Calculating Alpha Diversity
Suppose species diversity is equated with the effective number of species, and alpha diversity with the mean species diversity per subunit. Then alpha diversity can be calculated in two different ways that give the same result. The first approach is to calculate a weighted generalized mean of the within-subunit species proportional abundances, and then take the inverse of this mean. The second approach is to calculate the species diversity for each subunit separately, and then take a weighted generalized mean of these.
If the first approach is used, the equation is:
In the equation, N is the total number of subunits and S is the total number of species (species richness) in the dataset. The proportional abundance of the ith species in the jth subunit is . These proportional abundances are weighted by the proportion of data that each contributes to the dataset, which equals . The denominator hence equals mean proportional species abundance within the subunits (mean ) as calculated with the weighted generalized mean with exponent q - 1.
If the second approach is used, the equation is:
This also equals a weighted generalized mean but with exponent 1 - q. Here the mean is taken of the qDαj values, each of which represents the effective species density (species diversity per subunit) in one subunit j. The nominal weight of the jth subunit is, which equals the proportion of data that the subunit contributes to the dataset.
Large values of q lead to smaller alpha diversity than small values of q, because increasing q increases the effective weight given to those species with the highest proportional abundance and to those subunits with the lowest species diversity.
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