# Quasi-Experimental Design for Medical Studies with the Method of the Fuzzy Pseudo-Control Group
## 5 Validation of the Case Study Conclusions
### Case Study Solved with FDID  
In essence, we should apply the FDID four times to solve the four tasks formulated in Section 4.3. For the sake of comparability, we solved the four tasks at a significance level of \(\alpha\) = 0.05, and the Bootstrap CIs were calculated with \(N\) = 2000 pseudo-realities.  
3.1 FDID Solution of Task 1 (Effect of Annuloplasty over RF for \(A_{1}\) and \(B_{1}\) Patients)  
First, we will solve task 1, as described in Section 4.3, using the fuzzy DID method. We shall use the four samples given in Section 4.3.1. Table 14 shows the fuzzy mean values of the four samples. The \(ATE\) for task 1 is (-43.9%), given on line 4, column 4 of the table. Those results are illustrated in Figure 2, which shows the three temporal functions of the fuzzy means in \(A_{1}\), \(B_{1}\), and the counterfactual of \(A_{1}\), along with \(ATE\). After applying the algorithm of the fuzzy DID, we obtained four Reverse Percentile 95%-CIs for \(ATE\) in task 1:  
* For BM1, \(\Pr[-43.7<ATE<-26.1]=0.95\);
* For BM2, \(\Pr[-44.6<ATE<-27.1]=0.95\);
* For BM3, \(\Pr[-44.1<ATE<-26]=0.95\);
* For BM4, \(\Pr[-44.4<ATE<-26.9]=0.95\).  
\begin{table}
\begin{tabular}{l l l l} \hline  & \(A_{1}\) (\(E\)) & \(B_{1}\) (\(PC\)) & Difference (\(D\)) \\ \hline Preop (\(b\)) & \(M_{E,b}\) = +54, \% & \(M_{PC,b}\) = +35.5, \% & \(D_{b}\) = +18.5, \% \\ \hline Postop (\(e\)) & \(M_{E,e}\) = +7.79, \% & \(M_{PC,e}\) = +24.2, \% & \(D_{e}\) = \(-\)16.4, \% \\ \hline Temporal change (TC) & \(TC_{E}\) = \(-\)46.2, \% & \(TC_{PC}\) = \(-\)11.3, \% & \(ATE\) = \(-\)34.9, \% \\ \hline \end{tabular}
\end{table}
Table 14: Fuzzy means of \(RF\), % for subgroups \(A_{1}\) and \(B_{1}\).  
5.3.3. Fdid Solution of Task 3 (Effect of Annuloplasty over RF for \(A_{2}\) and \(B_{2}\) Patients)  
We shall use the four samples given in Section 4.3.3. Table 16 shows the fuzzy mean values of the four samples. The _ATE_ for task 3 is (-41.4%), given on line 4, column 4 of the table. Those results are illustrated in Figure 4, which shows the three temporal functions of the fuzzy means in \(A_{2}\), \(B_{2}\), and the counterfactual of \(A_{2}\), along with _ATE_. After applying the algorithm of the fuzzy DID, we obtained four Reverse Percentile 95%-CIs for _ATE_ in task 3:  
* For BM1, \(\Pr[-52.5<\textit{ATE}<-29.5]=0.95\);
* For BM2, \(\Pr[-54.5<\textit{ATE}<-29.7]=0.95\);
* For BM3, \(\Pr[-52.7<\textit{ATE}<-30.5]=0.95\);
* For BM4, \(\Pr[-54<\textit{ATE}<-29.9]=0.95\).  
5.3.4. Fdid Solution of Task 4 (Effect of Annuloplasty over MR for \(A_{2}\) and \(B_{2}\) Patients)  
We shall use the four samples given in Section 4.3.4. Table 17 shows the fuzzy mean values of the four samples. The _ATE_ for task 4 is (-3.87%), given on line 4, column 4 of the table. Those results are illustrated in Figure 5, which shows the three temporal functions of the fuzzy means in \(A_{2}\), \(B_{2}\), and the counterfactual of \(A_{2}\), along with _ATE_. After applying the algorithm of the fuzzy DID, we obtained four Reverse Percentile 95%-CIs for _ATE_ in task 4:  
* For BM1, \(\Pr[-4.62<\textit{ATE}<-3.17]=0.95\);
* For BM2, \(\Pr[-4.56<\textit{ATE}<-3.1]=0.95\);
* For BM3, \(\Pr[-4.63<\textit{ATE}<-3.15]=0.95\);
* For BM4, \(\Pr[-4.62<\textit{ATE}<-3.09]=0.95\).  
\begin{table}
\begin{tabular}{l l l l} \hline  & \(\mathbf{A_{2}}\) (\(\mathbf{E}\)) & \(\mathbf{B_{2}}\) (\(\mathbf{PC}\)) & **Difference (\(\mathbf{D}\))** \\ \hline Preop (\(b\)) & \(M_{E,b}=+59\), \% & \(M_{\text{PC,b}}=+42.6\), \% & \(D_{b}=+16.4\), \% \\ \hline Postop (\(e\)) & \(M_{E,e}=+7.89\), \% & \(M_{\text{PC,e}}=+32.8\), \% & \(D_{e}=-24.9\), \% \\ \hline Temporal change (TC) & \(TC_{E}=-51.1\), \% & \(TC_{\text{PC}}=-9.78\), \% & \(ATE=-41.4\), \% \\ \hline \end{tabular}
\end{table}
Table 16: Fuzzy means of \(RF\), % for subgroups \(A_{2}\) and \(B_{2}\).  
Figure 4: DID graphical illustration for task 3.