*3.2. Global ROSA*

Global ROSA can be computed using Fort contrast indices [25], which implicitly depend on parameters associated with the probability distribution. In engineering applications, it is primarily the probability *P<sup>f</sup>* [33,34], the design quantiles *F<sup>d</sup>* and *R<sup>d</sup>* [35], or the median [36].

Sensitivity indices subordinated to contrasts associated with probability (in short, contrast *P<sup>f</sup>* indices) are based on quadratic-type contrast functions [25]. However, contrast *P<sup>f</sup>* indices can be defined more easily based on the probability of failure and the conditional probabilities of failure [19]. A formula that does not require the evaluation of contrast functions can be used for practical computation. For practical use, the first-order probability contrast index *C<sup>i</sup>* can be rewritten in the form of [19]

$$\mathbf{C}\_{i} = \frac{P\_{f}\{\mathbf{1} - P\_{f}\} - E\{\left(P\_{f}|\mathbf{X}\_{i}\right)\{\mathbf{1} - P\_{f}|\mathbf{X}\_{i}\}\}}{P\_{f}\{\mathbf{1} - P\_{f}\}}.\tag{10}$$

The sensitivity index *C<sup>i</sup>* measures, on average, the effect of fixing *X<sup>i</sup>* on *P<sup>f</sup>* , where *P<sup>f</sup>* = *P*(Z < 0) is the failure probability and *P<sup>f</sup>* |*X<sup>i</sup>* = *P*((*Z*|*X<sup>i</sup>* ) < 0) is the conditional failure probability. The mean value E[·] is taken over *X<sup>i</sup>* . In Equation (10), the term *P<sup>f</sup>* (1 − *P<sup>f</sup>* ) is derived for probability estimator θ\* = Argmin ψ(θ) = *P<sup>f</sup>* from the minimum of contrast min θ ψ(θ):

$$\min\_{\theta} \psi(\theta) = \min\_{\theta} E(\psi(Z, \theta)) = \min\_{\theta} E(\mathbf{1}\_{Z \lhd 0} - \theta)^2 = \mathcal{V}(\mathbf{1}\_{Z \lhd 0}) = P\_f(\mathbf{1} - P\_f). \tag{11}$$

where V (1Z<0) is the variance in the case where there are only two outcomes of 0 and 1, with one having a probability of *P<sup>f</sup>* . The largest variance occurs if *P<sup>f</sup>* = 0.5, with each outcome given an equal chance. The contrast function ψ(θ) = *E*(1Z<<sup>0</sup> − θ) <sup>2</sup> vs. θ is convex and symmetrical in the interval across the vertical axis θ\*. The plot of *P<sup>f</sup>* (1 − *P<sup>f</sup>* ) vs. *P<sup>f</sup>* is a concave function with left-right symmetry. The contrast for conditional probability is expressed in a similar manner as (*P<sup>f</sup>* |*X<sup>i</sup>* )(1 − *P<sup>f</sup>* |*X<sup>i</sup>* ).

The second-order sensitivity index *Cij* is computed similarly:

$$\mathbf{C}\_{ij} = \frac{P\_f \{ \mathbf{1} - P\_f \} - E \{ \left( P\_f \left| \mathbf{X}\_{i\prime} \mathbf{X}\_j \right\rangle \left( \mathbf{1} - P\_f \left| \mathbf{X}\_{i\prime} \mathbf{X}\_j \right\rangle \right) \right)}{P\_f \{ \mathbf{1} - P\_f \}} - \mathbf{C}\_i - \mathbf{C}\_{j\prime} \tag{12}$$

where *P<sup>f</sup>* |*X<sup>i</sup>* ,*X<sup>j</sup>* = *P*((*Z*|*X<sup>i</sup>* ,*X<sup>j</sup>* ) < 0) is the conditional failure probability for fixed *X<sup>i</sup>* and *X<sup>j</sup>* . E[·] is taken over *X<sup>i</sup>* and *X<sup>j</sup>* . The index *Cij* measures the joint effect of *X<sup>i</sup>* and *X<sup>j</sup>* on *P<sup>f</sup>* minus the first-order effects of the same factors. The third-order sensitivity index *Cijk* is computed similarly:

