# Quasi-Experimental Design for Medical Studies with the Method of the Fuzzy Pseudo-Control Group
## 2 State of the Art in Bootstrap Statistical Testing with Fuzzy Samples  
Assume we have two one-dimensional (1D) samples of a continuous parameter \(Z\), with a total of \(n_{1}\) and \(n_{2}\) number of observations each, respectively. The observations \(z_{k}^{1}\) and \(z_{k}^{2}\) of the first and second samples, respectively, belong to Population 1 and Population 2 with degrees of membership \(\mu_{k}^{1}\) and \(\mu_{k}^{2}\), respectively. We can then form Fuzzy sample 1 (denoted \(Z^{1}\)) and Fuzzy sample 2 (denoted \(Z^{2}\)):  
\[Z^{1}=\Big{\{}\Big{(}z_{1}^{1}-\mu_{1}^{1}\Big{)},\Big{(}z_{2}^{1}-\mu_{2}^{1 }\Big{)},\ldots,\Big{(}z_{n_{1}}^{1}-\mu_{n_{1}}^{1}\Big{)}\Big{\}}, \tag{1}\]  
\[Z^{2}=\Big{\{}\Big{(}z_{1}^{2}-\mu_{1}^{2}\Big{)},\Big{(}z_{2}^{2}-\mu_{2}^{2 }\Big{)},\ldots,\Big{(}z_{n_{2}}^{2}-\mu_{n_{2}}^{2}\Big{)}\Big{\}}. \tag{2}\]  
We wish to explore how the different conditions to form the fuzzy samples influence the continuous 1D parameter \(Z\) values. In our setting, fuzzy samples are formed from two different populations, and we need to test whether \(Z\) from Population 1 has the same distribution as \(Z\) from Population 2. We can use every non-fuzzy sample that describes a given random variable (r.v.) to approximate the cumulative distribution function (CDF) of that r.v. using an _empirical cumulative distribution function_ (ECDF) [17]. The only condition to build ECDF is that the measurements in the sample should be independent and identically distributed (i.i.d. ):  
\[CDF_{j}(z)\approx ECDF_{j}(z)=\frac{1}{n_{j}}\sum_{ \begin{array}{c}k=1\\ z_{k}^{j}\leq\ z\end{array} }^{n_{j}}1,\text{ for }z\in(-\infty;+\infty)\text{ and }j=1,2. \tag{3}\]  
ECDF interprets the observations in \(Z^{1}\) and \(Z^{2}\) as non-fuzzy and neglects the information in the degrees of membership of the observations. Equation (3) assumes that we construct a discrete probability mass function (PMF), which approximates the density of the continuous r.v. \(Z\). In this PMF, the probability that the r.v. takes an arbitrary value equals the relative frequency of that value in the sample. We can use other CDF approximations under different assumptions for the observations in the sample (e.g., continuous linear, functional continuous linear, and granular continuous linear) [18]. Using the information in the degrees of membership from Fuzzy sample \(Z^{1}\) and Fuzzy sample \(Z^{2}\), we can derive a fuzzy empirical sample approximation of the CDF or an r.v. as a _fuzzy empirical distribution function_ (FECDF). This is a generalization of the ECDF [19]:  
\[CDF_{j}(z)\approx FECDF_{j}(z)= \sum_{ \begin{array}{c}k=1\\ z^{j}_{k}\leq\ z\end{array} }^{n_{j}}\mu_{k}^{i}\Big{/}\sum_{k=1}^{m_{j}}\mu_{k}^ {j},\ \text{for}\ z\in(-\infty;+\infty)\ \text{and}\ j=1,2. \tag{4}\]  
In an implicit form [20], FECDF uses the probability mass function, yet here, the probability for an arbitrary value of the r.v. is the relative weight of the degrees of membership of that value in the fuzzy sample. There are no analytical generalizations to calculate the \(p\)-value of most statistical tests over fuzzy samples (unlike crisp samples). An alternative approach is to use fuzzy Bootstrap simulation to calculate the conditional distributions of a given statistic if the null hypothesis, \(H_{0}\) (that the populations have the same statistics or distributions), is true [21]. Bootstrap simulation is a computer-intensive technique that uses \(N\) pseudo-realities [22], where \(N\) is a large natural. It has proven effective for hypothesis testing over fuzzy data [23; 24]. A fuzzy Bootstrap procedure only requires that the observations are i.i.d. The fuzzy Bootstrap procedures we discuss below have four modifications each (denoted BM1 through BM4).