Diversity and differentiation measures in ecology, genetics, and economics: There are many mathematical misconceptions in the sceintific literature about diversity indices. These misconceptions lead to invalid forms of reasoning about diversity. I develop a mathematical framework which which clears up several of the problems in the literature. I show that the current definitions of alpha and beta diversity are not correct except in special cases, and I derive the correct expressions. I illustrate the pitfalls of standard diversity and similarity measures, and introduce some general similarity and overlap measures that give the Jaccard index, Sorensen index, Morisita-Horn index of overlap, and Horn  index of overlap as special cases. I show how to use them properly by including lots of examples. I hope the examples will prove useful to ecology students, working ecologists, and geneticists. I also hope to receive feedback from users and especially anyone who disagrees with something I put on this site. I will put contrasting views on this site if they seem interesting.

These pages include excerpts from my articles in Oikos, Ecology, Molecular Ecology, and Ecological Economics, and also excerpts from the book Diversity Analysis which Anne Chao and I are currently writing for Taylor and Francis Publishers. The book should be out in early 2010.

Combining significance levels from multiple experiments or analyses: In the life sciences and social sciences we frequently do multiple experiments or analyses to test a hypothesis. Each experiment or analysis returns a probability value, the probability that the results could have occurred under the null hypothesis. This probability value assumes that the experiment or analysis was done only once. If multiple experiments or analyses are performed, sooner or later a significant probability value will be found even if the null hypothesis is true. In this paper I derive an exact formula for combining probability values for multiple experiments or analyses. This calculates the exact overall probability value of the group of experiments or analyses on the null hypothesis. The formula uses only the individual probabality values, so that experiments with dissimilar methodologies and statistics can be combined. The result, for the case of two probability values p and q, is pq-(pq)ln(pq). The formula is more sensitive than the Bonferroni correction.

A simple Point-centered Quarter Method for non-uniform forests: The Point-centered Quarter method (PCQ) has long been a standard technique for measuring plant density. Unfortunately the botanists who invented it were poor mathematicians, and they made assumptions which cause the standard density formula to be grossly invalid in nonuniform forests. Recent critics have recognized this problem and have recommended that the method be abandoned in spite of its other advantages. However, I have derived a new formula for PCQ data which is valid for nonuniform forests. I have tested the new formula both in computer-simulated and real forests and find it to be accurate. Therefore there is no need to abandon the PCQ method. The new formula uses exactly the same variables as the classical formula, so it can be used to reanalyze old data and find the correct density.