$$\mathbf{C}\_{ijk} = \frac{P\_f \{ \mathbf{1} - P\_f \} - \mathbf{E} \{ \left( P\_f \{ \mathbf{X}\_{i\prime} \mathbf{X}\_{j\prime} \mathbf{X}\_k \} \mathbf{1} - P\_f \{ \mathbf{X}\_{i\prime} \mathbf{X}\_{j\prime} \mathbf{X}\_k \} \right)}{P\_f \{ \mathbf{1} - P\_f \}} - \mathbf{C}\_i - \mathbf{C}\_j - \mathbf{C}\_k - \mathbf{C}\_{ij} - \mathbf{C}\_{ik} - \mathbf{C}\_{jk\prime} \tag{13}$$

where *P<sup>f</sup>* |*X<sup>i</sup>* ,*X<sup>j</sup>* ,*X<sup>k</sup>* = *P*((*Z*|*X<sup>i</sup>* ,*X<sup>j</sup>* ,*X<sup>k</sup>* ) < 0) is the conditional failure probability for fixed triples *X<sup>i</sup>* , *X<sup>j</sup>* , and *X<sup>k</sup>* . The other indices are computed analogously. All input random variables are considered statistically independent. The sum of all indices must be equal to one:

$$
\sum\_{i} \mathbb{C}\_{i} + \sum\_{i} \sum\_{j>i} \mathbb{C}\_{ij} + \sum\_{i} \sum\_{j>i} \sum\_{k>j} \mathbb{C}\_{ijk} + \dots + \mathbb{C}\_{123\dots M} = 1. \tag{14}
$$

Contrast *P<sup>f</sup>* indices can also be derived by rewriting Sobol indices in the context of ROSA [21]. Estimating all sensitivity indices in Equation (14) can be highly computationally challenging and difficult to evaluate. For a large number of input variables, it may be better to analyse the effects of input variables using the total effect index (in short, the total index) *CTi*.

$$\mathcal{C}\_{Ti} = 1 - \frac{P\_f \left(1 - P\_f\right) - E\left(\left(P\_f | \mathbf{X}\_{\sim i}\right) \left(1 - P\_f | \mathbf{X}\_{\sim i}\right)\right)}{P\_f \left(1 - P\_f\right)}\tag{15}$$

*Pf* |*X*~*<sup>i</sup>* = *P*((*Z*|*X*~*<sup>i</sup>* ) < 0) is the conditional failure probability evaluated for a input random variable *X<sup>i</sup>* and fixed variables (*X*1, *X*2, . . . , *Xi–*1, *Xi*+1, . . . , *XM*). The total index *CTi* measures the contribution of input variable *X<sup>i</sup>* , including all of the effects caused by its interactions, of any order, with any other input variable. The total index *CTi* can also be computed if all sensitivity indices in Equation (14) are computed. For example, *CT*<sup>1</sup> for *M* = 3 can be written as *CT*<sup>1</sup> = *C*<sup>1</sup> + *C*<sup>12</sup> + *C*<sup>13</sup> + *C*123.

The structural reliability can also be assessed using design quantiles (see, e.g., [37]). Sensitivity indices subordinated to contrasts associated with the α-quantile [25] (in short, contrast *Q* indices) are based on contrast functions of the linear type. The contrast function ψ associated with the α-quantile can be written with parameter θ as [25]

$$\Psi(\theta) = E(\psi(Y, \theta)) = E((Y - \theta)(\alpha - 1\_{Y < \theta})),\tag{16}$$

where *Y* is scalar (here, *F* or *R*). Equation (16) reaches the minimum if the argument θ is the α-quantile estimator θ\* (here, *F<sup>d</sup>* or *Rd*). The plot of contrast function ψ(θ) vs. θ is convex and, with some exceptions, asymmetric.

Equation (16) is not quadratic like the contrast associated with *P<sup>f</sup>* , because the distance (*Y* − θ) is considered linear. The first-order contrast *Q* index is defined, on the basis of Equation (16), as

$$Q\_i = \frac{\min\_{\theta} \psi(\theta) - E\left(\min\_{\theta} E(\psi(Y\_i, \theta) | \mathbf{X}\_i)\right)}{\min\_{\theta} \psi(\theta)},\tag{17}$$

where the first term in the numerator (and denominator) is the contrast computed for the estimator of α-quantile θ\* = Argmin ψ(θ). The second term in the numerator is computed analogously, but with the provision that *X<sup>i</sup>* is fixed. E[·] is taken over *X<sup>i</sup>* .

