  Sobol' indices measure the dependence of a high dimensional function on
groups of variables defined on the unit cube $[0,1]^d$. They are based on the
ANOVA decomposition of functions, which is an $L^2$ decomposition. In this
paper we discuss generalizations of Sobol' indices which yield $L^p$ measures
of the dependence of $f$ on subsets of variables. Our interest is in values
$p>2$ because then variable importance becomes more about reaching the extremes
of $f$. We introduce two methods. One based on higher order moments of the
ANOVA terms and another based on higher order norms of a spectral decomposition
of $f$, including Fourier and Haar variants. Both of our generalizations have
representations as integrals over $[0,1]^{kd}$ for $k\ge 1$, allowing direct
Monte Carlo or quasi-Monte Carlo estimation. We find that they are sensitive to
different aspects of $f$, and thus quantify different notions of variable
importance.
