Computing Fuzzy p-Value

Description

Statistical testing hypotheses has an important rule for making decision in practical and applied problems. In traditional testing methods, data, parameters, hypotheses and other elements of problem are consider crisp. But in applied sciences such as economics, agriculture and social sciences, it may be confront with vague definitions and fuzzy concepts. In such situations, the classical methods can not solve the vague test and they need to generalized for using in fuzzy environments. The vagueness entrance in testing hypotheses problem can be done via data or/and hypotheses. Therefore, the following three major problems can be usually considered for a fuzzy environment: (1) testing crisp hypotheses based on fuzzy data, (2) testing fuzzy hypotheses based on crisp data, and (3) testing fuzzy hypotheses based on fuzzy data. Similar to the classical testing hypotheses, one can consider different procedure methods for solving the above mentioned problems such as Neyman-Pearson, Bayes, likelihood ratio, minimax and p-value. Computing Fuzzy p-Value package, i.e. Fuzzy.p.value package, is an open source (LGPL 3) package for R which investigate on the above three problems on the basis of fuzzy p-value approach. All formulas and given examples are match with Springer Chapter book (2016) to easily show the performance of the proposed methods.

Details

In this package, We consider only two distributions Normal and Poisson as two suitable candidates from the class of all continues and discrete distributions. This program extendable for other distributions which users can do it if needed.

Author(s)

Abbas Parchami

Maintainer: Abbas Parchami <parchami@uk.ac.ir>

References

Filzmoser, P., and Viertl, R. (2004). Testing hypotheses with fuzzy data: the fuzzy p-value. Metrika 59: 21-29.

Holena, M. (2004). Fuzzy hypotheses testing in a framework of fuzzy logic. Fuzzy Sets and Systems 145: 229-252.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2010). Fuzzy p-value in testing fuzzy hypotheses with crisp data. Statistical Papers 51: 209-226.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2012). Testing fuzzy hypotheses based on vague observations: a p-value approach. Statistical Papers 53: 469-484.

Wang, X., Kerre, E. E. (2001). Reasonable properties for the ordering of fuzzy quantities (II). Fuzzy Sets and Systems 118: 387-405.

Yuan, Y. (1991). Criteria for evaluating fuzzy ranking methods. Fuzzy Sets Syst 43: 139-157.

See Also

FuzzyNumbers

Class of linear fuzzy sets to construct concept "approximately bigger"

Description

The are several famous fuzzy sets which have meaningful means and are user friendly for introducing. Several linear functions are constructed in this package for easily inspiration to the fuzzy concepts \ll approximately smaller \gg , \ll approximately bigger \gg and \ll approximately equal \gg. For example, function B is one of them which users can use it to construct the fuzzy concept \ll approximately bigger \gg in fuzzy hypotheses. As presented bellow, the membership function of fuzzy set B has two parameters a and b:

B(a,b)(x)=\left\{ \begin{array}{lcc} 1 &\ \ if & \ \ b \leq x \\ \frac{x-a}{b-a} &\ \ if & \ \ a \leq x <b \\ 0 &\ \ if & \ \ x <a \end{array} \right.

Usage

B(a, b)

Arguments

Value

This function convert a fuzzy hypothesis to its boundary which is a triangular fuzzy number. Therefore, although the input of this function is not triangular fuzzy number, but its output is triangular.

References

Parchami, A., Taheri, S. M., and Mashinchi, M. (2010). Fuzzy p-value in testing fuzzy hypotheses with crisp data. Statistical Papers 51: 209-226.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2012). Testing fuzzy hypotheses based on vague observations: a p-value approach. Statistical Papers 53: 469-484.

Examples

# Introducing the fuzzy null hypothesis H0
H0 = B(1275,1325)


## The function is currently defined as
function (a, b) 
(TriangularFuzzyNumber(a, b, b))

Class of linear fuzzy sets to construct concept "approximately smaller"

Description

The are several famous fuzzy sets which have meaningful means and are user friendly for introducing. Several linear functions are constructed in this package for easily inspiration to the fuzzy concepts \ll approximately smaller \gg , \ll approximately bigger \gg and \ll approximately equal \gg. For example, function S is one of them which can users can use it to construct the fuzzy concept \ll approximately smaller \gg in fuzzy hypotheses. As presented bellow, the membership function of fuzzy set S has two parameters a and b:

S(a,b)(x)=\left\{ \begin{array}{lcc} 1 &\ \ if & \ \ x \leq a \\ \frac{x-b}{a-b} &\ \ if & \ \ a < x \leq b \\ 0 &\ \ if & \ \ x >b \end{array} \right.

