Hi!

Just to further clarify the already excellent answer by @damarginal , although the sum of the first order indices cannot be greater than 1, their *estimate* can. This is because the sum of the first order indices is a random variable. Assuming that there are no interactions (so that the first order indice is equal to the total order indice), the exact sum is equal to 1. Now consider the random variable

B = 1 \textrm{ if } \sum_{i=1}^p \hat{S}_i > 1

and zero otherwise.

We have P(B=1)=1/2.

To check it, I created the linear model Y = 1 + 2X_1 + 3 X_2 + 4 X_3 where X_i\sim\mathcal{N}(0,1) and are independent. I used a sample size equal to 10000 to estimate Sobolâ€™ indices with Saltelli estimator. I counted 1 if the event occurs, and 0 otherwise. Then I repeated the experiment 500 times. Here is the distribution of the 0/1 random variable:

We check that the probability of having an estimate sum larger than 1 is close to 0.5.

Best regards,

MichaĂ«l