# BENK: The Beran Estimator with Neural Kernels for Estimating the Heterogeneous Treatment Effect
## 7 Numerical Experiments
### CATE Estimators for Comparison and Their Parameters  
For investigating BENK and its comparison, we use nine models, which can be united in three groups (the T-learner, the S-learner, the X-learner), such that each group is based on three base models for estimating SFs (the RSF, the Cox model, the Beran estimator with Gaussian kernels). The models are given below in terms of survival models:  
1. The T-learner [12] is a model which estimates the control SF \(S_{0}(t\mid\mathbf{z})\) and the treatment SF \(S_{1}(t\mid\mathbf{z})\) for every \(\mathbf{z}\). The CATE in this case is defined in accordance with (13);
2. The S-learner [12] is a model which estimates SF \(S(t\mid\mathbf{z},T)\) instead of \(S_{0}(t\mid\mathbf{z})\) and \(S_{1}(t\mid\mathbf{z})\), where the treatment assignment indicator \(T_{i}\in\{0,1\}\) is included as an additional feature to the feature vector \(\mathbf{z}_{i}\). As a result, we have a modified dataset \[\mathcal{D}=\{(\mathbf{z}_{1}^{*},\delta_{1},f_{1}),\ldots,(\mathbf{z}_{c}^{* },\delta_{c},f_{c}),(\mathbf{z}_{c+1}^{*},\gamma_{1},h_{1}),\ldots,(\mathbf{z} _{c+s}^{*},\gamma_{s},h_{s})\},\] (21) where \(\mathbf{z}_{i}^{*}=(\mathbf{x}_{i},T_{i})\in\mathbb{R}^{d+1}\) if \(T_{i}=0\), \(i=1,\ldots,c\), and \(\mathbf{z}_{c+i}^{*}=(\mathbf{y}_{i},T_{i})\in\mathbb{R}^{d+1}\) if \(T_{i}=1\), \(i=1,\ldots,t\). The CATE is determined as \[\tau(\mathbf{z})=\sum_{j=1}^{s}(t_{j}^{(1)}-t_{j-1}^{(1)})S^{(j)}(\mathbf{z}, 1)-\sum_{k=1}^{c}(t_{k}^{(0)}-t_{k-1}^{(0)})S^{(k)}(\mathbf{z},0);\] (22)
3. The X-learner [12] is based on computing the so-called imputed treatment effects and is represented in the following three steps. First, the outcome functions \(g_{0}(\mathbf{x})\) and \(g_{1}(\mathbf{y})\) are estimated using a regression algorithm. Second, the imputed treatment effects are computed as follows: \[D_{1}(\mathbf{y}_{i})=h_{i}-g_{0}(\mathbf{y}_{i}),\;D_{0}(\mathbf{x}_{i})=g_{ 1}(\mathbf{x}_{i})-f_{i}.\] (23) Third, two regression functions \(\tau_{1}(\mathbf{y})\) and \(\tau_{0}(\mathbf{x})\) are estimated for imputed treatment effects \(D_{1}(\mathbf{y})\) and \(D_{0}(\mathbf{x})\), respectively. The CATE for a point \(\mathbf{z}\) is defined as a weighted linear combination of the functions \(\tau_{1}(\mathbf{z})\) and \(\tau_{0}(\mathbf{z})\) as \(\tau(\mathbf{z})=\alpha\tau_{0}(\mathbf{z})+(1-\alpha)\tau_{1}(\mathbf{z})\), where \(\alpha\in[0,1]\) is a weight that is equal to the ratio of treated patients. The original X-learner does not deal with censored data. Therefore, we propose a simple survival modification of the X-learner. It is assumed that \(g_{0}(\mathbf{y}_{i})\) and \(g_{1}(\mathbf{x}_{i})\) are expectations \(E_{0}(\mathbf{y}_{i})\) and \(E_{1}(\mathbf{x}_{i})\) of the times to an event corresponding to control and treatment data, respectively. Expectations \(E_{0}(\mathbf{y}_{i})\) and \(E_{1}(\mathbf{x}_{i})\) are computed by means of one of the algorithms for determining estimates of SFs \(S_{0}(t\mid\mathbf{z})\) and \(S_{1}(t\mid\mathbf{z})\). The functions \(\tau_{1}(\mathbf{y})\) and \(\tau_{0}(\mathbf{x})\) are implemented using the random forest regression algorithm for all the basic models. Estimations of SFs \(S_{0}(t\mid\mathbf{z})\) and \(S_{1}(t\mid\mathbf{z})\) as well as \(S(t\mid\mathbf{z},T)\) are carried out by means o the following survival regression algorithms:  
1. The RSF parameters of random forests used in experiments are the following: * The numbers of trees are 10, 50, 100, 200; * The depths are 3, 4, 5, 6; * The smallest values of examples which fall in a leaf are 1 example, 1%, 5%, 10% of the training set. The above values for the hyperparameters are tested, choosing those leading to the best results;2. The Cox proportional hazards model [3], which is used with the elastic net regularization with the 3 to 1 ratio coefficient \(L_{1}/L_{2}\);
3. In contrast to the proposed BENK model, we use the Beran estimator with the standard Gaussian kernels.