Heterogeneous Treatment Effects with Continuous Treatment (e.g. using BART)

Question

Overview:

Most of the causal inference literature (both theoretical and applied), I have seen on heterogeneous treatment effects, only considers the case with a binary treatment $T\in\{0,1\}$. However, I would like to estimate heterogeneous treatment effects in a design with a continuous treatment $D\in \mathbb{R}$. I would like to my model to be nonparametric, such as the regression tree based model BART (Bayesian Additive Regression Trees). I have three questions:

1. Do you know any good references for basic theory on heterogeneous treatment effects, when the treatment is continuous?
2. Do you know if BART can handle continuous treatments? If so, do you know of any references to theoretical or applied research, where I can learn more about this?
3. Do you know of other nonparametric models that could be useful for estimating heterogeneous treatment effects, when the treatment is continuous?

Thank you very much for your help.

Technical details:

Let $D\in \mathbb{R}$ denote a real-valued treatment variable (sometimes also called a dose). Let $Y$ be some real-valued or binary outcome variable, and let $Y^d_i$ denote the potential outcome of a unit $i$, if $i$ were treated with dose level $d$. Let $X_i\in \mathbb{R}^p$ be a vector of covariates for unit $i$. Let $\mu^d(x) = E(Y^d_i | X_i=x)$ denote the conditional average dose response function. That is, $\mu^d(x)$ denotes the average potential outcome for individuals with covariates equal to $x$, if they were treated with dose level $d$.

I consider $\mu^d(x)$ to the fundamental quantity of interest, since we can use it to calculate e.g. the conditional average treatment effect of changing the dose level from $d_0$ to $d_1$ for any $d_0,d_1 \in \mathbb{R}$: $\mu^{d_1}(x) - \mu^{d_0}(x)$. Likewise, we can use it to calculate the conditional marginal treatment effect of an infinitesimal change in dose level for any dose level $d_0$: $\frac{\partial}{\partial d} \mu^{d_0}(x)$.

All in all, I am therefore interested in methods for estimating nonparametric models of $\mu^d(x)$. I am especially interested if BART can be used for this task.

Answer 1

1. The first part is to identify what is the exact estimand you are interested in. In the context of $T \in \{0, 1\}$, things are fairly simple. However, you can picture different variations for a continuous $T$. For example, you can imagine 'shifting' all the observed $T$ values by some constant $\alpha$. See Dia-Munoz & van der Laan 2011 for a detailed discussion of this problem (and an example using several different estimators). You can also look at 'setting' everyone to $T=t$. This option is usually referred to as estimating the dose-response curve. See Kennedy et al. 2017 for discussion of this approach. Based on your description, it sounds like you are more interested in the later.
2. I can't speak to BART, but the previous procedures allow for general functions to estimate the counterfactual quantity. The Kennedy et al. paper in particular describes a non-parametric approach.
3. See above for alternative approaches. The proposed estimators in the above linked papers are general. You can also use super-learner with a 'binning' approach, which allows for any classification algorithm you would want. This procedure is described in Chapter 14.

Answer 2

With respect to BART's ability to handle continuous treatments, I have found two useful references:

• (Hill, 2011): Section 6 of the paper (named "Estimating Dosage Effects") uses BART to estimate the causal effect of getting a dosage level $d\in \mathbb{R}$ compared to getting a dosage of 0.
• (Woody et al, 2020): The paper develops a BART model with a linear causal effect of a continuous variable, which is non-linearly moderated by a set of moderator variables.

References:

• Jennifer L. Hill (2011) Bayesian Nonparametric Modeling for Causal Inference, Journal of Computational and Graphical Statistics, 20:1, 217-240, DOI: 10.1198/jcgs.2010.08162
• Spencer Woody, Carlos M. Carvalho, P. Richard Hahn, & Jared S. Murray. (2020). Estimating heterogeneous effects of continuous exposures using Bayesian tree ensembles: revisiting the impact of abortion rates on crime. https://arxiv.org/abs/2007.09845