  Recently established, directed dependence measures for pairs $(X,Y)$ of
random variables build upon the natural idea of comparing the conditional
distributions of $Y$ given $X=x$ with the marginal distribution of $Y$. They
assign pairs $(X,Y)$ values in $[0,1]$, the value is $0$ if and only if $X,Y$
are independent, and it is $1$ exclusively for $Y$ being a function of $X$.
Here we show that comparing randomly drawn conditional distributions with each
other instead or, equivalently, analyzing how sensitive the conditional
distribution of $Y$ given $X=x$ is on $x$, opens the door to constructing novel
families of dependence measures $\Lambda_\varphi$ induced by general convex
functions $\varphi: \mathbb{R} \rightarrow \mathbb{R}$, containing, e.g.,
Chatterjee's coefficient of correlation as special case. After establishing
additional useful properties of $\Lambda_\varphi$ we focus on continuous
$(X,Y)$, translate $\Lambda_\varphi$ to the copula setting, consider the
$L^p$-version and establish an estimator which is strongly consistent in full
generality. A real data example and a simulation study illustrate the chosen
approach and the performance of the estimator. Complementing the
afore-mentioned results, we show how a slight modification of the construction
underlying $\Lambda_\varphi$ can be used to define new measures of
explainability generalizing the fraction of explained variance.