Usage

S(a, b)

Arguments

Value

This function convert a fuzzy hypothesis to its boundary which is a triangular fuzzy number. Therefore, although the input of this function is not triangular fuzzy number, but its output is triangular.

References

Parchami, A., Taheri, S. M., and Mashinchi, M. (2010). Fuzzy p-value in testing fuzzy hypotheses with crisp data. Statistical Papers 51: 209-226.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2012). Testing fuzzy hypotheses based on vague observations: a p-value approach. Statistical Papers 53: 469-484.

Examples

# Introducing the fuzzy alternative hypothesis H1
H1 = S(1275,1325)

## The function is currently defined as
function (a, b) 
(TriangularFuzzyNumber(a, a, b))

Class of triangular fuzzy numbers

Description

The are several famous fuzzy sets which have meaningful means and are user friendly for introducing. Several linear functions are constructed in this package for easily inspiration to the fuzzy concepts \ll approximately smaller \gg , \ll approximately bigger \gg and \ll approximately equal \gg. For example, function T is one of them which users can use it to construct the fuzzy concept \ll approximately equal \gg in fuzzy statistics and fuzzy hypotheses. As presented bellow, the membership function of fuzzy set T has three parameters a, b and c:

T(a,b,c)(x)=\left\{ \begin{array}{lcc} \frac{x-a}{b-a} &\ \ if & \ \ a < x \leq b \\ \frac{x-c}{b-c} &\ \ if & \ \ b < x \leq c \\ 0 &\ \ if & \ \ elsewhere \end{array} \right.

Usage

T(a, b, c)

Arguments

Value

This function easily introduce the membership function of a triangular fuzzy number.

References

Parchami, A., Taheri, S. M., and Mashinchi, M. (2010). Fuzzy p-value in testing fuzzy hypotheses with crisp data. Statistical Papers 51: 209-226.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2012). Testing fuzzy hypotheses based on vague observations: a p-value approach. Statistical Papers 53: 469-484.

Examples

# Introducing the membership function of triangular fuzzy number (for test statistics)
t = T(2,4,7)

## The function is currently defined as
function (a, b, c) 
(TriangularFuzzyNumber(a, b, c))

Class of trapezoidal fuzzy numbers

Description

Several linear functions are constructed in this package for easily inspiration to the fuzzy concepts \ll approximately smaller \gg , \ll approximately bigger \gg and \ll approximately equal \gg. For example, function Tr is one of them which users can use it to construct the fuzzy concept \ll approximately equal \gg for fuzzy statistics or fuzzy hypotheses. As presented bellow, the membership function of fuzzy set Tr has four parameters a, b, c and d:

Tr(a,b,c,d)(x)=\left\{ \begin{array}{lcc} \frac{x-a}{b-a} &\ \ if & \ \ a < x \leq b \\ 1 &\ \ if & \ \ b < x \leq c \\ \frac{x-d}{c-d} &\ \ if & \ \ c < x \leq d \\ 0 &\ \ if & \ \ elsewhere \end{array} \right.

Usage

Tr(a, b, c, d)

Arguments

Value

This function easily introduce the membership function of a trapezoidal fuzzy number.

References

Parchami, A., Taheri, S. M., and Mashinchi, M. (2010). Fuzzy p-value in testing fuzzy hypotheses with crisp data. Statistical Papers 51: 209-226.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2012). Testing fuzzy hypotheses based on vague observations: a p-value approach. Statistical Papers 53: 469-484.

Examples

# Introducing the membership function of trapezoidal fuzzy number (for test statistics t)
t = Tr(2,4,6,9)

## The function is currently defined as
function (a, b, c, d) 
(TrapezoidalFuzzyNumber(a, b, c, d))

Testing hypotheses based on fuzzy p-value for Normal distribution

Description

Function p_value.norm can draw the membership function of fuzzy p-value for the following three major problems which can be usually considered for the following tests in a fuzzy environment: (1) testing crisp hypotheses based on fuzzy data, (2) testing fuzzy hypotheses based on crisp data, and (3) testing fuzzy hypotheses based on fuzzy data. Also, one can consider a fuzzy significance of level for test by function p_value.norm. It must be noted that function p_value.norm is applied when the test statistics has Normal distribution. This idea can be extended for other test statistics which has other continue distribution. All needed formulas are presented in Springer chapter book 2016 and the considered examples are match with this reference.

