# Multidimensional Group Recommendations in the Health Domain
## 6 Evaluation
### Evaluation Measures  
To evaluate the similarity functions, we used the normalized Discounted Cumulative Gain [55]. The \(nDCG\) values for all users' recommendation lists can be averaged to get a measure of the average performance of a recommendation system. The \(nDCG\) can be calculated as follows,  
\[nDCG_{u}=\frac{DCG_{u}}{IDCG_{u}} \tag{17}\]  
where  
\[DCG_{u}=\sum_{i=1}^{k}\frac{2^{relevance(u,i)}-1}{log_{2}(i+1)} \tag{18}\]  
and  
\[IDCG_{u}=\sum_{i=1}^{k}\frac{2^{r(u,i)}-1}{log_{2}(i+1)} \tag{19}\]  
The \(DCG_{u}\) part of the equation calculates the relevance of the items that appear in the recommendation list of a user and the \(IDCG_{u}\) calculates the relevance of the items in an ideal scenario. Note that in an ideal recommendation list, \(DCG_{u}\) equals to \(IDCG_{u}\) producing an nDCG of 1.0. Then, nDCG scores are relative values on the interval 0.0 to 1.0.  
To evaluate the \(top\)-\(k\) results of each aggregation method, we counted the average of the distance between the \(top\)-\(k\) recommendation list produced for the group and the list produced for each user separately. For computing the distance, we used the Kendall tau distance that numerates the number of pairwise disagreements between two ranking lists [56].  
\[K(t_{1},t_{2})= |\{(i,j):i<j,(t_{1}(i)<t_{1}(j)\wedge t_{2}(i)>t_{2}(j))\] \[\vee(t_{1}(i)>t_{i}(j)\wedge t_{2}(i)<t_{2}(j))\}| \tag{20}\]  
\begin{table}
\begin{tabular}{c c l l} \hline \hline
**Partitions** & **Parameter Name** & **Explanation** & **Value** \\ \hline \multirow{3}{*}{Group Partition} & Group _occsional_ & \# of ratings given by patients in this group is 20 to 100 & 50\% of all patients \\  & Group _regular_ & \# of ratings given by patients in this group is 100 to 250 & 30\% of all patients \\  & Group _dedicated_ & \# of ratings given by patients in this group is 250 to 500 & 20\% of all patients \\ \hline \multirow{4}{*}{Education Levels} & \(EduHLt_{1}\) & Patients with Education Level 1 & 5\% of all patients \\  & \(EduHLt_{2}\) & Patients with Education Level 2 & 10\% of all patients \\  & \(EduHLt_{3}\) & Patients with Education Level 3 & 40\% of all patients \\  & \(EduHLt_{4}\) & Patients with Education Level 4 & 30\% of all patients \\  & \(EduHLt_{5}\) & Patients with Education Level 5 & 15\% of all patients \\ \hline \multirow{4}{*}{Anxiety Scores} & \(AnxCognCl_{2}\) & Patients with Anxiety Score 1 & 30\% of all patients \\  & \(AnxCognCl_{2}\) & Patients with Anxiety Score 2 & 40\% of all patients \\  & \(AnxCognCl_{3}\) & Patients with Anxiety Score 3 & 15\% of all patients \\  & \(AnxCognCl_{4}\) & Patients with Anxiety Score 4 & 10\% of all patients \\  & \(AnxCognCl_{5}\) & Patients with Anxiety Score 5 & 5\% of all patients \\ \hline \multirow{4}{*}{Scores Partition} & One & \# of ratings that have as score 1 & 20\% of all ratings \\  & Two & \# of ratings that have as score 2 & 10\% of all ratings \\ \cline{1-1}  & Three & \# of ratings that have as score 3 & 30\% of all ratings \\ \cline{1-1}  & Four & \# of ratings that have as score 4 & 20\% of all ratings \\ \cline{1-1}  & Five & \# of ratings that have as score 5 & 20\% of all ratings \\ \hline \multirow{4}{*}{Ratings Partition} & healthRelevant & \# of relevant to some health problems documents each user will & 40\% of ratings from \\ \cline{1-1}  & nonRelevant & \# of non relevant to any health problems documents each user & 60\% of ratings from \\ \cline{1-1}  & will rate. & \multicolumn{1}{c}{each user} \\ \hline \hline \end{tabular}
\end{table}
Table 4: Input parameters for generating the ratings dataset.  
where \(t_{1}(i)\) and \(t_{2}(i)\) are the rankings of the element \(i\) in \(t_{1}\) and \(t_{2}\), respectively.