*Symmetry* **2020**, *12*, 1720

The second-order α-quantile contrast index *Qij* is computed analogously, but with the fixing of pairs *X<sup>i</sup>* and *X<sup>j</sup>* :

$$Q\_{ij} = \frac{\min\_{\theta} \psi(\theta) - E\left(\min\_{\theta} E\{\psi(\mathbf{Y}, \theta) | \mathbf{X}\_{i\nu} \mathbf{X}\_{j}\}\right)}{\min\_{\theta} \psi(\theta)} - Q\_i - Q\_j. \tag{18}$$

The third-order sensitivity index *Qijk* is computed similarly:

$$Q\_{ijk} = \frac{\min\_{\theta} \psi(\theta) - E \Big( \min\_{\theta} \Big( \psi(\mathbf{y}, \theta) \big| \mathbf{X}\_i, \mathbf{X}\_j, \mathbf{X}\_k \big) \Big)}{\min\_{\theta} \psi(\theta)} - Q\_i - Q\_j - Q\_k - Q\_{ij} - Q\_{ik} - Q\_{jk} \tag{19}$$

All input random variables are considered statistically independent. The sum of all indices must be equal to one:

$$\sum\_{i} Q\_{i} + \sum\_{i} \sum\_{j>i} Q\_{ij} + \sum\_{i} \sum\_{j>i} \sum\_{k>j} Q\_{ijk} + \dots + Q\_{123\dots M} = 1. \tag{20}$$

The total index *QTi* can be written analogously to Equation (15) as:

$$Q\_{\rm Ti} = 1 - \frac{\min\_{\theta} \psi(\theta) - E \Big( \min\_{\theta} E(\psi(Y, \theta) | \mathbf{X}\_{i}) \Big)}{\min\_{\theta} \psi(\theta)},\tag{21}$$

where the second term in the numerator contains the conditional contrast evaluated for input random variable *X<sup>i</sup>* and fixed variables (*X*1, *X*2, . . . , *Xi–*1, *Xi*+1, . . . , *XM*). Equation (21) is analogous to Equation (15), but for the quantile. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 6 of 20 *3.3. Specific Properties of Contrasts Associated with Quantiles* 

#### *3.3. Specific Properties of Contrasts Associated with Quantiles* Can contrast indices *Q* be estimated more easily, without having to evaluate the contrast

1) = 0.386. In the case of the general Gaussian pdf *Y* ~ *N*(*μ*, *σ*<sup>2</sup>

specific form:

Can contrast indices *Q* be estimated more easily, without having to evaluate the contrast function from Equation (16)? Let us study Equation (16) using a simple case study, where *Y* has a Gaussian pdf: function from Equation (16)? Let us study Equation (16) using a simple case study, where *Y* has a Gaussian pdf: 

$$\phi(y,\,\mu,\,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(y-\mu)^2}{2\sigma}}.\tag{22}$$

2 \*

 

2

), function ψ(*θ*\*) can be written in a

(23)

*y*

2

Figure 1 depicts an example of the evaluation of the contrast function for the 0.4-quantile of the normalized Gaussian pdf—*Y* ~ *N*(0, 1)—where the 0.4-quantile is θ\* ≈ −0.253. The estimation of contrast function ψ(θ\*) is based on the dichotomy of the pdf into two parts, separated by the α-quantile. Figure 1 depicts an example of the evaluation of the contrast function for the 0.4-quantile of the normalized Gaussian pdf—*Y* ~ *N*(0, 1)—where the 0.4-quantile is *θ*\* ≈ −0.253. The estimation of contrast function ψ(*θ*\*) is based on the dichotomy of the pdf into two parts, separated by the *α*quantile.

**Figure 1.** Example of the evaluation of Equation (16) for the 0.4-quantile of the Gaussian pdf. **Figure 1.** Example of the evaluation of Equation (16) for the 0.4-quantile of the Gaussian pdf.

The value of the contrast function in Equation (16) is ψ(*−*0.253) = *E*((*Y −* (*−*0.253))(0.4−1*Y*<−0.253)) = 0.386, where the weight 0.6 favors the minority population over the 0.4-quantile and the weight 0.4 puts the majority population after the 0.4-quantile at a disadvantage. In this specific example, it can be observed that the function ψ(*θ*\*) vs. *θ*\* has an *N*(0, 1) course and therefore, ψ(*−*0.253) = *ϕ*(*−*0.253, 0, The value of the contrast function in Equation (16) is ψ(−0.253) = *E*((*Y* − (−0.253))(0.4−1*Y*<−0.253)) = 0.386, where the weight 0.6 favors the minority population over the 0.4-quantile and the weight 0.4 puts the majority population after the 0.4-quantile at a disadvantage. In this specific example, it can be observed that the function ψ(θ\*) vs. θ\* has an *N*(0, 1) course and therefore, ψ(−0.253) = φ(−0.253,

*e* .

