# Delving into Causal Discovery in Health-Related Quality of Life Questionnaires
## 3 Results  
### Simulation Findings  
The analytical results of all the assessment measures considered for causal structure estimation based on the PC, the Grow-Shrink, the Incremental Association, the Interleaved Incremental Association and the Fast Incremental Association algorithms are available in Appendix A (Tables A1-A5, respectively). By assessing Tables A1-A5, it was observed that the mean values corresponding to the Hamming distance and the structural Hamming distance increased in most cases as the structure complexity increased, and decreased as the sample size increased. For example, in Table A2 (GS algorithm), at \(n=100\), the HD increased from 1.48 to 11.89 as the complexity of the DAGs increased, at \(n=500\), it increased from 0.31 to 4.83, at \(n=1000\), the HD increased from 0.14 to 3.07 as the DAGs' complexity increased, and similarly at \(n=2000\), the HD increased from 0.13 to 2.16. At \(n=5000\), it increased from 0.15 to 1.55, and at \(n=10,000\), the HD increased from 0.11 to 1.37. It is easy to observe that the HD values were larger for smaller values of \(n\), while they gradually decreased as \(n\) increased. This behavior was observed in almost all cases, but it was less pronounced in the case of the Fast-IA algorithm (Table A5). At the same time, the values corresponding to the HD were, in general, lower than the ones corresponding to the SHD. In the same example of the GS algorithm, it is shown in Table A2 that, at \(n=100\), the SHD increased from 3.60 to 16.65 as the complexity of the DAGs increased, at \(n=500\), it increased from 1.03 to 10.37, at \(n=1000\), it increased from 0.37 to 6.16 as the DAGs' complexity increased, at \(n=2000\), the SHD increased from 0.35 to 3.26, at \(n=5000\), it increased from 0.41 to 2.13, and at \(n=10,000\), the SHD increased from 0.29 to 1.76. Similarly, as in the case of the HD, the SHD values were larger for smaller values of \(n\), and they decreased as \(n\) increased. In the case of the Fast-IA algorithm (Table A5), it was observed that the less complex DAGs #1, 2, 3, and 4 exhibited, for some values of the sample size \(n\), comparable or even higher SHD values, compared to the more complex DAGs #5, 6, and 7 (see, e.g., \(n=2000,\ 5000\) in Table A5). The assessment results regarding the mean relative HD and the mean relative SHD are visually displayed in Figures 2 and 3, respectively. In particular, by assessing the mean relative HD (Figure 2), it is shown that the PC (Figure 2A) and the Inter-IA (Figure 2D) algorithms exhibited the best performance, since both of them demonstrated a mean relative HD less than or equal to 0.15 for all DAGs, even at \(n=500\), and less than or equal to 0.10 for all DAGs at \(n=1000\) or higher. The PC algorithm exhibited slightly lower values compared to the Inter-IA algorithm for large sample sizes (\(n=2000,\ 5000,\ 10,000\)). The GS (Figure 2B) and IA (Figure 2C) algorithms both required the sample size to be at least 1000 in order for the mean relative HD to be less than or equal to 0.15 for all DAGs, and it was less than or equal to 0.10 for all DAGs at \(n=2000\) or higher (with the exception of the IA algorithm, DAG #6, at \(n=2000\)). The Fast-IA (Figure 2E) was the only algorithm that required at least \(n=2000\) in order for the mean relative HD to be less than or equal to 0.15 for all DAGs. By assessing the mean relative SHD (Figure 3), it was observed that the algorithms PC, GS, IA, and Inter-IA (Figure 3A-D) exhibited similar behavior across all values of \(n\) with a mean relative SHD being less than or equal to 0.30 for all DAGs at \(n=1000\), less than or equal to 0.20 for all DAGs at \(n=2000\), and less than or equal to 0.15 for all DAGs at \(n=10,000\). The Fast-IA algorithm (Figure 3E) exhibited large values of mean relative SHD even for \(n=5000\).