# Heterogeneous Treatment Effect with Trained Kernels of the Nadaraya-Watson Regression
## 4 The TNW-CATE Description
This model, along with other models, can be regarded as a basis for personalized medicine where the need for personalized treatment is tremendous. Taking into account all the characteristics of patients can make the treatment of many diseases more successful. When new treatments or new drugs are tested, the number of treatments may be very small. In this case, the proposed model can be effective. Another application of the model is a case when the feature vectors \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Number of Treatments**} \\ \hline
**Model** & **139** & **70** & **35** \\ \hline T-NW & \(0.482\pm 0.172\) & \(0.511\pm 0.138\) & \(0.696\pm 0.209\) \\ S-NW & \(0.277\pm 0.138\) & \(0.177\pm 0.079\) & \(0.304\pm 0.108\) \\ X-NW & \(0.276\pm 0.014\) & \(0.240\pm 0.096\) & \(0.348\pm 0.139\) \\ T-RF & \(0.099\pm 0.039\) & \(0.218\pm 0.082\) & \(0.231\pm 0.057\) \\ S-RF & \(0.062\pm 0.028\) & \(0.067\pm 0.015\) & \(0.075\pm 0.037\) \\ X-RF & \(0.046\pm 0.021\) & \(0.063\pm 0.014\) & \(0.079\pm 0.034\) \\ TNW-CATE & \(\mathbf{0.010}\pm 0.006\) & \(\mathbf{0.021}\pm 0.007\) & \(\mathbf{0.038}\pm 0.019\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The best MSE values and standard deviations of CATE for different models by studying the dataset IHDP. characterizing patients have a complex structure which cannot be appropriately represented by typical regression functions or kernels. One of the sources of the complex structure is the multimodal information about patients, for example, computer tomography, electronic health records, etc. The main limitation of the proposed model is the need to tune the hyperparameters of the neural network, which requires significant time spent on network training and validation. There are no strict heuristics that can significantly speed up this stage. However, this time is not crucial. For instance, the time of training TNW-CATE on the dataset IHDP (see Section 5.3) is 1704 s, whereas the times of training the models T-RF, S-RF, and X-RF are 49 s, 68 s, and 114 s, respectively. With this time complexity, we pay for the accuracy of the results. In spite of the apparent complexity of the whole neural network for training, TNW-CATE is actually simple because it can be realized as a single small subnetwork implementing the kernel. The numerical simulation experiments have illustrated the outperformance of TNW-CATE for several datasets. We used neural networks to learn kernels in the Nadaraya-Watson regression. However, different models can be applied to the kernel implementation, for example, the random forest [96] or the gradient boosting machine [99]. The study of different models for estimating CATE is a potential direction for further research. We also assumed the similarity between structures of control and treatment data. This assumption can be violated in some cases. Therefore, it would be interesting to modify TNW-CATE to account for these violations. This is another potential direction for further research. Another interesting direction for potential future research is to incorporate robust procedures and imprecise statistical models to deal with small datasets into TNW-CATE. The incorporation of the models can provide estimates for CATE, which are more robust than estimates obtained by using TNW-CATE. Conceptualization, L.U. and A.K. ; methodology, L.U. and A.K. ; software, S.K. ; validation, S.K. and A.K. ; formal analysis, L.U. and A.K; investigation, A.K. and S.K. ; resources, L.U. and S.K. ; data curation, A.K. and S.K. ; writing--original draft preparation, L.U. and S.K. ; writing--review and editing, A.K. ; visualization, S.K. ; supervision, L.U. ; project administration, L.U. ; funding acquisition, L.U. All authors have read and agreed to the published version of the manuscript. This work is supported by the Russian Science Foundation under grant 21-11-00116. Not applicable. The authors would like to express their appreciation to the anonymous referees whose very valuable comments have improved the paper.