# 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  
We shall demonstrate the effect of annuloplasty according to two of the most significant integral diagnostic parameters that summarize the IMR status of the patient: \(RF\) (regurgitation fraction, in %) and \(MR\) (mitral regurgitation in an 8-level scale). Both parameters are measured three times: preoperatively (when the patient is admitted to the Clinic for cardiovascular surgery), early postoperatively (7-10 days after surgery), and late postoperatively (ambulatory check-ups from 6 to 54 months after surgery). The \(RF\) (%) is interpreted through three separate parameters: preoperative regurgitation fraction (_Preop_RF_), early postoperative regurgitation fraction (_Early_Postop_RF_), and late postoperative regurgitation fraction (_Late_Postop_RF_). In each of its three forms, this parameter is a continuous variable measured in % and is calculated as the regurgitation volume divided by the total ejected volume (_LVEDV_-_LVESV_) [55]:  
\[RF=100\times RV/(LVEDV-LVESV). \tag{22}\]  
Here, _LVEDV_ is the left ventricular end-systolic volume in mL, _LVESV_ is the left ventricular end-diastolic volume in mL, and \(RV\) is the regurgitation volume through the MV in mL, which is the volume of blood that returns into the atrium through the mitral valve. _LVEDV_ and _LVESV_ are calculated using a modified Simpson's rule with an apical 2 nd and 4 th chamber position, described in [56]. RV is measured from the 4 th apical position using color and CW Doppler, and the radius of the proximal isovelocity surface area (_PISAr_) is manually measured using a magnified image and reduced Nyquist usually to 40-45, which allows us to outline the boundaries of _PISA_ clearly. Each of the three parameters is measured three times, as a preoperative (_Preop_LVEDV_, _Preop_LVESV_, _Preop_RV_), early postoperative (_Early_Postop_LVEDV_, _Early_Postop_LVESV_, _Early_Postop_RV_), and late postoperative (_Late_Postop_LVEDV_, _Late_Postop_LVESV_, _Late_Postop_RV_) parameter. In that sense, Equation (22) is three formulae, one per each period. According to Section 3.1, expert estimates were obtained as follows: \(X_{opt,A}=RF_{opt,A}=0\%\), \(X_{opt,up}=RF_{opt,up}=1\%\), and \(\Delta X=\Delta RF=2\%\). The grade of mitral regurgitation (\(MR\)) is measured by an 8-level scale: 0--lack of \(MR\); 1--(grade 0-I) trivial \(MR\); 2--(grade I) mild \(MR\); 3--(grade I-II) mild-to-moderate \(MR\); 4--(grade II) moderate \(MR\); 5--(grade II-III) moderate-to-high \(MR\); 6--(grade III) high \(MR\); 7--(grade above III) severe \(MR\). The grade of \(MR\) is presented through three parameters: preoperative grade of mitral regurgitation (_Preop_MR_), early postoperative grade of mitral regurgitation (_Early_Postop_MR_), and late postoperative grade of mitral regurgitation (_Late_Postop_MR_). Strictly speaking, _MR_ is a discrete parameter, yet the large count of discretes (eight) allows for an approximation by a continuous parameter at a negligible discretization error. In this way, we illustrate the ability of MFPCG to assess the influence of the effect using a continuous parameter \(X\), measured in an ordinal scale with five or more discretes. According to Section 3.1, expert estimates were obtained as follows: \(X_{opt,d}=MR_{opt,d}=0\), \(X_{opt,up}=MR_{opt,up}=1\), and \(\Delta X=\Delta MR=0.5\). On the one hand, we assess the effect of annuloplasty using MFPCG for patients with a relatively preserved medical state by comparing the values of both parameters for subgroups \(A_{1}\) and \(B_{1}\). We use four fuzzy samples for _RF_ and four fuzzy samples for _MR_, as Sections 4.3.1 and 4.3.2 demonstrate. The fuzzy samples are formed using all patients with the necessary characteristics, regardless of whether those are typical or outliers. On the other hand, we assess the same effect for patients with a relatively deteriorated medical state and compare the values of _MR_ and _RF_ for subgroups \(A_{2}\) and \(B_{2}\).