Heterogeneous treatment effects and optimal targeting policy evaluation

Abstract

We present a general framework to target customers using optimal targeting policies, and we document the profit differences from alternative estimates of the optimal targeting policies. Two foundations of the framework are conditional average treatment effects (CATEs) and off-policy evaluation using data with randomized targeting. This policy evaluation approach allows us to evaluate an arbitrary number of different targeting policies using only one randomized data set and thus provides large cost advantages over conducting a corresponding number of field experiments. We use different CATE estimation methods to construct and compare alternative targeting policies. Our particular focus is on the distinction between indirect and direct methods. The indirect methods predict the CATEs using a conditional expectation function estimated on outcome levels, whereas the direct methods specifically predict the treatment effects of targeting. We introduce a new direct estimation method called treatment effect projection (TEP). The TEP is a non-parametric CATE estimator that we regularize using a transformed outcome loss which, in expectation, is identical to a loss that we could construct if the individual treatment effects were observed. The empirical application is to a catalog mailing with a high-dimensional set of customer features. We document the profits of the estimated policies using data from two campaigns conducted one year apart, which allows us to assess the transportability of the predictions to a campaign implemented one year after collecting the training data. All estimates of the optimal targeting policies yield larger profits than uniform policies that target none or all customers. Further, there are significant profit differences across the methods, with the direct estimation methods yielding substantially larger economic value than the indirect methods.

Appendices

Appendix

A Proof that $\hat{\Pi }(d)$ is an unbiased profit estimator

In Section 3.3 we defined the inverse probability-weighted profit estimator,

$$\begin{aligned} \hat{\Pi }(d)=\sum _{i=1}^{N}\left( \frac{1-W_{i}}{1-e(X_{i})}(1-d(X_{i}))\cdot \pi _{i}(0)+\frac{W_{i}}{e(X_{i})}d(X_{i})\cdot \pi _{i}(1)\right) . \end{aligned}$$

(15)

$\hat{\Pi }(d)$ is an unbiased estimator for the expected profit $\mathbb {E}[\Pi (d)|X_{1},\dots ,X_{N}]$:

$$\begin{aligned} \mathbb {E}[\hat{\Pi }(d)|X_{1},\dots ,X_{N}]&=\sum _{i=1}^{N}\mathbb {E}\left[ \frac{1-W_{i}}{1-e(X_{i})}(1-d(X_{i}))\cdot \pi _{i}(0)+\frac{W_{i}}{e(X_{i})}d(X_{i})\cdot \pi _{i}(1)|X_{i}\right] \\&=\sum _{i=1}^{N}\left( \frac{1-e(X_{i})}{1-e(X_{i})}(1-d(X_{i}))\cdot \mathbb {E}[\pi _{i}(0)|X_{i}]+\frac{e(X_{i})}{e(X_{i})}d(X_{i})\cdot \mathbb {E}[\pi _{i}(1)|X_{i}]\right) \\&=\sum _{i=1}^{N}\left( (1-d(X_{i}))\cdot \mathbb {E}[\pi _{i}(0)|X_{i}]+d(X_{i})\cdot \mathbb {E}[\pi _{i}(1)|X_{i}]\right) \\&=\sum _{i=1}^{N}\mathbb {E}\left[ (1-d(X_{i}))\cdot \pi _{i}(0)+d(X_{i})\cdot \pi _{i}(1)|X_{i}\right] \\&=\mathbb {E}[{\Pi }(d)|X_{1},\dots ,X_{N}]. \end{aligned}$$

The second line follows because $\pi _{i}(0)$ and $\pi _{i}(1)$ are independent of $W_{i}$ conditional on $X_{i}$, which holds because each $W_{i}$ is a Bernoulli draw with success probability e(x), $x=X_{i}$. Further, because the N customers in the calculation of Eq. 3 are randomly drawn from the customer population, $\hat{\Pi }(d)$ is also an unbiased estimator of the expected profit of d in the whole customer population,

$$ \mathbb {E}[\hat{\Pi }(d)]=\mathbb {E}_{X}[\mathbb {E}[\hat{\Pi }(d)|X_{1},\dots ,X_{N}]]=\mathbb {E}_{X}[\mathbb {E}[\Pi (d)|X_{1},\dots ,X_{N}]]=\mathbb {E}[\Pi (d)]. $$