Usage

p_value.norm(kind, H0, H1, t, s.d, n, sig)

Arguments

Details

In this package, the users must introduce H0, H1, t and significance level only by fuzzy numbers T, Tr, S or B. Therefore, if each of them are precise (crisp), user must introduce it by special fuzzy sets (i.e. by indicator functions). For example the crisp null hypothesis "H_0: \ \mu > 0" must be introduced by H0 = B(0,0) to function p_value.norm. For another example, if the observed value of test statistics is crisp number t = 4.1, it must only be introduced by t = T(4.1,4.1,4.1) or t = Tr(4.1,4.1,4.1,4.1) to the function p_value.norm.

Value

The function p_value.norm is constructed for hypotheses testing in fuzzy environment where the test statistics has Normal distributions and the parameter which discussed by two hypotheses is the mean of Normal. The major target and the duty of function p_value.norm is: (1) draw the membership functions of p-value and significance level for a Normal test, (2) compute numerical measures of \Delta_{SP} and \Delta_{PS} (see Parchami and Mashinchi (2016) for formulas), (3) decide to accept null hypothesis or accept alternative hypothesis in the test, and finally (4) compute the degree of acceptance for the accepted hypothesis for a Normal test.

Author(s)

Abbas Parchami

References

Filzmoser, P., and Viertl, R. (2004). Testing hypotheses with fuzzy data: the fuzzy p-value. Metrika 59: 21-29.

Holena, M. (2004). Fuzzy hypotheses testing in a framework of fuzzy logic. Fuzzy Sets and Systems 145: 229-252.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2010). Fuzzy p-value in testing fuzzy hypotheses with crisp data. Statistical Papers 51: 209-226.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2012). Testing fuzzy hypotheses based on vague observations: a p-value approach. Statistical Papers 53: 469-484.

Wang, X., Kerre, E. E. (2001). Reasonable properties for the ordering of fuzzy quantities (II). Fuzzy Sets and Systems 118: 387-405.

Yuan, Y. (1991). Criteria for evaluating fuzzy ranking methods. Fuzzy Sets Syst 43: 139-157.

Examples

# Example 1 from Springer chapter book 2016
p_value.norm( kind=2, H0=T(1300,1300,1300), H1=B(1300,1300), t=T(1288,1313,1331), s.d=30, 
n=10, sig=T(0,0.05,0.1) )

# Solving Example 2 from Springer chapter book 2016
p_value.norm( kind=0, H0=T(1300,1300,1300), H1=B(1300,1300), t=T(1300,1313,1321), s.d=30, 
n=10, sig=T(0,0.15,0.3) )

# Solving Example 4 from Springer chapter book 2016
H0 = T(1275,1300,1325) #  Fuzzy null hypothesis
H1 = B(1275,1325) #  Fuzzy althernative hypothesis
t = T(1315,1327,1342) #  Fuzzy observed value for the test statistics
sig = T(0.0,0.15,0.3) #  Fuzzy significance level
p_value.norm( kind=2, H0, H1, t, s.d=120, n=36, sig)

# Solving Example 6 from Springer chapter book 2016
H0 = T(1275,1300,1325)
H1 = B(1275,1325)
t = T(1327,1327,1327) #  Crisp observed value for the test statistics
sig = T(0, .05, .1)
p_value.norm( kind=2, H0, H1, t, s.d=120, n=36, sig)


