# 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)
Since \(Pvalue_{5}=0\leq Pvalue_{4}=0\leq\alpha=0.05\) and \(MED_{E}=55.1\%>MED_{PC}=35.2\%\), then the preoperative median of the target population \(A_{1}\) is assumed to be statistically significantly greater than that in the pseudo-control population \(B_{1}\). Since \(\alpha=0.05<Pvalue_{7}=0.248\leq Pvalue_{6}=0.4995\), then the preoperative variance of the target population \(A_{1}\) is assumed statistically indistinguishable \begin{table}
\begin{tabular}{l l l l l l l} \hline
**Sample** & \(E_{b}\) & \(PC_{b}\) & \(R\) **Favorability** & \(E_{e}\) & \(PC_{e}\) & \(R\) **Favorability** \\ \hline No. of observations & 34 & 43 & N/A & 32 & 41 & N/A \\ \hline Fuzzy mean, \% & 54 & 35.5 & unfavorable & 7.79 & 24.2 & favorable \\ \hline Fuzzy median, \% & 55.1 & 35.2 & unfavorable & 0 & 20.7 & favorable \\ \hline Fuzzy STD, \% & 10.8 & 11.8 & N/A & 12.1 & 20.1 & N/A \\ \hline Fuzzy IQR, \% & 19.9 & 12.3 & N/A & 12.8 & 26.6 & N/A \\ \hline \end{tabular}
\end{table} 
Table 2: The numerical characteristics and change favorability for the \(RF\) fuzzy samples for groups \(A_{1}\) and \(B_{1}\). from that in the pseudo-control population \(B_{1}\). Since \(\alpha=0.05<Pvalue_{9}=0.0975\leq Pvalue_{8}=0.169\), then the preoperative interquartile interval of the target population \(A_{1}\) is assumed statistically indistinguishable from that in the pseudo-control population \(B_{1}\). We obtained detailed results for the other three Bootstrap Modifications. Their \(p\)-values are given in lines 4, 6, and 8 of Table 3.  
Let us categorize the differences between \(A_{1}\) and \(B_{1}\) using the BM1 Bootstrap. Section 3.4 defines the values of the five input variables of the discrete categorization function (20). The preoperative differences in the means of \(RF\) in the populations \(A_{1}\) and \(B_{1}\) are \(C_{M}=M_{-1}\) (the preoperative mean of \(RF\) in the target population \(A_{1}\) is statistically significantly less favorable than that in the pseudo-control population \(B_{1}\)). The preoperative differences in the medians of \(RF\) in the populations \(A_{1}\) and \(B_{1}\) are \(C_{MED}=MED_{1}\) (the preoperative median of \(RF\) in the target population \(A_{1}\) is statistically significantly less favorable than that in the pseudo-control population \(B_{1}\)). \(Cond_{1}=\text{T}\), since the distributions of \(RF\) in populations \(A_{1}\) and \(B_{1}\) are statistically different, and the two characteristics of dispersion in populations \(A_{1}\) and \(B_{1}\) are not statistically different. \(Cond_{2}=\text{T}\), since the distributions of \(RF\) in populations \(A_{1}\) and \(B_{1}\) are statistically indistinguishable. \(Cond_{3}=\text{F}\), since the distributions of \(RF\) in the populations \(A_{1}\) and \(B_{1}\) are not borderline statistically significantly different. According to the last line of Table 1, the preoperative differences in the continuous parameter \(RF\) between the two populations \(A_{1}\) and \(B_{1}\) are categorized as \(C_{t}=C_{-1}\). We can see from the same line of Table 1 that we have applied Rule 5: if one of the measures of location of \(RF\) in the target population \(A_{1}\) is statistically significantly less 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 more favorable than that in the pseudo-control population \(B_{1}\), then categorize in \(C_{-1}\). Columns 3 to 9 on line two 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 4, 6, and 8 of Table 4.  
Let us now assess the influence of annuloplasty on the late postoperative \(RF\) values. According to Section 3.2, from \(RF_{u,opt}+\Delta RF=1+2=3\%\leq M_{E}=7.79\%\leq(M_{PC}-\Delta RF)=24.2\% -2\%=22.2\%\), it follows that \(M_{E}\) is more favorable than \(M_{PC}\).