B    Approach to grow a causal tree

To understand how causal trees are grown using the approach proposed by Athey and Imbens (2016), we first review the approach to grow standard trees. Standard trees are trained to predict outcome levels. The algorithm to grow a tree employs recursive binary splits to obtain a partition of the feature space into regions (leaves) $R_{1},R_{2},\dots$ Let $\mathcal {R}_{k}=\{i:X_{i}\in R_{k}\}$ be the set of observations in leaf $R_{k}$, and let $N_{k}$ be the corresponding number of observations. For any feature $x\in R_{k}$, the predicted outcome is the average over all observations in $R_{k}$:

$$ \hat{\mu }(x)=\hat{\mu }_{R_{k}}=\frac{1}{N_{k}}\sum _{i\in \mathcal {R}_{k}}Y_{i}. $$

The algorithm adds a split to a current terminal node, $R_{k}$, based on a feature, $X_{l}$, and a cutoff, $\kappa$, such that $R_{k}$ is divided into the regions $R_{k1}=\{x\in R_{k}:x_{l}<\kappa \}$ and $R_{k2}=\{x\in R_{k}:x_{l}\ge \kappa \}$. The residual sum of squares that results from such a split is given by

$$\begin{aligned} \text {RSS}=\sum _{i\in \mathcal {R}_{k1}}(Y_{i}-\hat{\mu }_{R_{k1}})^{2}+\sum _{i\in \mathcal {R}_{k2}}(Y_{i}-\hat{\mu }_{R_{k2}})^{2}. \end{aligned}$$

(16)

The splitting rule chooses a feature and a cutoff such that the resulting split yields the smallest RSS among all possible binary splits. The residual sum of squares Eq. 16 can be written as

$$\begin{aligned} RSS=\sum _{i\in \mathcal {R}_{k}}Y_{i}^{2}-\sum _{i\in \mathcal {R}_{k}}\hat{\mu }(X_{i})^{2}=\sum _{i\in \mathcal {R}_{k}}Y_{i}^{2}-N_{k1}\hat{\mu }_{R_{k1}}^{2}-N_{k2}\hat{\mu }_{R_{k2}}^{2}. \end{aligned}$$

(17)

Here, $\hat{\mu }(X_{i})$ is the outcome prediction after splitting $R_{k}$ into $R_{k1}$ and $R_{k2}$. Hence, finding a split that minimizes the residual sum of squares Eq. 16 is equivalent to finding a split that maximizes the sum of the squared predictions, $\sum _{i\in \mathcal {R}_{k}}\hat{\mu }(X_{i})^{2}$. Furthermore, maximizing the sum of the squared predictions is equivalent to finding a split that maximizes the variance of the predictions $\hat{\mu }(X_{i})$ across the observations in the two new leaves, $i\in \mathcal {R}_{k1}\cup \mathcal {R}_{k2}$.

Like standard trees, causal trees are also grown using binary splitting rules. For a feature vector $x\in R_{k}$ we predict the CATE based on the mean difference between the outcome levels of the treated and untreated units,

$$ \hat{\tau }(x)=\hat{\tau }_{R_{k}}=\frac{1}{N_{k}(1)}\sum _{i\in \mathcal {R}_{k}(1)}Y_{i}-\frac{1}{N_{k}(0)}\sum _{i\in \mathcal {R}_{k}(0)}Y_{i}. $$

Here, $\mathcal {R}_{k}(w)$ is the set of observations i such that $X_{i}\in R_{k}$ and $W_{i}=w$, and $N_{k}(w)$ is the corresponding number of observations.

Table 8 2016 predicted conditional average treatment effects

Fig. 14

2016 predicted conditional average treatment effects

Note: Predictions obtained using ten-fold cross validation

Adding a split to a causal tree based on an analog of the residual sum of squares Eq. 16 appears infeasible, because the treatment effects are unobserved. However, for standard trees we can see from the equivalence between Eq. 16 and Eq. 17 that any split affects the RSS only through its impact on the sum of squared predictions, $\sum _{i\in \mathcal {R}_{k}}\hat{\mu }(X_{i})^{2}$. Hence, analogous to the approach to grow a standard tree, Athey and Imbens (2016) propose a splitting rule that maximizes $\sum _{i\in \mathcal {R}_{k}}\hat{\tau }(X_{i})^{2}$, which maximizes the variance of the predicted treatment effects $\hat{\tau }(X_{i})$ across the observations in the two new leaves. This splitting rule is feasible given the data and mimics the approach that would be used if the treatment effect was observed.

Causal trees differ from regression trees that add splits using the transformed outcome loss (Equation 10). For the latter, the treatment indicator only enters through the loss function. In contrast, causal trees use information about the treatment indicator at each split and will be more efficient. See Athey and Imbens (2016) for a discussion contrasting the two alternative tree methods.