# What equation does the causal tree optimize to partition the feature space?

I am reading the paper Athey & Imbens (2016) paper "Recursive partitioning for heterogeneous causal effects" from PNAS. I am not able to fully understand which criteria is used to construct (train) the causal tree. I have read this post How does a causal tree optimize for heterogenous treatment effects? but I still don't understand what is the criteria.(Also the link shared in the comment section in that post https://scholar.princeton.edu/sites/default/files/bstewart/files/lundberg_methods_tutorial_reading_group_version.pdf is not working.)

My doubts are as follows:

i] First in the section "Modifying Conventional CART for Treatment Effects", for the adaptive method they have modified the Conventional CART MSE to estimate heterogeneous treatment effects by below:

here I believe since Ste is used the estimate is unbiased. But I think this is for only estimating the performance on the test set (please correct me if I am wrong). I don't know how the tree is constructed using Str? Is it constructed like the traditional CART using the MSE between true(Yi) and predicted outcomes($$\hat{Y}$$i) from the splitting?

ii] Second, in the "Modifying the Honest Approach" section, they have used the expected mean square error defined as
which is rewritten using samples as

Is the above criteria the one used to make the splits in the tree? If so how is calculated? Is it the variance of the true(Yi) for the control group in the split?
If not, what is the optimizing criteria?

i] The paper is super confusing, imo. Let me try to convey my understanding. The problem with using traditional CART for estimating the treatment effect is that you never observe the true treatment effect $$\tau_i$$ so you cannot compute MSE. In "Modifying Conventional CART for Treatment Effects" they write the criterion function for honest trees in the prediction case (where the true $$Y_i$$ are observable) this is $$MSE_{\mu}$$, then by analogy they propose the same criterion for treatment effect but instead of using $$\hat{\mu}$$ they use $$\hat{\tau}$$, this is the formula for $$\widehat{MSE}_{\tau}(S^{te}, S^{tr}, \Pi)$$. Finally, they go back again and saying that if you don't wanna do the honest thing with splitting the sample (kinda traditional, adaptive CART but for treatment effect) then you can use $$-\widehat{MSE}_{\tau}(S^{tr}, S^{tr}, \Pi)$$
ii] Apparently, yes, the above criterion is used, and yes, $$S^2(\ell)$$ is the sample variance of $$Y$$ for control units within leaf $$\ell$$.
• thank you for your answer, I have much clarity now. Regarding point 1, I have an additional question, the $\hat{MSE}$$\tau$$(S$tr$,S$tr,$\pi$$)$ is used as a test criterion here, right? that is to evaluate the model after training? In the [Estimation and Inference of Heterogeneous Treatment Effects using Random Forests] (arxiv.org/pdf/1510.04342.pdf) have they used this same criterion for training the individual tree in the causal forest using procedure 1 algorithm, (remark 1 page 8, provides additional info)? Commented Aug 10, 2023 at 5:15
• For cross-validation (is that what you mean by evaluate the model after training?) they propose $-\widehat{MSE}_{\tau}(S^{tr, cv}, S^{tr, tr}, \Pi)$ so you use an extra sample (that you didn't use to grow a tree) to estimate the treatment effect. For the causal forest paper, my understanding is that they are using $\widehat{MSE}_{\tau}(S^{te}, S^{tr}, \Pi)$ to build trees. That is, they do use sample-splitting, one sample to construct the partition, the other for estimating the treatment effects.