## The function is currently defined as
function (kind, H0, H1, t, s.d, n, sig) 
{
    alpha_L = seq(0, 1, 0.01)
    alpha_U = seq(1, 0, -0.01)
    if (kind == 1) {
        p_L = pnorm(alphacut(t, alpha_L)[, "U"], alphacut(H0, 
            alpha_L)[, "L"], s.d/(n^0.5))
        p_U = pnorm(alphacut(t, alpha_U)[, "L"], alphacut(H0, 
            alpha_U)[, "U"], s.d/(n^0.5))
    }
    else if (kind == 2) {
        p_L = 1 - pnorm(alphacut(t, alpha_L)[, "U"], alphacut(H0, 
            alpha_L)[, "L"], s.d/(n^0.5))
        p_U = 1 - pnorm(alphacut(t, alpha_U)[, "L"], alphacut(H0, 
            alpha_U)[, "U"], s.d/(n^0.5))
    }
    else if (supp(t)[1] >= supp(H0)[2]) {
        p_L = 2 * (1 - pnorm(alphacut(t, alpha_L)[, "U"], alphacut(H0, 
            alpha_L)[, "L"], s.d/(n^0.5)))
        p_U = 2 * (1 - pnorm(alphacut(t, alpha_U)[, "L"], alphacut(H0, 
            alpha_U)[, "U"], s.d/(n^0.5)))
    }
    else if (supp(t)[2] <= supp(H0)[1]) {
        p_L = 2 * pnorm(alphacut(t, alpha_L)[, "U"], alphacut(H0, 
            alpha_L)[, "L"], s.d/(n^0.5))
        p_U = 2 * pnorm(alphacut(t, alpha_U)[, "L"], alphacut(H0, 
            alpha_U)[, "U"], s.d/(n^0.5))
    }
    else {
        return("The fuzzy p-value can not defined for this example, since the fuzziness of 
		the problem is very high. This case may be accured for the testing equality 
		fuzzy null hypothesis, against the non-equality fuzzy althenative hypothesis")
    }
    alpha = cbind(alpha_L, alpha_U)
    p = cbind(p_L, p_U)
    plot(p, alpha, type = "l", lwd = 3, col = 1, xlim = c(-0.04, 
        1.02))
     s1 = supp(sig)[1]
     s2 = core(sig)[1]
     s3 = core(sig)[1]
     s4 = supp(sig)[2]
     lines( c(0,s1,s2,s3,s4,1), c(0,0,1,1,0,0), type='l', lty=3, lwd=2, col=2 )
# plot(sig, lty=3, lwd=2, col=2, add=TRUE) # Conflict importing 2 plot from different packages
    legend("topright", c("Fuzzy p-value", "Significance level"), 
        col = c(1, 2), text.col = 1, lwd = c(3, 2), lty = c(1, 
            3))
     if( class(sig) == "numeric" ){
            sig <- TriangularFuzzyNumber(sig, sig, sig)
            }
     P_L = p_L
     P_U = p_U
     knot.n = 100
     S_L = alphacut(sig, round(seq(0, 1, .01), 5))[,"L"]
     S_U = alphacut(sig, round(seq(0, 1, .01), 5))[,"U"]
     Int1 = ( P_U - S_L ) * ( P_U > S_L )
     Int2 = ( P_L - S_U ) * ( P_L > S_U )
     Arz = 1 / (knot.n - 1)  #Arze Mostatilha baraye mohasebe-ye Integral
     Integral1 <- ( sum( Int1 ) - Int1[1]/2 - Int1[length(Int1)]/2 ) *Arz
     Integral2 <- ( sum( Int2 ) - Int2[1]/2 - Int2[length(Int2)]/2 ) *Arz
     Delta_PS = Integral1 + Integral2
     Int3 = ( S_U - P_L ) * ( S_U > P_L )
     Int4 = ( S_L - P_U ) * ( S_L > P_U )
     Integral3 <- ( sum( Int3 ) - Int3[1]/2 - Int3[length(Int3)]/2 ) *Arz
     Integral4 <- ( sum( Int4 ) - Int4[1]/2 - Int4[length(Int4)]/2 ) *Arz
     Delta_SP = Integral3 + Integral4
    print("Delta_SP = ")
    print(Delta_SP)
    print("Delta_PS = ")
    print(Delta_PS)
    Degree_P_biger_than_S = Delta_PS/(Delta_PS + Delta_SP)
    Degree_S_biger_than_P = 1 - Degree_P_biger_than_S
    if (Degree_P_biger_than_S >= Degree_S_biger_than_P) {
        a = "The null hypothesis (H0) is accepted with degree D(P>S)="
        b = round(Degree_P_biger_than_S, 4)
        c = ", at  the considered significance level."
        noquote(sprintf("%s %s %s ", a, b, c))
    }
    else {
        if (Degree_P_biger_than_S < Degree_S_biger_than_P) {
            a = "The althernative hypothesis (H1) is accepted with degree D(S>P)="
            b = round(Degree_S_biger_than_P, 4)
            c = ", at  the considered significance level."
            noquote(sprintf("%s %s %s ", a, b, c))
        }
        else {
            return(noquote(paste0("Impossible case")))
        }
    }
  }

