# 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
Depending on how we form the synthetic samples, we have quasi-equal information generation (i.e., the synthetic samples have almost the same quantity of information as the original sample) and equal-size generation (i.e., the synthetic samples have the same number of fuzzy observations as the original sample). Regarding the type of CDF approximation, we can use either ECDF or FECDF. As a result, we can define the four Bootstrap Modifications:  
* BM1: Fuzzy Bootstrap with quasi-equal-information generation using an ECDF. In each pseudo-reality, any synthetic sample is generated from the ECDF (constructed using (3) from the original sample) so that the degree of membership sum is almost identical to the same sum for the original sample. It is unlikely that the synthetic and original samples will have the same cardinality. * BM2: Fuzzy Bootstrap with quasi-equal-information generation using a FECDF. In each pseudo-reality, any synthetic sample is generated from the FECDF (constructed using (4) from the original sample) so that the degree of membership sum is almost identical to the same sum at the original sample. It is unlikely that the synthetic and original samples will have the same cardinality. * BM3: Fuzzy Bootstrap with equal-size generation using an ECDF. In each pseudo-reality, any synthetic sample is generated from the ECDF (constructed using (3) from the original sample) with cardinality equal to the cardinality of the original sample. It is unlikely that the synthetic and original samples will have the same degree of membership sums. * BM4: Fuzzy Bootstrap with equal-size generation using a FECDF. In each pseudo-reality, any synthetic sample is generated from the FECDF (constructed using (4) from the original sample) with cardinality equal to the cardinality of the original sample. It is unlikely that the synthetic and original samples will have the same degree of membership sums. We reiterate that these modifications are valid for all Bootstrap tests we present below. The test statistic measures the difference between two sample CDFs when testing the equality of two population distributions. This assumes that the two populations have identical underlying continuous distributions. There are three statistics classes when testing the equality of two continuous distributions. A typical representative of the quadratic class is the quadratic Anderson-Darling statistic and the Kramer-von Mises statistic [25]. The rank class uses metrics such as the Mann-Whitney \(U\) statistic and the Wilcoxon \(T\) statistic [26]. The most frequently used ones from the supremum class are the Kolmogorov-Smirnov [27] and its improved version--the Kuiper statistic (\(Ku\)) [28]. \(Ku\) is the sum of the supremum of positive differences and the supremum of the negative differences between two approximations of CDF based on the available samples:  
\[Ku=\underset{z}{sup}(CDF_{1}(z)-CDF_{2}(z))+\underset{z}{sup}(CDF_{2}(z)-CDF_{ 1}(z)). \tag{5}\]  
The statistic (5) has the same sensitivity to deviations for all values of \(Z\). For continuous CDF, the estimate of the supremum requires non-trivial optimization and takes considerable time. The supremum in (5) is often replaced with a maximum, as suggested in [29]. In the case of fuzzy samples with FEDF approximation of \(CDF_{1}\) and \(CDF_{2}\), we can represent (5) as  
\[Ku=\underset{k=1,2,\ldots,n_{1}}{max}\Big{(}FEDF_{1}\Big{(}z_{k}^{1}\Big{)}- FEDF_{2}\Big{(}z_{k}^{1}\Big{)}\Big{)}+\underset{k=1,2,\ldots,n_{2}}{max}\Big{(}FEDF_{2} \Big{(}z_{k}^{2}\Big{)}-FEDF_{1}\Big{(}z_{k}^{2}\Big{)}\Big{)}. \tag{6}\]  
The work [30] proves that in the case of FEDF calculated using (4)-(6), we can calculate \(Ku\) directly from the sample observations without the need to construct the FEDF:  
\[