# Quasi-Experimental Design for Medical Studies with the Method of the Fuzzy Pseudo-Control Group
## 1 Definition and Necessity of Pseudo-Control Groups
The new medication, as a rule, is more expensive than its previous version(s), yet due to its favorable effect, it soon becomes the treatment norm, while at the same time, the higher price does not directly affect the patients as the medical insurance covers it. Modern research assumes that independent labs can replicate results from every published experiment. If discrepancies with the original study are identified, then they are immediately shared with the research community. This assumption is not valid for this new medication, though. Anyone with doubts about the new medication's favorable effect has no opportunity to experiment with a control group, as each patient should be given the best possible treatment. Again, the only possibility is to compare the effect of the medication in patients from the experimental group with that in patients with counterindications for that medication. The latter form a pseudo-control group because their medical condition differs from that of the patients in the experimental group. Let us reformulate the problem at hand. We explore the effect of a given intervention (impact, influence) \(R\) in conjunction with another base intervention, \(V\), over a given parameter \(X\), which describes the outcome of objects in a target population, \(P\). Assume that before and after interventions \(V\) and \(R\), we have measured the values of \(X\) for a given group of objects from \(P\), called the _experimental group_. We can identify the effect in question compared to the effect of the base intervention \(V\) over parameter \(X\), which characterizes the status of objects from a population \(Q\), similar to \(P\), called a _pseudo-control population_. We assume that before and after intervention \(V\), we have measured the values of \(X\) for a given group of objects from \(Q\), called the _pseudo-control group_. 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. \(PC_{b}\) is the sample that contains the values of parameter \(X\) for the pseudo-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 \(V\) and the investigated intervention \(R\). 4. \(PC_{e}\) is the sample that contains the values of parameter \(X\) for the pseudo-control group at the end of the experiment after the group has been subjected to the base intervention \(V\). Scientific experiments that require the assessment of an intervention effect over a target population, without a control group that can be allocated initially by the random assignment of the study units, are called quasi-experiments [6]. Most published studies using that approach never even mention that they have performed quasi-experiments [7]. The fact that pseudo-control groups are traditionally called control groups does not help in that regard. There are numerous quasi-experimental designs (panel data analysis, nonequivalent control group designs, case-control design, etc. ), but the most well-known and widely used pretest-posttest approaches are the difference-in-differences design (DID) and the regression discontinuity design (RDD). DID, in its basic form, is equivalent to regression analysis over two dummy variables--one for the time period and one for the group membership. It is an easy-to-estimate quantitative method with understandable ideas [8]. It will always assess (correctly or incorrectly) the effect of intervention \(R\) over parameter \(X\). However, the DID requires that all the assumptions of the least-squares model (the ordinary or the weighted one) hold [9]. One of the additional assumptions of DID is \"the parallel trends assumption\". The latter claims that the expected outcome for the treated and untreated populations would have been parallel if, counterfactually, no treatment was applied to both populations.