# 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)
\(Cond_{1}=\mathrm{F}\), since the variances in the populations \(A_{1}\) and \(B_{1}\) are statistically different. \(Cond_{2}=\mathrm{F}\), since the variances in the populations \(A_{1}\) and \(B_{1}\) are statistically distinguishable. \(Cond_{3}=\mathrm{F}\), since the distributions of \(RF\) in the populations \(A_{1}\) and \(B_{1}\) are not borderline statistically significantly discernible. According to the first line of Table 1, the late postoperative differences in the continuous parameter \(RF\) between the populations \(A_{1}\) and \(B_{1}\) are categorized as \(C_{t}=C_{+1}\). The \begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline
**Modification** & **Time** & \(\mathbf{C_{M}}\) & \(\mathbf{C_{MED}}\) & \(\mathbf{Cond_{1}}\) & \(\mathbf{Cond_{2}}\) & \(\mathbf{Cond_{3}}\) & \(\mathbf{C_{t}}\) & **Rule** &
\begin{tabular}{l} **Result of** \\ **MFPCG** \\ \end{tabular} & **Class** \\ \hline BM1 & Preop & \(M_{-1}\) & \(MED_{-1}\) & \(\mathrm{T}\) & \(\mathrm{T}\) & \(\mathrm{F}\) & \(C_{-1}\) & 5 & \((C_{b},C_{e})=\) \\  & & & & & & & & & \\ \hline BM1 & Late\_Postop & \(M_{+1}\) & \(MED_{+1}\) & \(\mathrm{F}\) & \(\mathrm{F}\) & \(\mathrm{F}\) & \(C_{+1}\) & 1 & \((C_{-1},C_{+1})\) & YES+ \\ \hline BM2 & Preop & \(M_{-1}\) & \(MED_{-1}\) & \(\mathrm{T}\) & \(\mathrm{T}\) & \(\mathrm{F}\) & \(C_{-1}\) & 5 & \((C_{b},C_{e})=\) \\  & & & & & & & & & \\ \hline BM2 & Late\_Postop & \(M_{+1}\) & \(MED_{+1}\) & \(\mathrm{F}\) & \(\mathrm{F}\) & \(\mathrm{F}\) & \(C_{+1}\) & 1 & \((C_{-1},C_{+1})\) & YES+ \\ \hline BM3 & Preop & \(M_{-1}\) & \(MED_{-1}\) & \(\mathrm{T}\) & \(\mathrm{T}\) & \(\mathrm{F}\) & \(C_{-1}\) & 5 & \((C_{b},C_{e})=\) \\  & & & & & & & & & \\ \hline BM3 & Late\_Postop & \(M_{+1}\) & \(MED_{+1}\) & \(\mathrm{F}\) & \(\mathrm{F}\) & \(\mathrm{F}\) & \(C_{+1}\) & 1 & \((C_{-1},C_{+1})\) & YES+ \\ \hline BM4 & Preop & \(M_{-1}\) & \(MED_{-1}\) & \(\mathrm{T}\) & \(\mathrm{T}\) & \(\mathrm{F}\) & \(C_{-1}\) & 5 & \((C_{b},C_{e})=\) \\  & & & & & & & & & \\ \hline BM4 & Late\_Postop & \(M_{+1}\) & \(MED_{+1}\) & \(\mathrm{F}\) & \(\mathrm{F}\) & \(\mathrm{F}\) & \(C_{+1}\) & 1 & \((C_{-1},C_{+1})\) & YES+ \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The MFPCG diagnostics of the influence of \(R\) on a continuous parameter \(RF\), comparing subgroups \(A_{1}\) and \(B_{1}\). same line in Table 1 shows that we have applied Rule 1: if one of the measures of location of \(RF\) in the target population \(A_{1}\) is statistically significantly more favorable than that in the pseudo-control population \(B_{1}\), whereas the other measure of location in the target population \(A_{1}\) is neither statistically significantly nor borderline statistically significantly less favorable than that in the pseudo-control population \(B_{1}\), then categorize in \(C_{+1}\). Columns 3 to 9 on line three of Table 4 show the input variables' values and the categorization function's values. We obtained similar results for the remaining three Bootstrap Modifications, shown in lines 5, 7, and 9 of Table 4.  
According to Section 3.5, the result from MFPCG for BM1 Bootstrap is (\(C_{b}\), \(C_{e}\)) = (\(C_{-1}\), \(C_{+1}\)). That result is in line two of column 10 of Table 4. Hence, \(b=-1\), and \(e=1\). According to Equation (21), the result (\(C_{b}\), \(C_{e}\)) is classified in class 'YES+' since \(b-e=(-1)-1=-2<-1/2\). The resulting class shows that according to BM1 Bootstrap, the influence of annuloplasty has a statistically significantly favorable influence over parameter \(RF\) in populations \(A_{1}\) and \(B_{1}\). This is indicated in line two of the last column of Table 4.  
Similar results can be obtained for the other three Bootstrap Modifications. The results are in lines 4, 6, and 8 of Table 4.