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:
- Do you know any good references for basic theory on heterogeneous treatment effects, when the treatment is continuous?
- 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?
- 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.