# Causal inference for additive multiple treatments

I encountered a causal inference problem in practice and want to find if there is a previously established statistical toolset that can be applied to my problem.

My problem is characterized as follows:

• My goal is to characterize the causal effects of each $$T$$ treatment $$X_1, \cdots X_T$$ on outcome $$Y$$ where the values of treatments and outcomes are binary (0 or 1).
• Unlike a typical multiple treatment setting, multiple treatments can be active simultaneously. For example, it is possible for a sample to have $$X_1 =1$$ and $$X_2 =1$$.
• If necessary, the additivity assumption can be introduced. For example, the causal effect of having $$(X_1, X_2)=(1, 1)$$ is the sum of the causal effects of the two cases: $$(X_1, X_2)=(1,0)$$ and $$(X_1, X_2)=(0, 1)$$.
• As in typical causal inference settings such as that of propensity score matching, there can be multiple common cause variables.

Q1. Is there a specific term describing the problem setting described above? If there is such a term, could you cite a few pedagogical materials?

Q2. If the problem is not studied before, how can it be tackled using existing causal inference methods?

## 1 Answer

A1: A term sometimes used is "joint interventions". However, joint interventions explicitly refers to multiple treatments, not multiple treatments with the additivity assumption. This chapter may be a useful resource to refer to. Ideally, we would avoid the additivity assumption, and I wouldn't recommend using a different term to shorten the phrase (it is clearer to describe the assumption being made rather than use some singular term).

A2: This problem has been tackled through a variety of approaches. As detailed in the above chapter. One of my favorite applied examples is Taubman et al. 2009. The authors apply the parametric g-formula in the time-varying treatments/exposures (which is more complicated but demonstrates the concept). McCaffrey et al. 2013 describe multiple treatments in the context of propensity scores