# 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)  
In task 1, we use four fuzzy samples for _RF_, as described below. \(E_{b}\) is a fuzzy sample that contains 34 values of _Preop_RF_ and their degrees of membership to the experimental group \(A_{1}\):  
\(E_{b}\) = \(\chi^{RF}_{A_{1},prop}\) = {(57.0.630), (63,0.900), (42,0.630), (42,0.630), (57,0.360), (64,0.810), (39,0.810), (47,0.630), (55,0.900), (65,0.630), (37,0.630), (67,0.630), (65,0.490), (62,0.630), (65,0.700), (31,0.490), (53,0.630), (64,0.630), (44,0.630), (56,0.630), (68,0.700), (43,0.900), (55,0.630), (36,0.630), (55,0.490), (55,0.900), (68,0.630), (37,0.490), (64,0.630), (56,0.490), (49,0.630), (69,0.700), (45,0.900), (56,0.700)). \(PC_{b}\) is a fuzzy sample that contains 43 values of _Preop_RF_ and their degrees of membership to the pseudo-control group \(B_{1}\):  
\(PC_{b}=\chi_{B_{1},\textit{prepop}}^{RF}=\{\)(34,0.900), (62,0.810), (28,0.900), (38,0.900), (59,0.630), (51,0.810), (38,0.630), (39,0.900), (36,0.810), (28,0.900), (24,0.900), (59,0.630), (38,0.490), (49,0.810), (40,0.900), (59,0.630), (35,0.630), (47,0.630), (24,0.490), (48,0.810), (25,0.900), (38,0.490), (34,0.900), (32,0.900), (21,0.490), (37,0.810), (22,0.630), (26,0.900), (34,0.810), (29,0.630), (36,0.900), (36,0.630), (60,0.490), (20,0.630), (21,0.900), (32,0.900), (15,0.900), (19,0.900), (31,0.700), (41,0.700), (36,0.700), (39,0.700), (21,0.357)). \(E_{e}\) is a fuzzy sample that contains 32 values of _Late_Postop_RF_ and their degrees of membership to the experimental group \(A_{1}\):  
\(E_{e}=\chi_{A_{1},\textit{LPpostp}}^{RF}=\{\)(0,0.630), (37,0.900), (27,0.630), (43,0.630), (0,0.360), (24,0.810), (18,0.810), (12,0.630), (0,0.900), (0,0.630), (0,0.630), (0,0.630), (0,0.630), (0,0.630), (13,0.700), (0,0.630), (0,0.630), (0,0.630), (0,0.700), (0,0.900), (17,0.630), (12,0.630), (0,0.490), (0,0.900), (0,0.630), (0,0.490), (0,0.630), (18,0.490), (0,0.630), (8,0.700), (7,0.900), (0,0.700)). \(PC_{e}\) is a fuzzy sample that contains 41 values of _Late_Postop_RF_ and their degrees of membership to the pseudo-control group \(B_{1}\):  
\(PC_{e}=\chi_{B_{1},\textit{LPpostp}}^{RF}=\{\)(0,0.900), (43,0.810), (6,0.900), (13,0.900), (88,0.630), (39,0.810), (48,0.630), (12,0.900), (65,0.810), (14,0.900), (0,0.900), (48,0.490), (41,0.810), (11,0.900), (38,0.630), (12,0.630), (14,0.630), (36,0.490), (58,0.810), (27,0.900), (0,0.490), (19,0.900), (24,0.900), (65,0.490), (23,0.630), (12,0.900), (43,0.810), (10,0.630), (18,0.900), (30,0.630), (41,0.490), (23,0.630), (0,0.900), (21,0.900), (33,0.900), (12,0.900), (0,0.700), (22,0.700), (0,0.700), (24,0.700), (0,0.357)]. Table 2 presents the numerical characteristics of the four fuzzy samples. First, let us assess the influence of annuloplasty on the preoperative \(RF\) values. According to Section 3.2, from \(RF_{u,opt}+\Delta RF=1+2=3\%\leq M_{\textit{PC}}=35.5\%\leq(M_{E}-\Delta RF)=54 -2=52\%\), it follows that \(M_{E}\) is less favorable than \(M_{\textit{PC}}\). This is indicated in column four of Table 2. From \(RF_{u,opt}+\Delta RF=1+2=3\%\leq MED_{PC}=35.2\%\leq(MED_{E}-\Delta RF)=55.1-2=53.1\%\), it follows that \(MED_{E}\) is less favorable than \(MED_{PC}\). This is indicated in column four of Table 2.  
Line two of Table 3 presents the \(p\)-values for the nine Bootstrap statistical tests for the BM1 Bootstrap. According to the Stage 3 Algorithm, since \(Pvalue_{1}=0\leq\alpha=0.05\), then the preoperative population distributions of \(RF\) are assumed to be statistically significantly different. Since \(Pvalue_{3}=0\leq Pvalue_{2}=0\leq\alpha=0.05\) and \(M_{E}=54\%>M_{\textit{PC}}=35.5\%\), then the preoperative mean in the target population \(A_{1}\) is assumed to be statistically significantly greater than that in the pseudo-control population \(B_{1}\).