Patil & Taillie

Both Simpson's index and Shannon's index are affected by the number of species and the evenness of species abundances, but they are affected differently. A rare species contributes much less to diversity in Simpson's index than in Shannon's. Patil & Taillie point out that all of these diversity measures can be subsumed in a single diversity spectrum. Let $R(\pi_k) = {1 - \pi_k^\beta \over \beta}$. $\beta$ scales the relative importance of richness and evenness. Then

$$\Delta_\beta(C) = \sum_{k=1}^s\pi_k\left({1 - \pi_k^\beta \over \beta}\right) \quad .$$

A little algebra (and l'Hopitals rule from calculus) shows that if $\beta = -1$,

$$\begin{eqnarray*} \copy\tempbox \Delta_\beta &=& \hbox{species richness} \cr \hb... ...box{if $\beta = 1$,\hfil}\Delta_\beta &=& \hbox{Simpson's index} \end{eqnarray*}$$

Thus, ``diversity profiles'' can be plotted to compare two or more communities over a range of evenness emphasis from no emphasis at all (species richness) to high emphasis (Simpson's index).

Notice that all of these measure treat species as if they were completely interchangable with one another. A community in which there were 100 species of sedges, and nothing else, all in equal frequency would receive the same diversity score as one in which there were 10 species of sedges, 10 species of grasses, 10 species of legumes, 10 species of roses, 10 species of buttercups, and 50 species of composites, all in equal frequency. How do we feel about that?

Again the tyranny of numbers. Don't forget why we started calculating these things in the first place. They are a guide, not a formula, and you shouldn't hesitate to trust your gut instincts if the formula seems to be giving you the wrong answer. If it does conflict with your gut, however, you'd be well-advised to figure out why. Identify what's wrong with the formula, what it's leaving out that's important, how it measures the wrong thing, etc.