Testing hypotheses based on fuzzy p-value for Poisson distribution

Description

Function p_value.pois can draw the membership function of fuzzy p-value for the following three major problems which can be usually considered for the following tests in a fuzzy environment: (1) testing crisp hypotheses based on fuzzy data, (2) testing fuzzy hypotheses based on crisp data, and (3) testing fuzzy hypotheses based on fuzzy data. Also, one can consider a fuzzy significance of level for each of above tests by function p_value.pois. It must be noted that function p_value.pois is applied when the test statistics has Poisson distribution. This idea can be extended for other test statistics which has other discrete distribution. All needed formulas are presented in Springer Chapter book 2016 and the considered examples are match with this reference for the readers.

Usage

p_value.pois(kind, H0, H1, t, n, sig)

Arguments

Details

In this package, the users must introduce H0, H1, t and significance level only by fuzzy numbers T, Tr, S or B. Therefore, if each of them are precise (crisp), user must introduce it by special fuzzy sets (i.e. by indicator functions). For example the crisp null hypothesis "H_0: \ \lambda > 5.2" must be introduced by H0 = B(5.2, 5.2) to function p_value.pois.

Value

The function p_value.pois is constructed for hypotheses testing in fuzzy environment where the test statistics has Poisson distributions and the parameter which discussed by two hypotheses is the mean of Poisson. The major duty of function p_value.pois is: (1) draw the membership functions of p-value and significance level for a Poisson test, (2) compute numerical measures of \Delta_{SP} and \Delta_{PS} (see Parchami and Mashinchi (2016) for formulas), (3) decide to accept null hypothesis or accept alternative hypothesis in the test, and finally (4) compute the degree of acceptance for the accepted hypothesis for a Poisson test.

Author(s)

Abbas Parchami

References

Filzmoser, P., and Viertl, R. (2004). Testing hypotheses with fuzzy data: the fuzzy p-value. Metrika 59: 21-29.

Holena, M. (2004). Fuzzy hypotheses testing in a framework of fuzzy logic. Fuzzy Sets and Systems 145: 229-252.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2010). Fuzzy p-value in testing fuzzy hypotheses with crisp data. Statistical Papers 51: 209-226.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2012). Testing fuzzy hypotheses based on vague observations: a p-value approach. Statistical Papers 53: 469-484.

Wang, X., Kerre, E. E. (2001). Reasonable properties for the ordering of fuzzy quantities (II). Fuzzy Sets and Systems 118: 387-405.

Yuan, Y. (1991). Criteria for evaluating fuzzy ranking methods. Fuzzy Sets Syst 43: 139-157.

Examples

# Solving Example 5 from Springer Chapter book 2016
# Considering the assumptions of this example, it must be note that H0 and H1 are not about 
#   the parameter of the test statistics, and in fact hypotheses must be 12*H0 and 12*H1
H0 = B( 12*2.75 ,  12*3.25 )
H1 = S( 12*2.75 ,  12*3.25 )
t = T( 27, 27, 27)
sig = T(0.0,0.05,0.1)
p_value.pois( kind=1, H0, H1, t, n=12, sig)

# Solving Example 7 from Springer Chapter book 2016
H0 = B( 12*2.75 ,  12*3.25 )
H1 = S( 12*2.75 ,  12*3.25 )
t = T( 24, 27, 30)
sig = T(0.0,0.05,0.1)
p_value.pois( kind=1, H0, H1, t, n=12, sig)


