# ANCOVA indices-How to have robust sensitivity analysis?

Dear UQLab

First of all, thank you very much for the implementation of the ANCOVA indices to establish sensitivity analysis with dependent marginals.
Second, I have two problems:

1. What’s the meaning of NEGATIVE first order ANCOVA indices?
2. Is it “wrong” to have big (~10^14) total variance in the sensitivity analysis using the first order ANCOVA indices?

Finally, Thanks for your enjoyable UQLab.
Best regards
Ali

Dear @ali

You can find a detailed analysis and comparisons for sensitivity indices in case of dependent inputs in the master thesis by P. Wiederkehr (2018) and a nice literature review. This is still a cumbersome problem, for which certain indices, although mathematically well-defined, cannot really be interpreted. This master’s thesis gives some kind of recommendations.

To understand the negative indices, just think about the variance of a sum of correlated variables (linear case), where the latter have negative covariances: the total variance may be smaller than the sum of the variances of each individual variable. The cross-terms are negative, which would translate, after normalization, into negative indices. Of course it is a bit more tricky, but this simple example gives at least an intuition.

Regarding high variance of the output, it depends on your problem and the units of your quantity of interest. In mechanical/civil engineering, if you compute stresses in Pascal (Pa), they can easily get values of 3\cdot 10^8, i.e 300 MPa. With 10% coefficient of variance, this would lead to a standard deviation of 3\cdot 10^7 Pa and a variance of 9\cdot 10^{14} Pa \!{}^2. So it is totally possible.

Best regards
Bruno

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Dear Prof. @bsudret

I read the mentioned thesis before asking this question. It studied good examples.
Unfortunately, I have the problem of interpretation of my sensitivity analysis. I really appreciate your recommendations.
For instance, I get the value of -9.819E-06 in one of my random variables (RV) by the first order ANCOVA sensitivity analysis. Can I interpret and substitute this value with ZERO or the absolute value?!
Besides, based on the mentioned study in p. 15, there is still some lack of knowledge in the context of ANCOVA indices!.

In a sentence, what are your recommendations for us to do sensitivity analysis with dependent RVs?

I am most thankful for the sustainable supports of UQTeam.
Best regards
Ali

Dear UQLab

One of the main purposes of my study is to do a comparative analysis between the sensitivity values which are derived by the “INDEPENDENT” and “DEPENDENT” marginals with the same limit state function. The copula formalism is applied to demonstrate the nonlinear dependency between the marginals. I was wondering to know which combination is true?

1. Both dependent and independent marginals are compared in a sensitivity analysis by terms of the ANCOVA first-order indices.
2. The ANCOVA first-order sensitivity analysis is applied to dependent marginals and the first-order Sobol’s sensitivity analysis is implemented to investigate the sensitivity values of Independent random variables.
3. Both of the mentioned conditions are true! (I think based on the Sensitivity Analysis in UQLab discussion)

Best regards
Ali

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