Equation (23) can only be used for estimates of contrast *Q* indices if *Y* has a Gaussian pdf; otherwise, Equation (23) has the form of an approximate relation. Another form of the sensitivity indices in Equation (20) derived from Equation (23) would be very practical; however, the conditional Gaussian pdf of *Y*, Gaussian pdf of *Y*|*Xi*, etc., makes the use of Equation (23) problematic in black

Due to the left-right symmetry of the Gaussian pdf in Figure 1, the same contrast function value

**Figure 2.** Example of the evaluation of Equation (16) for the 0.6-quantile of the Gaussian pdf.

ψ ,,

box tasks, where skewness and kurtosis can have non-Gaussian values.

can be obtained for the 0.6-quantile (see Figure 2).

Gaussian pdf:

quantile.

0, 1) = 0.386. In the case of the general Gaussian pdf *Y* ~ *N*(µ, σ 2 ), function ψ(θ\*) can be written in a specific form: 1) = 0.386. In the case of the general Gaussian pdf *Y* ~ *N*(*μ*, *σ*<sup>2</sup> ), function ψ(*θ*\*) can be written in a specific form:

be observed that the function ψ(*θ*\*) vs. *θ*\* has an *N*(0, 1) course and therefore, ψ(*−*0.253) = *ϕ*(*−*0.253, 0,

**Figure 1.** Example of the evaluation of Equation (16) for the 0.4-quantile of the Gaussian pdf.

The value of the contrast function in Equation (16) is ψ(*−*0.253) = *E*((*Y −* (*−*0.253))(0.4−1*Y*<−0.253)) = 0.386, where the weight 0.6 favors the minority population over the 0.4-quantile and the weight 0.4

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 6 of 20

,,

Can contrast indices *Q* be estimated more easily, without having to evaluate the contrast function from Equation (16)? Let us study Equation (16) using a simple case study, where *Y* has a

2 1

Figure 1 depicts an example of the evaluation of the contrast function for the 0.4-quantile of the normalized Gaussian pdf—*Y* ~ *N*(0, 1)—where the 0.4-quantile is *θ*\* ≈ −0.253. The estimation of contrast function ψ(*θ*\*) is based on the dichotomy of the pdf into two parts, separated by the *α*-

2

 

2

*y e* . (22)

 

*y*

*3.3. Specific Properties of Contrasts Associated with Quantiles* 

$$
\Psi(\theta^\*) = \sigma^2 \cdot \phi(\theta^\*, \mu, \sigma) = \frac{\sigma}{\sqrt{2\pi}} e^{-\frac{(\theta^\* - \mu)^2}{2\sigma}}.\tag{23}
$$

$$
\text{where } \text{norm}(\text{norm}(\text{norm}(\text{dom}(\theta)))) = \frac{\sigma}{\sqrt{2\pi}} e^{-\frac{(\theta^\* - \mu)^2}{2\sigma}}.\tag{23}
$$

Equation (23) can only be used for estimates of contrast *Q* indices if *Y* has a Gaussian pdf; otherwise, Equation (23) has the form of an approximate relation. Another form of the sensitivity indices in Equation (20) derived from Equation (23) would be very practical; however, the conditional Gaussian pdf of *Y*, Gaussian pdf of *Y*|*X<sup>i</sup>* , etc., makes the use of Equation (23) problematic in black box tasks, where skewness and kurtosis can have non-Gaussian values. Equation (23) can only be used for estimates of contrast *Q* indices if *Y* has a Gaussian pdf; otherwise, Equation (23) has the form of an approximate relation. Another form of the sensitivity indices in Equation (20) derived from Equation (23) would be very practical; however, the conditional Gaussian pdf of *Y*, Gaussian pdf of *Y*|*Xi*, etc., makes the use of Equation (23) problematic in black box tasks, where skewness and kurtosis can have non-Gaussian values.

Due to the left-right symmetry of the Gaussian pdf in Figure 1, the same contrast function value can be obtained for the 0.6-quantile (see Figure 2). Due to the left-right symmetry of the Gaussian pdf in Figure 1, the same contrast function value can be obtained for the 0.6-quantile (see Figure 2).

