# ANCOVA and Kucherenko sensitivity analysis methods

Hi,
I have a general question about ANCOVA and Kucherenko methods. As I understood after a read through that these two methods are dedicated for dependent input random variables. However, both of these methods are applicable regardless of the dependence since we can get the output of uncorrelated input variables and show the independent contributions.

So, if I apply these methods for a computational model that does not have any dependence or correlations for the inputs, does that mean that the sensitivity indices results are similar from the Sobol’ sensitivity indices method since I read that the ANCOVA is a way to generalize the formulation of Sobol’ indices for dependent input case?

Basically, can I actually do a comparison between the results from Sobol’, ANCOVA and Kucherenko methods?

Thank you for clarifying my question!

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Hi @Vishashalu,

Thanks for asking here. Yes, your general understanding is correct: ANCOVA and Kucherenko indices are designed for dependent input variables, but they are generally applicable to any input structures. Nevertheless, these two types of indices are not equivalent for dependent inputs: the definition of ANCOVA relies on the functional decomposition, while the Kucherenko indices are defined directly by conditional expectation/variance. Moreover, for ANCOVA indices, only those of the first-order are well-defined (and implemented in UQLab), and the definition of higher-order and total ANCOVA indices remains open (as clearly pointed out in the sensitivity manual).

If the input variables are independent, the two types of indices by definition are identical to Sobol’ indices. However, numerical results are generally different:

1. Since the calculation of ANCOVA indices depends on PCE, their values correspond to the Sobol’ indices of the PCE. If you use PCE to evaluate the Sobol’ indices of the computational model, then the values of these two indices coincide in theory. However, UQLab uses Monte Carlo simulations to calculate the ANCOVA indices while it makes use of the independence and the properties of PCE to compute the Sobol’ indices by post-processing the coefficients without sampling.
2. For Kucherenko indices, UQLab provides two sampling-based methods which are different from the estimator based on Monte Carlo simulation designed for Sobol’ indices. More precisely, UQLab uses the so-called pick-and-freeze algorithm to efficiently evaluate the Sobol’ indices.

In summary, if your input variables are independent, ANCOVA and Kucherenko indices coincide with the definition of Sobol’ indices. However, I would recommend using Sobol’ indices in this case for the sake of efficiency and accuracy.