# Delving into Causal Discovery in Health-Related Quality of Life Questionnaires
## 4 Discussion
### Impact, Limitations, and Future Work
All authors have read and agreed to the published version of the manuscript. The research project was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the \"2 nd Call for H.F.R.I. Research Projects to support Post-Doctoral Researchers\" (Project Number: 553). Resources and code related to the analysis are available at the GitHub repository: [[https://github.com/teomoi/Causal-HRQoL](https://github.com/teomoi/Causal-HRQoL)]([https://github.com/teomoi/Causal-HRQoL](https://github.com/teomoi/Causal-HRQoL)). Author [PERSON] was employed by the company Foodpairing NV. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**GS** & **DAG \#** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline  & \# of nodes & 5 & 10 & 11 & 15 & 21 & 26 & 26 \\ \hline Sample size \(n\) & \# of edges & 5 & 9 & 11 & 16 & 17 & 21 & 19 \\ \hline  & HD & 1.48 & 2.27 & 4.42 & 9.04 & 8.62 & 11.89 & 10.02 \\  & rHD & 0.30 & 0.25 & 0.40 & 0.56 & 0.51 & 0.57 & 0.53 \\
100 & 0’s in HD & 49 & 17 & 0 & 0 & 0 & 0 & 0 \\  & SHD & 3.60 & 5.68 & 8.76 & 13.37 & 11.62 & 16.65 & 14.41 \\  & rSHD & 0.72 & 0.63 & 0.80 & 0.84 & 0.68 & 0.79 & 0.76 \\  & 0’s in SHD & 28 & 1 & 0 & 0 & 0 & 0 & 0 \\ \hline  & HD & 0.31 & 0.40 & 1.15 & 2.22 & 3.17 & 4.83 & 3.58 \\  & rHD & 0.06 & 0.04 & 0.10 & 0.14 & 0.19 & 0.23 & 0.19 \\
500 & 0’s in HD & 714 & 645 & 241 & 65 & 11 & 2 & 18 \\  & SHD & 1.03 & 2.45 & 3.69 & 7.73 & 6.28 & 10.37 & 7.58 \\  & rSHD & 0.21 & 0.27 & 0.34 & 0.48 & 0.37 & 0.49 & 0.40 \\  & 0’s in SHD & 703 & 283 & 38 & 2 & 0 & 0 & 0 \\ \hline  & HD & 0.14 & 0.18 & 0.50 & 0.70 & 1.92 & 3.07 & 2.26 \\  & rHD & 0.03 & 0.02 & 0.05 & 0.04 & 0.11 & 0.15 & 0.12 \\
1000 & 0’s in HD & 870 & 832 & 580 & 498 & 94 & 52 & 116 \\  & SHD & 0.37 & 1.05 & 1.71 & 3.76 & 4.44 & 6.16 & 4.32 \\  & rSHD & 0.07 & 0.12 & 0.16 & 0.24 & 0.26 & 0.29 & 0.23 \\  & 0’s in SHD & 866 & 602 & 228 & 101 & 3 & 20 & 62 \\ \hline  & HD & 0.13 & 0.17 & 0.32 & 0.38 & 1.23 & 2.16 & 1.67 \\  & rHD & 0.03 & 0.02 & 0.03 & 0.02 & 0.07 & 0.10 & 0.09 \\
2000 & 0’s in HD & 877 & 841 & 731 & 679 & 264 & 145 & 195 \\  & SHD & 0.35 & 0.54 & 0.96 & 1.28 & 3.25 & 3.26 & 2.59 \\  & rSHD & 0.07 & 0.06 & 0.09 & 0.08 & 0.19 & 0.16 & 0.14 \\  & 0’s in SHD & 877 & 789 & 565 & 444 & 43 & 130 & 175 \\ \hline  & HD & 0.15 & 0.18 & 0.26 & 0.35 & 0.94 & 1.55 & 1.39 \\  & rHD & 0.03 & 0.02 & 0.02 & 0.02 & 0.06 & 0.07 & 0.07 \\
5000 & 0’s in HD & 857 & 820 & 769 & 701 & 414 & 203 & 225 \\  & SHD & 0.41 & 0.52 & 0.54 & 0.55 & 2.02 & 2.13 & 1.97 \\  & rSHD & 0.08 & 0.06 & 0.05 & 0.03 & 0.12 & 0.10 & 0.10 \\  & 0’s in SHD & 857 & 816 & 760 & 674 & 268 & 201 & 220 \\ \hline  & HD & 0.11 & 0.18 & 0.31 & 0.36 & 0.75 & 1.37 & 1.26 \\  & rHD & 0.02 & 0.02 & 0.03 & 0.02 & 0.04 & 0.07 & 0.07 \\
10,000 & 0’s in HD & 889 & 836 & 732 & 687 & 475 & 245 & 275 \\  & SHD & 0.29 & 0.48 & 0.63 & 0.50 & 1.50 & 1.76 & 1.67 \\  & rSHD & 0.06 & 0.05 & 0.06 & 0.03 & 0.09 & 0.08 & 0.09 \\  & 0’s in SHD & 889 & 831 & 727 & 680 & 375 & 326 & 345 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The mean Hamming distance (HD) across 1000 iterations, the mean relative HD (rHD) across 1000 iterations, the number of cases the HD was zero across 1000 iterations (0’s in HD), the mean structural Hamming distance (SHD) across 1000 iterations, the mean relative SHD (rSHD) across 1000 iterations, and the number of cases the SHD was zero across 1000 iterations (0’s in SHD) based on estimations with the Grow–Shrink (GS) algorithm are displayed. The notations DAG #1, 2, 3, 4, 5, 6, and 7 correspond to the seven synthetic directed acyclic graphs that are described in Section 2.2.1 and are displayed in Figure 1.