**Figure 2. Figure 2.**  Example of the evaluation of Equation (16) for the 0.6-quantile of the Gaussian pdf. Example of the evaluation of Equation (16) for the 0.6-quantile of the Gaussian pdf. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 7 of 20

The following approach is more powerful. The value of contrast function ψ(θ\*) can be expressed using the centers of gravity of the green and yellow areas (see Figure 3). The following approach is more powerful. The value of contrast function ψ(*θ*\*) can be expressed using the centers of gravity of the green and yellow areas (see Figure 3).

**Figure 3.** Graphical representation of the contrast in Equation (16) for the 0.4-quantile of *N*(0, 1). **Figure 3.** Graphical representation of the contrast in Equation (16) for the 0.4-quantile of *N*(0, 1).

In the specific case of *Y* ~ *N*(0, 1), the dependence between *l* and *θ*\* is a hyperbola *l* <sup>2</sup> − (*θ*\*)<sup>2</sup> ≈ 1.6<sup>2</sup> with asymptotes *l = ±θ*\* (see Figure 4). In a more general case of *Y* ~ *N*(*μ*, *σ*2), the dependence between *l* and *θ*\* is a hyperbola *l* <sup>2</sup> − (*θ*\* − *μ*) <sup>2</sup> ≈ *σ*2·1.6<sup>2</sup> with asymptotes *l = ±*(*θ*\* − *μ*). The intersection of two asymptotes is at the center of symmetry of the hyperbola, which is the mean value *μ* = *E*(*Y*). The skewness and kurtosis (departure from the Gaussian pdf) lead to asymmetric and symmetric deviations from this hyperbola, but asymptotes of such a curve remain *l = ±*(*θ*\* − *μ*). Figure 4 illustrates an example with the so-called Hermite pdf with a mean value of 0, standard deviation of 1, skewness of 0.9, and kurtosis of 2.9. Although deviations from the hyperbola are significant around the mean value, the dependence *l* vs. *θ*\* approaches the asymptotes *l = ±*(*θ*\* − *μ*) in the regions of design quantiles (see Figure 4b). The observation can be generalized to any pdf or histogram of *Y*. In the specific case of *Y* ~ *N*(0, 1), the dependence between *l* and θ\* is a hyperbola *l* <sup>2</sup> <sup>−</sup> (θ\*)<sup>2</sup> <sup>≈</sup> 1.6<sup>2</sup> with asymptotes *l* = ±θ\* (see Figure 4). In a more general case of *Y* ~ *N*(µ, σ 2 ), the dependence between *l* and θ\* is a hyperbola *l* <sup>2</sup> <sup>−</sup> (θ\* <sup>−</sup> <sup>µ</sup>) <sup>2</sup> <sup>≈</sup> <sup>σ</sup> 2 ·1.6<sup>2</sup> with asymptotes *<sup>l</sup>* <sup>=</sup> <sup>±</sup>(θ\* <sup>−</sup> <sup>µ</sup>). The intersection of two asymptotes is at the center of symmetry of the hyperbola, which is the mean value µ = *E*(*Y*). The skewness and kurtosis (departure from the Gaussian pdf) lead to asymmetric and symmetric deviations from this hyperbola, but asymptotes of such a curve remain *l* = ±(θ\* − µ). Figure 4 illustrates an example with the so-called Hermite pdf with a mean value of 0, standard deviation of 1, skewness of 0.9, and kurtosis of 2.9. Although deviations from the hyperbola are significant around the mean value, the dependence *l* vs. θ\* approaches the asymptotes *l* = ±(θ\* − µ) in the regions of design quantiles (see Figure 4b). The observation can be generalized to any pdf or histogram of *Y*.

(**a**) (**b**) **Figure 4.** Plot of parameter *l* and function ψ(*θ*\*) vs. the *α*-quantile *θ*\*: (**a**) The Gaussian and non-

For any pdf of *f*(*y*) of *Y,* an alternative form of the contrast function to Equation (16) can be

<sup>ψ</sup> <sup>1</sup> 

 

\*

where *l* is the distance of the centers of gravity of the two areas before and after the *α*-quantile (see the example in Figure 3). Sensitivity indices reflect change around the *α*-quantile estimator *θ*\* using *l* while *α* is constant. Equation (24) is general for any pdf and offers new possibilities for evaluating

*yfyl dy yfy dy*

*l* , (24)

(25)

Gaussian pdf; (**b**) The same asymptotes of hyperbolic and non-hyperbolic function.

