# Quasi-Experimental Design for Medical Studies with the Method of the Fuzzy Pseudo-Control Group
## 4 Annuloplasty Favorability Case Study
### Assessing the Effect of Annuloplasty for Patients with Severe IMR Using Fuzzy Pseudo-Control Groups
#### 4.3.1 Effect of Annuloplasty over RF for Patients with Relatively Preserved Medical State (Task 1)
This is indicated in column seven of Table 2. From \(RF_{d,opt}-\Delta RF=0-2=-2\%<MED_{E}=0\%<RF_{u,opt}+\Delta RF=1+2=3\%\leq max \big{\{}RF_{u,opt}-\Delta RF,MED_{E}+\Delta RF\big{\}}=0\%<RF_{u,opt}+\Delta RF =1+2=3\%\leq max\big{\{}RF_{u,opt}-\Delta RF,MED_{E}+\Delta RF\big{\}}=0\%\), and the corresponding \(\Delta RF\) is \(C_{-1}\). \begin{table}
\begin{tabular}{l l l l l l l l l l l} \hline
**Modification** & **Time** & **FBT1** & **FBT2** & **FBT3** & **FBT4** & **FBT5** & **FBT6** & **FBT7** & **FBT8** & **FBT9** \\ \hline BM1 & Preop & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.4995 & 0.2480 & 0.1690 & 0.0975 \\ \hline BM1 & Late\_Postop & 0.0010 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0335 & 0.0270 & 0.1870 & 0.1800 \\ \hline BM2 & Preop & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.5360 & 0.2565 & 0.1785 & 0.1170 \\ \hline BM2 & Late\_Postop & 0.0010 & 0.0000 & 0.0000 & 0.0005 & 0.0000 & 0.0285 & 0.0160 & 0.1745 & 0.1495 \\ \hline BM3 & Preop & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.5310 & 0.2670 & 0.2075 & 0.1155 \\ \hline BM3 & Late\_Postop & 0.0010 & 0.0000 & 0.0000 & 0.0010 & 0.0000 & 0.0340 & 0.0320 & 0.1735 & 0.1670 \\ \hline BM4 & Preop & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.5410 & 0.2420 & 0.1720 & 0.1135 \\ \hline BM4 & Late\_Postop & 0.0000 & 0.0000 & 0.0000 & 0.0005 & 0.0000 & 0.0320 & 0.0195 & 0.1500 & 0.1300 \\ \hline \end{tabular}
\end{table} 
Table 3: The \(p\)-values of the fuzzy Bootstrap statistical tests in MFFCG for parameter \(RF\), comparing subgroups \(A_{1}\) and \(B_{1}\). Line three of Table 3 presents the \(p\)-values for the nine Bootstrap statistical tests for the first Bootstrap Modification. The \(p\)-values of the nine Bootstrap statistical tests for BM1 Bootstrap are given on line three of Table 3. According to Section 3.3, since \(Pvalue_{1}=0.01\leq\alpha=0.05\), then the late postoperative population distributions of \(RF\) are assumed to be statistically significantly different. Since \(Pvalue_{3}=0\leq Pvalue_{2}=0\leq\alpha=0.05\) and \(M_{E}=7.79\%<M_{PC}=24.2\%\), then the late postoperative mean of the target population \(A_{1}\) is assumed to be statistically significantly smaller than that in the significantly smaller than that in the pseudo-control population \(B_{1}\). Since \(Pvalue_{5}=0\leq Pvalue_{4}=0\leq\alpha=0.05\) and \(MED_{E}=0\%<MED_{PC}=20.7\%\), then the late postoperative median of the target population \(A_{1}\) is assumed to be statistically significantly smaller than that in the pseudo-control population \(B_{1}\). Since \(Pvalue_{7}=0.027\leq Pvalue_{6}=0.0335\leq\alpha=0.05\) and \(VAR_{E}=12.1^{2}<VAR_{PC}=20.1^{2}\), then the late postoperative variance of the target population \(A_{1}\) is assumed to be statistically significantly smaller than that in the pseudo-control population \(B_{1}\). Since \(\alpha=0.05<Pvalue_{9}=0.18\leq Pvalue_{8}=0.187\), the late postoperative interquartile range of the target population \(A_{1}\) is assumed statistically indistinguishable from that in the pseudo-control population \(B_{1}\). Detailed results can be obtained for the other three Bootstrap Modifications. Their \(p\)-values are in lines 5, 7, and 9 of Table 3.  
Let us categorize the difference between \(A_{1}\) and \(B_{1}\) using BM1 Bootstrap. According to Section 3.4, we can define the values of the five input variables of the discrete function of categorization (20). The late postoperative differences in the means of \(RF\) in the populations \(A_{1}\) and \(B_{1}\) are \(C_{M}=M_{+1}\) (the late postoperative mean of \(RF\) in the target population \(A_{1}\) is statistically significantly more favorable than that in the pseudo-control population \(B_{1}\)). The late postoperative differences in the medians of \(RF\) in the populations \(A_{1}\) and \(B_{1}\) are \(C_{MED}=MED_{+1}\) (the late postoperative median of \(RF\) in the target population \(A_{1}\) is statistically significantly more favorable than that in the pseudo-control population \(B_{1}\)).