# 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
#### 4.3.2 Effect of Annuloplasty over MR for Patients with Relatively Preserved Medical State (Task 2)  
In task 2, we use four fuzzy samples for \(MR\), as described below. \(E_{b}\) is a fuzzy sample that contains 34 values of _Preop_MR_ and their degrees of membership to the experimental group \(A_{1}\):  
\(E_{b}=\chi_{A_{1},prop}^{MR}\) = {(5,0.630), (6,0.900), (4,0.630), (6,0.630), (4,0.360), (6,0.810), (5,0.810), (4,0.630), (6,0.900), (6,0.630), (4,0.630), (5,0.630), (5,0.630), (5,0.490), (5,0.630), (6,0.700), (4,0.490), (4,0.630), (6,0.630), (5,0.630), (5,0.630), (6,0.700), (6,0.900), (6,0.630), (4,0.630), (4,0.490), (6,0.900), (4,0.630), (5,0.490), (5,0.630), (5,0.630), (7,0.700), (6,0.900), (6,0.700)). \(PC_{b}\) is a fuzzy sample that contains 43 values of _Preop_MR_ and their degrees of membership to the pseudo-control group \(B_{1}\):  
\(PC_{b}=\chi_{B_{1},prop}^{MR}\) = {(3,0.900), (3,0.810), (3,0.900), (3,0.900), (4,0.630), (4,0.810), (4,0.630), (3,0.900), (4,0.810), (3,0.900), (4,0.810), (3,0.900), (4,0.630), (4,0.630), (4,0.630), (4,0.490), (4,0.810), (3,0.900), (4,0.490), (3,0.900), (3,0.900), (4,0.490), (4,0.810), (4,0.630), (3,0.900), (4,0.810), (4,0.630), (3,0.900), (4,0.630), (4,0.630), (3,0.900), (3,0.900), (3,0.900), (3,0.900), (3,0.900), (3,0.700), (3,0.700), (4,0.700), (4,0.700), (4,0.357)). \(E_{e}\) is a fuzzy sample that contains 32 values of _Late_Postop_MR_ and their degrees of membership to the experimental group \(A_{1}\):  
\(E_{e}=\chi_{A_{1},L,postp}^{MR}\) = {(0,0.630), (2,0.900), (2,0.630), (4,0.630), (2,0.360), (4,0.810), (2,0.810), (2,0.630), (0,0.900), (0,0.630), (1,0.630), (0,0.630), (0,0.630), (1,0.630), (0,0.630), (0,0.630), (0,0.630), (0,0.700), (0,0.900), (2,0.630), (2,0.630), (0,0.490), (0,0.900), (0,0.630), (0,0.490), (0,0.630), (2,0.490), (0,0.630), (1,0.700), (1,0.900), (0,0.700). \(PC_{e}\) is a fuzzy sample that contains 41 values of _Late_Postop_MR_ and their degrees of membership to the pseudo-control group \(B_{1}\):  
\(PC_{e}=\chi_{B_{1},L,postp}^{MR}\) = {(2,0.900), (2,0.810), (2,0.900), (2,0.900), (5,0.630), (4,0.810), (4,0.630), (2,0.900), (6,0.810), (1,0.900), (1,0.900), (3,0.490), (2,0.810), (2,0.900), (4,0.630), (1,0.630), (2,0.630), (4,0.490), (4,0.810), (4,0.900), (1,0.490), (1,0.900), (2,0.900), (5,0.490), (4,0.630), (2,0.900), (4,0.810), (2,0.630), (2,0.900), (2,0.630), (4,0.490), (3,0.630), (0,0.900), (2,0.900), (4,0.810), (2,0.630), (2,0.900), (2,0.630), (4,0.490), (3,0.630), (0,0.900), (2,0.900), (4,0.900), (2,0.900), (0,0.700), (1,0.700), (0,0.700), (2,0.700), (0,0.357)). The numerical realization of MFPCG for task 2 is given in Tables 5-7, which have a similar structure to Tables 2-4 described in Section 4.3.1. \begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \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 & 0.0000 & 0.0000 & 1.0000 & 0.9985 \\ \hline BM1 & Late\_Postop & 0.0000 & 0.0000 & 0.0000 & 0.0115 & 0.0010 & 0.3015 & 0.1735 & 1.0000 & 0.5240 \\ \hline BM2 & Preop & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0005 & 0.0000 & 1.0000 & 0.9875 \\ \hline BM2 & Late\_Postop & 0.0010 & 0.0000 & 0.0000 & 0.0080 & 0.0010 & 0.3480 & 0.1570 & 1.0000 & 0.5735 \\ \hline BM3 & Preop & 0.0000 & 0.0000 & 0.0000 & 0.0005 & 0.0000 & 0.0000 & 0.0000 & 1.0000 & 0.9980 \\ \hline BM3 & Late\_Postop & 0.0010 & 0.0000 & 0.0000 & 0.0080 & 0.0020 & 0.3000 & 0.1715 & 1.0000 & 0.4940 \\ \hline BM4 & Preop & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.0000 & 0.9945 \\ \hline BM4 & Late\_Postop & 0.0000 & 0.0000 & 0.0000 & 0.0080 & 0.0005 & 0.3165 & 0.1450 & 1.0000 & 0.5945 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The \(p\)-values of the fuzzy Bootstrap statistical tests in MFPCG for parameter _MR_, comparing subgroups \(A_{1}\) and \(B_{1}\).