\*

\*

derived in a new form:

contrast via *l*.

*l* and *θ*\* is a hyperbola *l*

The following approach is more powerful. The value of contrast function ψ(*θ*\*) can be expressed

**Figure 3.** Graphical representation of the contrast in Equation (16) for the 0.4-quantile of *N*(0, 1).

asymptotes is at the center of symmetry of the hyperbola, which is the mean value *μ* = *E*(*Y*). The skewness and kurtosis (departure from the Gaussian pdf) lead to asymmetric and symmetric deviations from this hyperbola, but asymptotes of such a curve remain *l = ±*(*θ*\* − *μ*). Figure 4 illustrates an example with the so-called Hermite pdf with a mean value of 0, standard deviation of 1, skewness of 0.9, and kurtosis of 2.9. Although deviations from the hyperbola are significant around the mean

In the specific case of *Y* ~ *N*(0, 1), the dependence between *l* and *θ*\* is a hyperbola *l*

quantiles (see Figure 4b). The observation can be generalized to any pdf or histogram of *Y*.

using the centers of gravity of the green and yellow areas (see Figure 3).

with asymptotes *l = ±θ*\* (see Figure 4). In a more general case of *Y* ~ *N*(*μ*, *σ*<sup>2</sup>

<sup>2</sup> ≈ *σ*2·1.6<sup>2</sup>

<sup>2</sup> − (*θ*\* − *μ*)

**Figure 4.** Plot of parameter *l* and function ψ(*θ*\*) vs. the *α*-quantile *θ*\*: (**a**) The Gaussian and non-Gaussian pdf; (**b**) The same asymptotes of hyperbolic and non-hyperbolic function. **Figure 4.** Plot of parameter *l* and function ψ(θ\*) vs. the α-quantile θ\*: (**a**) The Gaussian and non-Gaussian pdf; (**b**) The same asymptotes of hyperbolic and non-hyperbolic function.

For any pdf of *f*(*y*) of *Y,* an alternative form of the contrast function to Equation (16) can be derived in a new form: For any pdf of *f*(*y*) of *Y,* an alternative form of the contrast function to Equation (16) can be derived in a new form:

$$
\psi(\theta^\*) = l \cdot a \cdot (1 - a),
\tag{24}
$$

<sup>2</sup> − (*θ*\*)<sup>2</sup> ≈ 1.6<sup>2</sup>

), the dependence between

with asymptotes *l = ±*(*θ*\* − *μ*). The intersection of two

where *l* is the distance of the centers of gravity of the two areas before and after the *α*-quantile (see the example in Figure 3). Sensitivity indices reflect change around the *α*-quantile estimator *θ*\* using *l* while *α* is constant. Equation (24) is general for any pdf and offers new possibilities for evaluating contrast via *l*. where *l* is the distance of the centers of gravity of the two areas before and after the α-quantile (see the example in Figure 3). Sensitivity indices reflect change around the α-quantile estimator θ\* using *l* while α is constant. Equation (24) is general for any pdf and offers new possibilities for evaluating contrast via *l*.

$$d = \int\_{\theta^\*}^{\infty} y \cdot f(y) dy - \int\_{-\infty}^{\theta^\*} y \cdot f(y) dy \tag{25}$$

In general, SSA is relevant to the mean value of *Y*, while the SA of the quantile (QSA) is relevant to the α-quantile of *Y*. However, in many cases, there is a strong similarity between the conclusions of QSA and SSA if all or at least the total sensitivity indices are examined. It can be shown in a simple example of *Y* = *X*<sup>1</sup> + *X*<sup>2</sup> that corr(*Q*(*Y*|*X<sup>i</sup>* ), *E*(*Y*|*X<sup>i</sup>* )) ≈ 1, where *Q*(*Y*|*X<sup>i</sup>* ) is the conditional α-quantile and *E*(*Y*|*X<sup>i</sup>* ) is the conditional mean value. Changing *X<sup>i</sup>* causes synchronous changes in the α-quantile *Q*(*Y*|*X<sup>i</sup>* ) and mean value *E*(*Y*|*X<sup>i</sup>* ).

Although contrasts are of a different type, similarities between the results of QSA and SSA have been observed in the task of SA of the resistance of a building load-bearing element [35]. Other numerical illustrations of contrast *Q* indices are presented in [38,39].
