Much ado about nothings: using zero similarity points in distance-decay curves

Distance decay is used to describe the (usually exponential) decay in ecological similarity of assemblages between two sites as a function of their distance apart along an environmental gradient. Exponential distance-decay curves are routinely fitted by calculating the ecological similarity between each pair of sites, and fitting a linear regression to the points on a scatter plot of log-similarity vs. distance (x-axis). However, pairs of sites where the assemblages have no species in common pose a problem, because the similarity is zero, and the log transformation cannot be applied. Common fixes to this problem (i.e., either removing or transforming the zero values) are shown to have undesirable consequences and to give widely disparate estimates. A new method is presented as a special case of a generalized dissimilarity model. It is fitted very quickly and easily using existing software, and it does not require removal or transformation of the zero similarity points. Its simplicity makes it convenient for use in conjunction with the resampling methods that are routinely employed to test hypotheses, to obtain standard errors of estimated parameters, or to compare distance-decay curves. A word of caution about standard application of the bootstrap is noted, and modified bootstrap and jackknife alternatives are demonstrated.