# 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
For the test statistic, the works [32; 33] use the difference,  
\[\Delta_{1-2,r}^{mean}=\widehat{M}_{1}-\widehat{M}_{2} \tag{9}\]  
The difference (9) is a realization of the random variable \(\Delta_{1-2}^{mean}\). The work [32] presents a Bootstrap statistical test in eight variants to explore the difference in the means of two populations using fuzzy samples. The null hypothesis \(H_{0}\) for all tests is that the populations have equal means. The alternative hypothesis \(H_{1}\) varies depending on the test. The tests calculate the \(p\)-value, i.e., the probability of observing a difference between the means of the fuzzy samples as least as great as the measured one, if \(H_{0}\) is true. The algorithms to calculate the \(p\)-values of those Bootstrap variants are also presented. Assume that the elements of the fuzzy samples \(Z^{1}\) and \(Z^{2}\) from (1) and (2) are sorted to derive the sorted fuzzy samples, \(Z^{1,sort}\) and \(Z^{2,sort}\):  
\[Z^{1,sort}=\Big{\{}\Big{(}z_{1}^{1,sort}-\mu_{1}^{1,sort}\Big{)},\Big{(}z_{2 }^{1,sort}-\mu_{2}^{1,sort}\Big{)},\ldots,\Big{(}z_{n_{1}}^{1,sort}-\mu_{n_{ 1}}^{1,sort}\Big{)}\Big{\}} \tag{10}\]  
where \(z_{1}^{1,sort}\leq z_{2}^{1,sort}\leq\ldots\leq z_{n_{1}}^{1,sort}\),  
\[Z^{2,sort}=\Big{\{}\Big{(}z_{1}^{2,sort}-\mu_{1}^{2,sort}\Big{)},\Big{(}z_{2 }^{2,sort}-\mu_{2}^{2,sort}\Big{)},\ldots,\Big{(}z_{n_{1}}^{2,sort}-\mu_{n_{ 1}}^{2,sort}\Big{)}\Big{\}} \tag{11}\]  
where \(z_{1}^{2,sort}\leq z_{2}^{2,sort}\leq z_{3}^{2,sort}\leq\ldots\leq z_{n_{2 }}^{2,sort}\). The work [33] offers a generalized procedure to calculate a fuzzy \(p\)-quantile of a distribution using sorted data from a fuzzy sample. The procedure uses the real function \(q^{j}(. )\) from (12) for \(p\in[0;1]\), which is linearly approximated on the nodes \(\Big{(}p_{i}^{j},q_{i}^{j}\Big{)}\) given in (13) (where \(j=1,2\) refers to the sorted samples (10) and (11)):  
\[q^{j}(p)=\left\{ \begin{array}{ll}q_{i}^{j}&\text{for }p=p_{i}^{j}\\ q_{i}^{j}+\frac{\big{(}q_{i+1}^{j}-q_{i}^{j}\big{)}\big{(}p-p_{i}^{j}\big{)}} {\Big{(}p_{i+1}^{j}-p_{i}^{j}\big{)}}&\text{for }p_{i}^{j}<p<p_{i+1}^{j}\end{array} \right.,\text{ for }j=1,2, \tag{12}\]\[\left(p_{i^{\prime}}^{j},q_{i}^{j}\right)=\left\{ \begin{array}{ll}\left(0,z_{1}^ {jsort}\right)&\text{for $i=0$}\\ \left(\frac{\mu_{1}^{jsort}}{2},z_{1}^{jsort}\right)&\text{for $i=1$}\\ \left(\sum\limits_{k=1}^{i-1}\mu_{k}^{jsort}+\frac{\mu_{i}^{jsort}}{2},z_{i}^ {jsort}\right)&\text{for $i=2,3,\ldots,n_{j}$}\\ \left(1,z_{n_{j}}^{jsort}\right)&\text{for $i=n_{j}+1$}\end{array} \right.,\text{ for $j=1,2$.} \tag{13}\]  
The functions (12) assess the fuzzy \(p\)-quantile of the continuous r.v. \(Z\) using the sorted fuzzy samples (10) and (11). Assume that \(MED_{1}\) and \(MED_{2}\) are the medians for Populations 1 and 2, respectively. We can calculate the fuzzy sample medians, \(\widehat{MED_{1}}\) and \(\widehat{MED_{2}}\), using the function (12) as follows:  
\[\widehat{MED_{j}}=q^{j}(0.5),\text{ for $j=1,2$} \tag{14}\]  
The test statistic in [33; 35; 36] is the difference:  
\[\Delta_{1-2,r}^{med}=\widehat{MED_{1}}-\widehat{MED_{2}}. \tag{15}\]  
The difference (15) is a realization of the random variable \(\Delta_{1-2}^{med}\). The Bootstrap algorithms to test the equality of medians over non-fuzzy samples are presented in [35; 36]. Assume that \(VAR_{1}\) and \(VAR_{2}\) are the variances for Populations 1 and 2, respectively. We can calculate the fuzzy sample variances, \(\widehat{VAR_{1}}\) and \(\widehat{VAR_{2}}\), as  
\[\widehat{VAR_{j}}=\frac{\sum\limits_{k=1}^{n_{j}}\mu_{k}^{j}\left(z_{k}^{j}- \widehat{M}_{j}\right)^{2}}{\sum\limits_{k=1}^{n_{j}}\mu_{k}^{j}-max\left\{ \mu_{1}^{j},\mu_{2}^{j},\ldots,\mu_{n_{j}}^{j}\right\}},\text{ for $j=1,2$.} \tag{16}\]  
In (16), \(\widehat{M}_{j}\) are the sample fuzzy means calculated using (8). For the test statistic, we can use the ratio  
\[R_{1/2,r}^{var}=\widehat{VAR_{1}}/\widehat{VAR_{2}}. \tag{17}\]  
The ratio (17) is a realization of the random variable \(R_{1/2}^{var}\).