Distribution of Sobol's Indices

Question

Some background: Given a linear regression model (or any other GLM), we all know how to test the null hypothesis $\hat\beta_i=0$. The lm function in R (or any equivalent in other languages) would even conduct this test automatically. We can then relate to the contribution of the $i^{th}$ variable to the model, or in other words discuss its importance.

---

Sobol indices are a model-agnostic method of quantifying variable importance. There are two variants of indices: Given a model of the shape $y=f(X)$, the importance of the $i^{th}$ variable $X_i$ is described by the first-order Sobol index $$S_i=\frac{Var(E[y|X_i])}{Var(y)}$$ and the total Sobol index $$T_i=\frac{Var(E[y|X_{-i}])}{Var(y)}$$ where $X_{-i}$ is the dataset $X$ without the $i^{th}$ variable in discussion.

---

I am willing to conduct some hypothesis tests regarding Sobol indices. They obviously lie in $[0,1]$ so they might be uniform or Beta. But, is there a known distribution for them? Any known statistical procedure for such metrics? Is there some literature regarding hypothesis testing for global sensitivity analysis?

Answer 1

I would say the "standard approach" is to use bootstrap confidence intervals. The reason why formal tests may not be that easy:

1. The distribution of indices depends on your estimation procedure.
2. No matter what estimator you use the distributions are complicated.
3. You most likely would like to rank the variables, which means you have to deal with their dependency.

None of the computational libraries/packages I am aware of (such as sensobol or sensitivity) do hypothesis tests, neither is this discussed in the standard textbooks on the topic.

That said, the paper "Statistical inference for Sobol pick freeze Monte Carlo method" by authors F. Gamboa and A. Janon and T. Klein and A. Lagnoux and C. Prieur (link to journal and link to arxiv) covers pick-freeze estimators pretty comprehensively.