## The function is currently defined as
function (kind, H0, H1, t, n, sig) 
{
    alpha_L = seq(0, 1, 0.01)
    alpha_U = seq(1, 0, -0.01)
    if (kind == 1) {
        p_L = ppois(alphacut(t, alpha_L)[, "U"], alphacut(H0, 
            alpha_L)[, "L"])
        p_U = ppois(alphacut(t, alpha_U)[, "L"], alphacut(H0, 
            alpha_U)[, "U"])
    }
    else if (kind == 2) {
        p_L = 1 - ppois(alphacut(t, alpha_L)[, "U"], alphacut(H0, 
            alpha_L)[, "L"])
        p_U = 1 - ppois(alphacut(t, alpha_U)[, "L"], alphacut(H0, 
            alpha_U)[, "U"])
    }
    else if (supp(t)[1] >= supp(H0)[2]) {
        p_L = 2 * (1 - ppois(alphacut(t, alpha_L)[, "U"], alphacut(H0, 
            alpha_L)[, "L"]))
        p_U = 2 * (1 - ppois(alphacut(t, alpha_U)[, "L"], alphacut(H0, 
            alpha_U)[, "U"]))
    }
    else if (supp(t)[2] <= supp(H0)[1]) {
        p_L = 2 * ppois(alphacut(t, alpha_L)[, "U"], alphacut(H0, 
            alpha_L)[, "L"])
        p_U = 2 * ppois(alphacut(t, alpha_U)[, "L"], alphacut(H0, 
            alpha_U)[, "U"])
    }
    else {
        return("The fuzzy p-value can not defined for this example, since the fuzziness of 
		the problem is very high. This case may be accured for the testing equality 
		fuzzy null hypothesis, against the non-equality fuzzy althenative hypothesis")
    }
    alpha = cbind(alpha_L, alpha_U)
    p = cbind(p_L, p_U)
    plot(p, alpha, type = "l", lwd = 3, col = 1, xlim = c(-0.04, 
        1.02))
     s1 = supp(sig)[1]
     s2 = core(sig)[1]
     s3 = core(sig)[1]
     s4 = supp(sig)[2]
     lines( c(0,s1,s2,s3,s4,1), c(0,0,1,1,0,0), type='l', lty=3, lwd=2, col=2 )
# plot(sig, lty=3, lwd=2, col=2, add=TRUE) # Conflict importing 2 plot from different packages
    legend("topright", c("Fuzzy p-value", "Significance level"), 
        col = c(1, 2), text.col = 1, lwd = c(3, 2), lty = c(1, 
            3))
     if( class(sig) == "numeric" ){
            sig <- TriangularFuzzyNumber(sig, sig, sig)
            }
     P_L = p_L
     P_U = p_U
     knot.n = 100
     S_L = alphacut(sig, round(seq(0, 1, .01), 5))[,"L"]
     S_U = alphacut(sig, round(seq(0, 1, .01), 5))[,"U"]
     Int1 = ( P_U - S_L ) * ( P_U > S_L )
     Int2 = ( P_L - S_U ) * ( P_L > S_U )
     Arz = 1 / (knot.n - 1)  #Arze Mostatilha baraye mohasebe-ye Integral
     Integral1 <- ( sum( Int1 ) - Int1[1]/2 - Int1[length(Int1)]/2 ) *Arz
     Integral2 <- ( sum( Int2 ) - Int2[1]/2 - Int2[length(Int2)]/2 ) *Arz
     Delta_PS = Integral1 + Integral2
     Int3 = ( S_U - P_L ) * ( S_U > P_L )
     Int4 = ( S_L - P_U ) * ( S_L > P_U )
     Integral3 <- ( sum( Int3 ) - Int3[1]/2 - Int3[length(Int3)]/2 ) *Arz
     Integral4 <- ( sum( Int4 ) - Int4[1]/2 - Int4[length(Int4)]/2 ) *Arz
     Delta_SP = Integral3 + Integral4
    print("Delta_SP = ")
    print(Delta_SP)
    print("Delta_PS = ")
    print(Delta_PS)
    Degree_P_biger_than_S = Delta_PS/(Delta_PS + Delta_SP)
    Degree_S_biger_than_P = 1 - Degree_P_biger_than_S
    if (Degree_P_biger_than_S >= Degree_S_biger_than_P) {
        a = "The null hypothesis (H0) is accepted with degree D(P>S)="
        b = round(Degree_P_biger_than_S, 4)
        c = ", at  the considered significance level."
        noquote(sprintf("%s %s %s ", a, b, c))
    }
    else {
        if (Degree_P_biger_than_S < Degree_S_biger_than_P) {
            a = "The althernative hypothesis (H1) is accepted with degree D(S>P)="
            b = round(Degree_S_biger_than_P, 4)
            c = ", at  the considered significance level."
            noquote(sprintf("%s %s %s ", a, b, c))
        }
        else {
            return(noquote(paste0("Impossible case")))
        }
    }
  }