# Delving into Causal Discovery in Health-Related Quality of Life Questionnaires
## 4 Discussion  
### Causal Discovery  
In order to achieve reliable results in a real-data HRQoL application, several aspects should be taken into account. The simulation results in this study indicated that the performance of different constraint-based algorithms was influenced by both the complexity of the underlying causal structures and the sample size. Overall, the mean values corresponding to both the HD and the SHD (and the corresponding relative measures) generally increased as the number of nodes/edges increased, and decreased as the sample size increased, with the HD values being lower, in general, compared to the corresponding SHD values. Similarly, the number of zero values in both HD and SHD across the 1000 iterations was observed to be lower for more complex structures and increased as the sample size increased, with the numbers of zero values in the SHD case being in most cases lower than the corresponding ones in the HD case. The pattern evolution of the measures' values over the sample size was similar in each algorithm across all DAGs considered. In addition, the algorithms PC, GS, IA, and Inter-IA exhibited a similar pattern evolution among each other in the cases of both HD and SHD. In the more complex DAGs #5, 6, and 7, which demonstrated a range of 21-26 in the number of nodes and 17-21 in the number of edges, compared to the simpler DAGs #1, 2, 3, and 4, which demonstrated a range of 5-25 in the number of nodes and 5-16 in the number of edges, higher mean values were exhibited in both HD and SHD in almost all cases, and for all algorithms, excluding the Fast-IA algorithm. Similarly, by assessing the number of successful estimations of the whole causal structure across the 1000 iterations, the algorithms exhibited the best performance in the simpler DAGs #1 and 2, and the worst in the more complex DAGs #5, 6, and 7. On the other hand, by increasing the sample size, the performance of the algorithms, considering all measures of assessment, improved and usually started to converge for values of \(n\) equal to or higher than 2000. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cause Domain** & **Effect Domain** & **Count** \\ \hline Environment & Physical health & 3 \\ \hline Environment & Psychological & 2 \\ \hline Environment & Social relationships & 2 \\ \hline Physical health & Environment & 1 \\ \hline Physical health & Social relationships & 1 \\ \hline Psychological & Environment & 1 \\ \hline Psychological & Social relationships & 1 \\ \hline Social relationships & Environment & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Intensities of the inter-domain associations based on sample DAG #6. These results were expected to some extent, since, intuitively, a more complex underlying causal structure is harder to estimate in an efficient manner compared to a less complex structure, while a larger sample size is connected, in general, to more robust estimation. A similar result regarding the performance improvement when the sample size is larger was observed by [PERSON] et al. [47], although in that study the impact of the sample size was assessed for continuous variables in the presence/absence of latent confounders. Regarding the complexity of the structure/network, [PERSON] et al. [53] concluded that by comparing small networks (less than 50 nodes) versus large networks (more than 50 nodes), no systematic differences in the accuracy of network reconstruction were observed in the rankings of the learning algorithms that were assessed. Although this seems to be contradictory to the results in our study, it should be noted that networks of similar size may demonstrate large differences in the number of parameters and thus large differences in their levels of complexity, as is stated as well by [PERSON] et al. [53]. Therefore, the distinction based on the number of nodes does not represent an explicit distinction in structure complexity. On the other hand, in our study, the complexity of the structure is reflected in both the number of nodes and edges.