# Recommendation Algorithm Based on Survival Action Rules
## 4 Experiments and Results
The \(Cs_{0,05}\) column shows the average percentage of examples for which the \(p\)-value is less than 0.05, testing the null hypothesis that the curve for the target part of the recommendation is identical to the curve derived from the arbiter model for examples with mutated attribute values. The log-rank test was found to be inappropriate for the comparison because it requires raw survival data. Therefore, an alternative approach was taken using the two-sample Kolmogorov-Smirnov test (two-sample K-S test) [74]. This method was chosen specifically for cases where at least one of the survival curves being compared is derived from an arbiter model using a single test example, resulting in the output of callable step functions. These step functions, although not compatible with the analysis criteria of the log-rank test, are effectively analyzed using the two-sample K-S test. The two-sample K-S test evaluates the maximum distance between the cumulative distribution functions (CDFs) of two samples. The test statistic \(D_{n,m}\) is defined as:  
\[D_{n,m}=\sup_{x}|F_{1,n}(x)-F_{2,m}(x)|\]  
where \(F_{1,n}(x)\) and \(F_{2,m}(x)\) are empirical distribution functions of the first and second samples, with \(n\) and \(m\) observations, respectively, and \(\sup_{x}\) represents the supremum of the set of distances over all possible values of \(x\). In the context of survival analysis, this approach allows a non-parametric comparison of the empirical survival distributions. The quantitative analysis of the determined rules (Table 3) has shown that most of them are statistically significant. The recommendation algorithm is based on the set of induced rules. The datasets used do not allow for verifying whether the value changes suggested by the recommendation algorithm lead to an increase/decrease in survival time for test examples. This verification would be possible if it were possible to \"track the fate\" of a given test example over time. Therefore, an independent arbiter model was used to \begin{table}
\begin{tabular}{l r r} \hline \hline
**Dataset** & \(Cs_{0,05}\) & \(Cs_{0,01}\) \\ \hline FD001 & 80 & 66 \\ FD002 & 97 & 95 \\ FD003 & 93 & 86 \\ FD004 & 98 & 95 \\ GBSG2 & 100 & 100 \\ melanoma & 100 & 100 \\ actg320 & 96 & 96 \\ bmt-ch & 100 & 100 \\ follicle & 100 & 99 \\ hd & 100 & 100 \\ lung & 100 & 100 \\ maintenance & 92 & 84 \\ mgus & 99 & 99 \\ pbc & 100 & 100 \\ std & 100 & 100 \\ uis & 100 & 100 \\ whas1 & 99 & 98 \\ whas500 & 100 & 100 \\ zinc & 100 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Predictive capabilities of the recommendation algorithm. Columns Cs represent the average values of the consistency score at different significance levels. verify whether an alternative model for estimating survival time would be consistent with the estimation made by the recommendation algorithm. The results of this verification are presented in Table 4. As can be seen, there is a high correspondence between the recommendation algorithm and the XGBoost algorithm running on mutated test examples. It is important to note that the recommendation algorithm and the XGBoost algorithm do not necessarily base their decisions on the same set of attributes. For example, in an extreme case, the recommendation algorithm might suggest changing the values of attributes \(a\) and \(b\), while the XGBoost-based model uses a different set of features. In such a case, there will be a discrepancy between the decisions of the recommendation algorithm and the XGBoost algorithm. For the XGBoost algorithm, an example before changing the attribute values--where only attributes \(a\) and \(b\) are modified--is \"identical\" to the example after the change. This situation occurred in some examples from the FD001, FD003, and maintenance datasets. These discrepancies do not necessarily imply that one of these algorithms is wrong; they simply make decisions in different ways.