What do the specific values of the Sobol' indices mean?

Question

I understand that first order and total effect Sobol' indices demonstrate the relative importance of the input parameters on the output of a given model. My question is, do the specific values of each index provide any quantifiable information about the outcome of a model or can they only be used to compare the relative importance of each parameter?

For example, say one calculated first and total indices of an input parameter to be 0.1 and 0.6, respectively, using Monte-Carlo sampling. That means that 10% of the outcome variance is due to that individual parameter. Now let's say that my input variable has a value of 2, I run the given model and receive an output. If I were to change my input variable by 20% for example, would I be able to conclude anything about the outcome based off of the value of the Sobol' indices?

Answer 1

In a Sobol type sensitivity analysis we consider a function $y=f(x)$ where $x=(x_1,x_2,..,x_q)$. As long as the components of $x$ are probabilistically independent then $f$ has the unique ANOVA decomposition. To perform the SA I need a probability distribution over $X$, I.e. I need to formulate my uncertainty in $X$. The uncertainty in $X$ induces uncertainty in $Y$, SA tells me how uncertainty in model inputs influences uncertainty in model outputs. The entire sensitivity analysis depends on this distribution so please make sure it is sensible prior to proceeding.

$$ f(x) = f_0 + \sum_i f_i (x_i) + \sum_{i<j} f_{ij}(x_{ij}) + \ldots + f_{12 \ldots , q } (x)$$

$f_0$ is the mean value of the function, the $f_i$ are the main effects, $f_{ij}$ are the second order interactions and so on.

Then let $V = var(Y)$ and $V_J = Var_{X_J}( E(f(x) | X_{-J} ) )$ for any set $ J \subseteq \{ 1, 2 , \ldots , q \}$. In general $S_J = V_J / V$.

Then $S_i = V_i / V$ is the proportion of uncertainty in $Y$ attributed to the main effect of variable $i$. Some authors (e.g. Tony O'Hagan) believe that this is the single most important variable.

Other authors prefer to measure the most important variable by

$$S_{T_i} = \sum_{J \supseteq i } (V_J / V) = \sum_{J \supseteq i} S_J $$

This quantity represents the total uncertainty induced by variable $i$. E.g. for a three parameter input we have

$$ S_{T_1} = S_1 + S_{12} + S_{13} + S_{123}. $$

So from your analysis $10\%$ of output uncertainty is due to the main effect of the first variable; it is however related to $60\%$ of total output uncertainty due to how it interacts with the other variables.