# Quasi-Experimental Design for Medical Studies with the Method of the Fuzzy Pseudo-Control Group
## 1 Definition and Necessity of Pseudo-Control Groups  
We explore the effect of a given intervention \(R\) over a continuous parameter \(X\), which describes the status of objects in a population \(P\), called a _target population_. Parameter \(X\) should be selected to be a proxy variable for the outcome of the intervention. To solve such a task, we would usually conduct an experiment to measure the values of \(X\) before and after intervention \(R\) for a set of objects from \(P\), called an _experimental group_[1]. If we are only interested in the change in the parameter in the experimental group following the intervention, then we are at risk of reaching misleading conclusions. Some of the main reasons for this could be:  
* If we observed a favorable change in \(X\) that is due to the cumulative effect of factors unaccounted for in the study rather than to intervention \(R\) (that only weakly contributed to the situation, did not contribute, or even contributed unfavorably), then our conclusion that the effect of \(R\) over \(X\) for objects from \(P\) was favorable would be incorrect. * If we observed an unfavorable change in \(X\) that is due to the cumulative effect of factors unaccounted for in the study and not due to intervention \(R\) (that only weakly worsened the situation, did not contribute, or even contributed favorably), then our conclusion that the effect of \(R\) over \(X\) for objects from \(P\) was unfavorable would be incorrect. * If we observed a negligible change in \(X\) that combined the unfavorable effect of \(R\) over \(X\) and the cumulative favorable effect of the factors unaccounted for in the study, then our conclusion that there was no effect of \(R\) over \(X\) for objects from \(P\) would be incorrect. Therefore, experiments use a _control_ group comprising objects from \(P\) that are not subjected to the investigated intervention \(R\)[2]. Several methodologies generate adequate conclusions from experiments that use control and experimental groups. Often, if intervention \(R\) acts in conjunction with another base intervention, \(V\), whose effect cannot be paused during the experiment (e.g., time), then parameter \(X\) is measured before and after the intervention. As a result, we can form four samples as follows:  
1. \(E_{b}\) is the sample that contains the values of parameter \(X\) for the experimental group at the beginning of the experiment. 2. \(K_{b}\) is the sample that contains the values of parameter \(X\) for the control group at the beginning of the experiment. 3. \(E_{e}\) is the sample that contains the values of parameter \(X\) for the experimental group at the end of the experiment, after the group has been subjected to the base intervention  
of control groups [3]. We can define the effect of the investigated intervention \(R\) over the selected parameter \(X\) as follows:  
1. We test the samples \(E_{e}\) and \(K_{e}\) for equality: * If the values \(X\) in \(E_{e}\) are statistically significantly more favorable than in \(K_{e}\), then intervention \(R\) over parameter \(X\) is proven to be statistically favorable. * If the values \(X\) in \(E_{e}\) are statistically significantly less favorable than in \(K_{e}\), then intervention \(R\) over parameter \(X\) is proven to be statistically unfavorable. * If \(E_{e}\) and \(K_{e}\) have statistically indistinguishable values of \(X\), then the effect of intervention \(R\) over parameter \(X\) is considered statistically unproven. 2. We test, for nullity, the change in \(X\) in the paired samples \(E_{b}\) and \(E_{e}\) and the change in \(X\) in the paired samples \(K_{b}\) and \(K_{e}\): * If the temporal change (TC) in the experimental group is statistically significantly favorable, whereas the TC in the control group is statistically significantly unfavorable, then the effect of intervention \(R\) over parameter \(X\) is proven to be statistically favorable.