  Recognizing, quantifying and visualizing associations between two variables
is increasingly important. This paper investigates how a new function-valued
measure of dependence, the quantile dependence function, can be used to
construct tests for independence and to provide an easily interpretable
diagnostic plot of existing departures from the null model. The dependence
function is designed to detect general dependence structure between variables
in quantiles of the joint distribution. It gives an insight into how the
dependence structures changes in different parts of the joint distribution. We
define new estimators of the dependence function, discuss some of their
properties, and apply them to construct new tests of independence. Numerical
evidence is given on the test's benefits against three recognized independence
tests introduced in the previous years. In real-data analysis, we illustrate
the use of our tests and the graphical presentation of the underlying
dependence structure.
