A Comparison of Three Measures of Species Diversity

Sununesar Two popular diversity indices, Simpson 's index Y and the InJoirinationl index H, are compared with a new measure Q based onl the inter-quartile slope oJ the cumulative species abundance curve. It is assumed that interest extends to characteriZing the site environlnent: the population present at the instant of sampling is considered to be on/i' one oJ a range oJ possible populations which the site could support. Expressions are derived Jor the expectations and variances of the three sample statistics a, aH, and a} wvhen the species abundances are gainmna variates. Q is a mnore in/cbrinative measure than H or Y. Both H and Y depend greatly onl the ab'unidancces ot the commonest species, which may fluctuate wvidel/ Jroi i'ear to iear. Q depends onl the more stable species with median abundance and discrim.inilates better between sites than H or Y: it can he recommended when sites are to be compared. The expected value oQ/Q is expressed in terns of the parameters of the gainina, lognormal and log-series models and is shown to he mnuch more robust than H or Y to the particular choice ot'model. For the log-series model, E(Q) is represented h ' the parameter a, while Jor the lognormnal E( Q) = 0.3 71 T/u, where T is the number of species in the population and u is the standard deviation oJ logged abundances. aU inay he biased in sinall samples, though the bias should bejairlY sinall whenl more thamn 50% of a species are present in the sample. A bsence oJ sinall sample bias should not he an overriding criterion in selecting a diversityv index since sinall samples wt'ill at best onl/ allow the crudest comparisons bet ween communities. The use of a single index to characterise the pattern of the abundances of different species in a community has obvious appeal and several such measures have been formulated. In practice the diversity is measured for a sample drawn from the community, so it is important that any proposed index is independent of sample size, at least for large samples; this is achieved if the index is based on the species relative abundances. We here study the behaviour of the two most popular measures of diversity, Simpson's index and the Information index, and propose an alternative index which provides a better characterisation of the community. To study the behavior of the diversity sample statistics theoretically, we must make assumptions about the mathematical form of the distribution of species abundances in the community and the nature of the sampling variability. Our assumptions can of course be no more than approximations to reality, but we expect our main conclusions to be fairly robust to deviations from the chosen model. We shall assume the population contains T species (denoted by S* in many ecological