Dependence maps: Local dependence in practice

Abstract

There is often more structure in the way two random variables are associated than a single scalar dependence measure, such as correlation, can reflect. Local dependence functions such as that of Holland and Wang (1987) are, therefore, useful. However, it can be argued that estimated local dependence functions convey information that is too detailed to be easily interpretable. We seek to remedy this difficulty, and hence make local dependence a more readily interpretable practical tool, by introducing dependence maps.Via local permutation testing, dependence maps simplify the estimated local dependence structure between two variables by identifying regions of (significant) positive, (not significant) zero and (significant) negative local dependence. When viewed in conjunction with an estimate of the joint density, a comprehensive picture of the joint behaviour of the variables is provided. A little theory, many implementational details and several examples are given.