A new relationship for rarefaction
All diversity indices are functions of the vector of the numbers of individuals in different species in a statistical population. So they are also functions of the number of species. It is well known, from the species-area curve and from collector's curves, that the number of species is a function of sampling effort. The rarefaction and Coleman functions are both functions that allow comparisons to be made at the same number of individuals, but have different mathematical forms. We show that the numerical difference between them, in the samples we have studied, is negligibly small. We show how to modify the Coleman function to allow for sampling without replacement, and show that the modified function is identical to the hypergeometric rarefaction function. Rarefaction should always be used, with any index, when comparing diversity in different size samples, but the number of species is the preferred index. Suggestions for comparing rarefaction curves from different samples are made.