# BENK: The Beran Estimator with Neural Kernels for Estimating the Heterogeneous Treatment Effect
## 4 CATE with Censored Data  
Before considering the CATE estimation problem with the censored data, we introduce basic statements of survival analysis. Let us define the training set \(D_{0}\), which consists of \(c\) triplets \((\mathbf{x}_{i},\delta_{i},f_{i})\), \(i=1,\ldots,c\), where \(\mathbf{x}_{i}^{T}=(x_{i1},\ldots,x_{id})\) is the feature vector characterizing the \(i\)-th patient from the control group, \(f_{i}\) is the time to the event concerning the \(i\)-th control patient and \(\delta_{i}\in\{0,1\}\) is the indicator of censored or uncensored observations. If \(\delta_{i}=1\), then the event of interest is observed (the uncensored observation). If \(\delta_{i}=0\), then we have the censored observation. Only the right-censoring is considered when the observed survival time is less than or equal to the true survival time. Many applications of survival analysis deal with the right-censored observations [2]. The main goal of survival machine learning modeling is to use set \(D_{0}\) to estimate probabilistic characteristics of time \(F\) to the event of interest for a new patient with the feature vector \(\mathbf{z}\). In the same way, we define the training set \(D_{1}\), which consists of \(d\) triplets \((\mathbf{y}_{i},\gamma_{i},h_{i})\), \(i=1,\ldots,s\), where \(\mathbf{y}_{i}^{T}=(y_{i1},\ldots,y_{id})\) is the feature vector characterizing the \(i\)-th patient from the treatment group, \(h_{i}\) is the time to the event concerning the \(i\)-th treatment patient and \(\gamma_{i}\in\{0,1\}\) is the indicator of censoring. The survival function (SF), denoted \(S(t\mid\mathbf{z})\), can be regarded as an important concept in survival analysis. It represents the probability of survival of a patient with the feature vector \(\mathbf{z}\) up to time \(t\), that is, \(S(t\mid\mathbf{z})=\Pr\{T>t\mid\mathbf{z}\}\). The hazard function, denoted \(\lambda(t\mid\mathbf{z})\), can be viewed as another concept in survival analysis. It is defined as the rate of an event at time \(t\) given that no event occurred before time \(t\). It is expressed through the SF as follows:  
\[\lambda(t\mid\mathbf{z})=-\frac{\mathrm{d}}{\mathrm{d}t}\ln S(t\mid\mathbf{z}). \tag{7}\]  
The integral of the hazard function, denoted \(H(t\mid\mathbf{x})\), is called the cumulative hazard function and can be interpreted as the probability of an event at time \(t\) given survival until time \(t\), i.e.,  
\[\Lambda(t\mid\mathbf{z})=\int_{0}^{t}\lambda(r\mid\mathbf{z})dr. \tag{8}\]  
It is expressed through the SF as follows:  
\[\Lambda(t\mid\mathbf{z})=-\ln(S(t\mid\mathbf{z})). \tag{9}\]  
The above functions for controls and treatments are written with indices 0 and 1, respectively, for instance, \(S_{0}(t\mid\mathbf{z})=\Pr\{F>t\mid\mathbf{z}\}\) and \(S_{1}(t\mid\mathbf{z})=\Pr\{H>t\mid\mathbf{z}\}\). In order to compare survival models, [PERSON]'s concordance index, or the C-index [57], is usually used. The C-index measures the probability that, in a randomly selected pair of examples, the example that failed first had a worst predicted outcome. It is calculated as the ratio of the number of pairs correctly ordered by the model to the total number of admissible pairs. A pair is not admissible if the events are both right-censored or if the earliest time in the pair is censored. The corresponding survival model is supposed to be perfect when the C-index is 1. The case when the C-index is 0.5 says that the survival model is the same as random guessing. The case when the C-index is less than 0.5 says that the corresponding model is worse than random guessing. In contrast to the standard CATE estimation problem statement given in the previous section, the CATE estimation problem with censored data has another statement, which is due to the fact that outcomes in survival analysis are random times to an event of interest having some conditional probability distribution.