A significance test for empty corners in scatter diagrams

Abstract

Regression analysis is usually the statistical tool of choice in hydrological studies when there is a strong correlation between two variables. However, weak correlations can also be of interest if a region within the scatter plot is data-free. This could direct attention to seeking some underlying physical process that might create regions with low probability of generating data points. A necessary prior requirement here is to verify that the data-free area in the plot is sufficiently large to be a real effect and not a visual illusion. This check can be most simply carried out in a hypothesis-testing framework. A permutation approach to hypothesis testing is suggested for the particular case where a data-free region occupies one of the corners of a scatter plot, and a test statistic Δ is presented for testing the statistical significance of the size of this “empty corner”. Application to some rainfall data from southern Israel shows that the new test can sometimes yield higher levels of statistical significance than linear regression when applied to the same data.