# Delving into Causal Discovery in Health-Related Quality of Life Questionnaires
## 3 Results
### Simulation Findings
The number for successfully estimating the whole causal structure across the 1000 iterations, represented by the number of zero values in either the HD or the SHD, was observed to be higher for simpler structures and increased as the number of participants \(n\) increased. For instance, in the case of the PC algorithm (Table A1), even at \(n=500\), for the simpler DAGs #1 and 2, the number of zero values in HD across 1000 iterations was 532 and 491, respectively, while for the more complex DAGs #5, 6, and 7, the corresponding numbers were 45, 84, and 142, respectively (Table A1). At \(n=10,000\), DAGs #1 and 2 exhibited 905 and 885, respectively, as the number of zero values in HD across 1000 iterations, while DAGs #5, 6 and 7 exhibited 445, 249, and 232, respectively. The numbers of zero values in the SHD case were in general lower than the corresponding ones in the HD case. In the previous example, it is shown in Table A1 that at \(n=500\), for the simpler DAGs #1 and 2, the numbers of zero values in SHD across 1000 iterations were 526 and 479, respectively, while for the more complex DAGs #5, 6, and 7, the corresponding numbers were 8, 20, and 48, respectively. At \(n=10,000\), however, the values were almost identical with the HD case, with DAGs #1 and 2 exhibiting 905 and 885, respectively, and DAGs #5, 6, and 7 exhibiting 390, 249, and 232, respectively. Similar results apply for all the algorithms. Additionally, it was observed that the difference in zero values among the least complex and most complex DAGs was in most cases very large. For example, when assessing the SHD considering the Inter-IA algorithm, at \(n=500\), DAGs #1 and 2 exhibited 737 and 479 for the number of zeros in SHD across 1000 iterations, while for DAGs #5, 6, and 7, the corresponding numbers were 8, 0, and 6, respectively (Table A4). At \(n=1000\), DAGs #1 and 2 exhibited 871 and 797, respectively, while the number of zeros in SHD in DAGs #5, 6, and 7 was 46, 22, and 85, respectively. Finally, at \(n=10,000\), DAGs #1 and 2 exhibited 864 and 885, respectively, while the number of zeros in SHD in DAGs #5, 6, and 7 was 390, 177, and 238, respectively. This level of difference was observed across all algorithms for both HD and SHD. In addition, more complex structures, such as DAGs #5, 6, and 7, exhibited higher values of relative HD/SHD compared to less complex structures such as DAGs #1, 2, and 3, or even #4. This was particularly evident in the case of the relative HD (Figure 2), in which the values corresponding to DAGs #5, 6, and 7 were in almost all cases higher than those corresponding to the remaining DAGs for all algorithms apart from the Fast-IA algorithm. In particular, at \(n=1000\) or higher, a clear distinction is observed at the level of the values corresponding to the group of DAGs #5, 6, and 7 and the group of DAGs #1, 2, 3, and 4. For instance, in the case of the GS algorithm (Figure 2B, Table A2), at \(n=1000\), the relative HD values for DAGs #1, 2, 3, and 4 were 0.03, 0.03, 0.05, and 0.04, while for DAGs #5, 6, and 7 the corresponding values were much higher (0.11, 0.15, and 0.12). At \(n=2000\), the relative HD values for DAGs #1, 2, 3, and 4 were 0.03, 0.02, 0.03, and 0.02, while for DAGs #5, 6, and 7 the corresponding values were 0.07, 0.10, and 0.09. Even at \(n=10,000\), the values' level differences between these two groups were still observed, since the relative HD values for DAGs #1, 2, 3, and 4 were 0.02, 0.02, 0.03, and 0.02, while for DAGs #5, 6, and 7 the corresponding values were 0.04, 0.07, and 0.07. Similar differences in the relative HD values between the groups of DAGs #1, 2, 3, and 4 and DAGs #5, 6, and 7 were observed in the case of the PC, IA, and Inter-IA algorithms (Figure 2A,C,D). In the case of the relative SHD (Figure 3), the difference between the groups of more complex and less complex DAGs was less pronounced. Still, the values corresponding to DAGs #5, 6, and 7 were in general higher than those corresponding to the remaining DAGs for all algorithms apart from the Fast-IA algorithm.