Global sensitivity analysis: Can the first order effect be greater than the total order effect?

I am using the Global Sensitivity Analysis app from the File Exchange (link). For one of my results of a parameter, the first order Sobol sensitivity index is greater than the total order Sobol sensitivity index. Is this possible mathematically? I thought that the total order index is always greater than or equal to the first order index. They are equal if there are no parameter interactions. The total is greater than the first if there are parameter interactions.

Answer 1

[Florian Augustin]

Hi,

You are right that first order Sobol indices are always smaller than or equal to the corresponding total order Sobol indices. However, sample-approximations of Sobol indices can be susceptible to approximation errors. This is especially the case when the total number of sensitivity input parameters is not small, or the number of samples used for the approximation is not sufficiently large.

Adding more samples should help to reduce approximation errors. If you use SimBiology's sbiosobol function, you can add samples to existing results using the addsamples function. Unfortunately, this feature is not available in the app you mentioned.

If the computational cost of the Sobol analysis is too high, you could try a multi-parametric global sensitivity analysis with the sbiompgsa function. Using a classifier like

"mean(yourModelResponseName, 1) < mean(yourModelResponseName, 'all')"

gives you an alternative sensitivity measure to Sobol indices.

If the issue of first order Sobol indices being larger than the total order Sobol indices persists, then we would need to look at the particular analysis you are conducting. Please let me know if it is an option for you to share yor model (possibly offline).

Best,

Florian