From star charts to stoneflies: detecting relationships in continuous bivariate data

Ecology, March, 1998 by James E. Garvey, Elizabeth A. Marschall, Russell A. Wright

INTRODUCTION

Within many ecological systems, complex interactions underlie observed patterns. In many cases, functional relationships exist between controlling factors and associated response variables. However, the strength of these relationships often differ greatly over the range of observed values. For example, in many systems, we expect the response to vary little when the controlling factor exerts strong effects. Conversely, when the effect of the controlling factor is weak or absent, the response may vary greatly with the influence of other factors. Though we often attempt to characterize these relationships statistically using correlations and regressions, these unsatisfactory approaches clearly violate the assumption that variance of the response does not change with the magnitude of the controlling factor. These techniques fail to capture an important and interesting characteristic of these relationships: Correlations between the response and controlling factor change over the range of observations.

To illustrate some common features of these types of data sets, we use density data from two co-occurring benthic insects. Leuctra tenuis (a stonefly) and Ameletus ludens (a mayfly) were sampled in pools in head-water streams in Ohio (see Dingledine 1996 for details). Though experimental results indicated that Ameletus distributions were likely strongly controlled by presence/absence of fish, Leuctra densities were not as easily explained by these fish treatments. To answer the post hoc question of whether Leuctra might be responding to presence of Ameletus rather than to fish, we plotted the densities of Leuctra vs. Ameletus from each benthic sample ([ILLUSTRATION FOR FIGURE 1 OMITTED]; Dingledine 1996). Clearly, at high Ameletus densities, Leuctra were rare, whereas at low Ameletus densities, Leuctra occurred at both high and low densities [ILLUSTRATION FOR FIGURE 1 OMITTED]. Though Leuctra and Ameletus densities may have been negatively related when Ameletus were abundant, a regression (implying a functional relationship between Leuctra and Ameletus) over the entire range of data would be inappropriate if other factors drove Leuctra densities when Ameletus were rare. A traditional measure of correlation (e.g., Pearson product-moment correlation) also would not be appropriate because of its underlying assumption of a linear relationship between variables, which is clearly of little relevance in these data.

Rather than asking how much variance can be explained by these two factors, we suggest asking if the distributions of Ameletus and Leuctra densities could arise independently. Our initial willingness to interpret the absence of data points in the upper right quadrant of Fig. 1 as indicating that Ameletus had a negative effect on Leuctra was tempered by the fact that neither species occurred in high numbers very frequently. Even if density distributions of the two species were independent, the probability of observing them jointly at high densities was low, given that high densities of either species rarely occurred. Clearly, a test is required in which we assume that the joint bivariate distribution of densities arose independently, consisting of chance pairings of these data. We then compare the observed joint distribution to this null distribution, determining whether the densities were related.

Recall that the familiar Kolmogorov-Smirnov test compares two univariate cumulative probability distributions by determining if [D.sub.KS], the maximum difference between the two distributions, is significantly large. We have borrowed a technique from the astronomy literature to extend this approach to bivariate distributions. Because cumulative probability distributions are not well defined beyond one dimension (Press et al. 1992), Fasano and Franceschini (1987) modified an approach by Peacock (1983) to compare two-dimensional (bivariate) distributions. Their solution was to find the maximum difference, [D.sub.BKS], in integrated probabilities for four quadrants around each point in the plane. If [D.sub.BKS] between an observed and theoretical bivariate distribution (one-sample test) or two observed bivariate distributions (two-sample test) exceeds that expected randomly, we conclude that they differ. To demonstrate the utility of the two-dimensional Kolmogorov-Smirnov (2DKS) test in exploring the types of relationships described above, we provide several examples. First, we use the 2DKS test to detect dependence between two variables in simulated data sets. By then comparing this test to traditional techniques using ecological examples, we demonstrate its potential widespread utility for delineating patterns in ecological data.

METHODS

Two-dimensional Kolmogorov-Smirnov test (2DKS)

To test for independence in a single bivariate data set (one-sample test), we used a 2DKS test (Peacock 1983, Fasano and Franceschini 1987, Press et al. 1992, 1996a, b). For each pair of coordinates ([X.sub.i], [Y.sub.i], i = 1, . . ., n), we counted the points in each of four surrounding quadrants, with [X.sub.i], [Y.sub.i] at the origin,