**6. Discussion**

Low quantiles represent a significant part of the analysis of reliability of load-bearing structures. SA of design quantiles can be used wherever reliability can be judged by comparing two statistically independent variables.

Both types of quantile-oriented sensitivity analysis identified a very similar sensitivity order to Sobol SA. Identical identification of probabilistically insignificant variables can serve to reliably decrease the dimension of random design space by introducing noninfluential variables as deterministic. On the contrary, the probability distribution of dominant variables should be identified with the greatest possible accuracy.

In all cases, the most important output information is the sensitivity order obtained using total indices. Total indices identify approximately the same sensitivity order for all types of SA; see Figures 11, 14 and 15b. Sobol total indices can be a good proxy of quantileoriented total indices in cases where changes in the quantile are primarily influenced by changes in the variance and less by the shape of the probability distribution of the output variable *R*. It can be noted that although the case of strong discrepancy between quantile-oriented indices and Sobol indices has not yet been observed, some atypical (in practice, less real) tasks have not yet been solved, e.g., Sobol SA with strong interaction effects or SA of quantiles close to the mean.

Quantile-oriented contrast *Q* indices are based on the quantile deviation *l*, which is the absolute distance of two average values of the population below and above the quantile; see Equations (4)–(6). Quantile deviation *l* has the same unit and is similar to *σR*, because it measures the variability of the population around the quantile. Quantile deviation *l* has good resistance to outlier values around the quantile. When *X<sup>i</sup>* changes deterministically, the quantile deviation *l* is found to change, but not always monotonically, despite a monotonic variation in the standard deviation *σR*. The correlation between *l* and *σ<sup>R</sup>* may or may not be strong even though a dependence exists; see Figure 16.

By applying the quantile deviation *l* as a new measure of sensitivity, contrast *Q* indices defined by Fort [63] can be rewritten in a new form; see Equations (8)–(12). By substituting *l* with *l* 2 , *Q* indices can be rewritten as the new *K* indices, which are based on the decomposition of *l* 2 , similarly to the way Sobol indices are based on the decomposition of variance; see Equations (13)–(17).

Contrast *Q* indices have an unpleasantly relatively high proportion of higher-order indices (interaction effects), which makes it difficult to interpret SA results. The new *K* indices have characteristics close to Sobol indices because they have a smaller proportion of interaction effects than *Q* indices. Although *K* and Sobol indices are similar, they are not the same because the key variable *l* depends not only on the variance but also on the shape of the distribution (variance, skewness, kurtosis).

The comparison of contrast *Q* indices, *K* indices and Sobol indices was performed using a non-linear function *R*, which includes both non-linear and non-monotonic effects of five input variables *X<sup>i</sup>* on the output. In the case study, four input variables *X<sup>i</sup>* , *i* = 2, 3, 4, 5 have approximately linear dependence *l*|*X<sup>i</sup>* vs. *σR*|*X<sup>i</sup>* , where *l* and *σ<sup>R</sup>* are computed for fixed *X<sup>i</sup>* while the other *X~i* are considered as random. However, this does not apply to input variable *X*<sup>1</sup> (yield strength *fy*), which leads to a non-linear concave dependence *l*|*X<sup>i</sup>* vs. *σR*|*X<sup>i</sup>* . The example shows the strong influence of the shape of the distribution (skewness and kurtosis) on the quantile deviation *l* as one of the causes leading to differences in *K* indices from Sobol indices.

The findings presented here correlate very well with the results of SA of a beam under bending exposed to lateral-torsional buckling, where contrast *Q* indices and Sobol indices identified very similar sensitivity rank of input random variables [78,80].

Although other types of quantile-oriented sensitivity indices exist [98–100], they are not of a global type with the sum of all indices equal to one, so they were not used in this paper, because mutual comparison would be difficult.

The *K* indices and *Q* indices presented here are as computationally demanding as the sensitivity indices derived in [62], but with the advantage that they are not limited to small and large quantiles. For small 0.001-quantile, the estimates of asymptotic *QE* indices [62] are practically the same as the results published in this article; therefore, the asymptotic form [62] does not provide any immediately apparent application advantage when the Monte Carlo method is applied.

It can be noted that civil engineering has numerous reliability tasks in which interactions can be a significant part of the design of structural members or systems, see e.g., [101–105]. Another goal of SA may be the examination of the design quantiles of these tasks.
