment interaction effect, derives identification condition and the estimation strategy, and generalizes this approach to interactions of any higher order. In evaluating a two-way causal interaction, there is an assumption of a full factorial design, with randomization of treatment assignment and a non-zero probability of any given treatment combination.

**Part 3**:

There are two non-interaction effects of interest: the *average treatment combination effect*** (ATCE)**, which is the average effect of a treatment combination relative to the baseline condition, and the *average marginal treatment effect*** (AMTE)**, which is the average effect of the treatment relative to the baseline condition, averaging over the treatment. The conventional approach to causal interaction is to look at the *average treatment interaction effect***(ATIE)**, with the estimation through the use of linear regression with interaction terms. There are limits to the ability of interaction plots in showing us the interaction effects themselves. Turning to regression coefficients, one finds that the relative magnitude of interaction effects varies greatly when changing the baseline condition. This means that comparison of any two ATIE’s are affected by the choice of baseline conditions. To overcome this problem, the authors introduce instead the *average marginal treatment interaction effect***(AMTIE)**, which unlike ATIE’s are interval and order invariant. The AMTIE is interpreted as the additional effect induced by the combination beyond the effects induced by each separate component. The tradeoff to this alternative approach is that the results will depend on how the treatment is assigned. Interaction effects play an essential role in making causal inferences, and evaluating the **AMTIE** allows for evaluation of interaction effects in high dimension, enables effect decomposition, and unlike other approaches is invariant to the baseline condition.

**References**

Imai, K. and Egami, N. 2015. Causal Interaction in High Dimension

Presentation Slides, EGAP 13. Kosuke Imai

Hainmueller, J., Hopkins, D. and Yamamoto, T. 2014. Causal Inference in Conjoint Analysis: Understanding Multidimensional Choices via Stated Preference Experiments. Political Analysis. 22:1–30