# Quasi-Experimental Design for Medical Studies with the Method of the Fuzzy Pseudo-Control Group
## 3 The Method of the Fuzzy Pseudo-Control Group
### A Favorability Assessment of the Difference Between Populations  
For the experimental group, we can calculate the sample fuzzy numerical characteristics of the distribution of \(X\) from Fuzzy sample 1, denoted as follows: \(M_{E}\)--sample fuzzy mean value in the experimental group; \(MED_{E}\)--sample fuzzy median in the experimental group; \(VAR_{E}\)--sample fuzzy variance in the experimental group; \(IQR_{E}\)--sample fuzzy interquartile range in the experimental group. Similarly, for the pseudo-control group, we can calculate the sample fuzzy numerical characteristics of the distribution of \(X\) from Fuzzy sample 2, denoted as follows: \(M_{PC}\)--sample fuzzy mean in the pseudo-control group; \(MED_{PC}\)--sample fuzzy median in the pseudo-control group; \(VAR_{PC}\)--sample fuzzy variance in the pseudo-control group; \(IQR_{PC}\)--sample fuzzy interquartile range in the pseudo-control group. Each distribution has multiple measures of location and multiple measures of dispersion. For the sake of simplicity, from this point forward in the paper, the term \"measures of location\" will refer only to the mean and/or the median. Similarly, \"measures of dispersion\" will refer only to the variance (the squared standard deviation) and/or the interquartile range. We will assume hill or monotonic preferences over the \(X\) values, as discussed in Section 3.1. Below, we present an algorithm to assess the favorability of differences between the fuzzy measures of location in the two populations. Stage 2 Algorithm: A Favorability Assessment of the Difference Between the Fuzzy Central Tendencies of Two Populations  
**For fuzzy means:**  
If \(\left|M_{E}-M_{PC}\right|<\Delta X\), or \(\left(X_{d,opt}-\Delta X\right)<M_{E}<M_{PC}<\left(X_{u,opt}+\Delta X\right)\), or \(\left(X_{d,opt}-\Delta X\right)<M_{PC}<M_{E}<\left(X_{u,opt}+\Delta X\right)\), then \(M_{E}\) is assumed to be neutral to \(M_{PC}\). If \(M_{PC}\leq min\Big{\{}X_{d,opt}-\Delta X,M_{E}-\Delta X\Big{\}}\leq X_{d,opt}- \Delta X<M_{E}<X_{u,opt}+\Delta X\), or \(\left(X_{d,opt}-\Delta X\right)<M_{E}<X_{u,opt}+dX\leq max\big{\{}X_{u,opt}+ dX,M_{E}+dX\big{\}}\leq M_{PC}\), or \(\left(X_{u,opt}+\Delta X\right)\leq M_{E}\leq\left(M_{PC}-\Delta X\right)\), or \(\left(M_{PC}+\Delta X\right)\leq M_{E}\leq\left(M_{dep}-\Delta X\right)\), then \(M_{E}\) is assumed to be more favorable than \(M_{PC}\). If \(M_{E}\leq min\Big{\{}X_{d,opt}-\Delta X,M_{PC}-\Delta X\Big{\}}\leq X_{d,opt}- \Delta X<M_{PC}<X_{u,opt}+\Delta X\), or \(\left(X_{d,opt}-\Delta X\right)<M_{PC}<X_{u,opt}+dX\leq max\big{\{}X_{u,opt}+ dX,M_{PC}+dX\big{\}}\leq M_{E}\), or \(\left(X_{u,opt}+\Delta X\right)\leq M_{PC}\leq\left(M_{E}-\Delta X\right)\), or \(\left(M_{E}+\Delta X\right)\leq M_{PC}\leq\left(M_{E}-\Delta X\right)\), then \(M_{E}\) is assumed to be less favorable than \(M_{PC}\). If \(M_{E}\leq\left(X_{d,opt}-\Delta X\right)<\left(X_{u,opt}+\Delta X\right)\leq M_ {PC}\), or \(M_{PC}\leq\left(X_{d,opt}-\Delta X\right)<(X_{u,opt}+\Delta X)\leq M_{E}\), then the favorability of \(M_{E}\) compared to \(M_{PC}\) is problem-specific and should be defined by an expert in the respective field. Such cases only rarely occur, as they indicate excessive medical intervention. **For fuzzy medians:**  
If either \(\left|MED_{E}-MED_{PC}\right|<\Delta X\), or \(\left(X_{d,opt}-\Delta X\right)<MED_{E}<MED_{PC}<\left(X_{u,opt}+\Delta X\right)\), or \(\left(X_{d,opt}-\Delta X\right)<MED_{PC}<MED_{E}<\left(X_{u,opt}+\Delta X\right)\), then \(MED_{E}\) is assumed to be neutral to \(MED_{PC}\). If \(MED_{PC}\leq min\Big{\{}X_{d,opt}-\Delta X,MED_{E}-\Delta X\Big{\}}\leq X_{d, opt}-\Delta X<MED_{E}<X_{u,opt}+\Delta X\), or \(MED_{PC}\), or \(\left(X_{u,opt}+\Delta X\right)\leq MED_{E}\leq\left(MED_{PC}-\Delta X\right)\), or \(\left(MED_{PC}+\Delta X\right)\leq MED_{E}\leq\left(X_{d,opt}-\Delta X\right)\), then \(MED_{E}\) is assumed to be more favorable than \(MED_{PC}\).