Diversity Index - True Diversity (The Effective Number of Types)

True Diversity (The Effective Number of Types)

True diversity, or the effective number of types, refers to the number of equally-abundant types needed for the average proportional abundance of the types to equal that observed in the dataset of interest (where all types may not be equally abundant). The true diversity in a dataset is calculated by first taking the weighted generalized mean of the proportional abundances of the types in the dataset, and then taking the inverse of this. The equation is:

The denominator equals average proportional abundance of the types in the dataset as calculated with the weighted generalized mean with exponent q - 1. In the equation, R is richness (the total number of types in the dataset), and the proportional abundance of the ith type is . The proportional abundances themselves are used as the nominal weights. When q = 1, the above equation is undefined, so the corresponding mean is calculated with the following equation instead:

The value of q is often referred to as the order of the diversity. It defines the sensitivity of the diversity value to rare vs. abundant species by modifying how the mean of the species proportional abundances is calculated. With some values of the parameter q, the generalized mean with exponent q - 1 gives familiar kinds of mean as special cases. In particular, q = 0 corresponds to the harmonic mean, q = 1 to the geometric mean and q = 2 to the arithmetic mean. As q approaches infinity, the generalized mean with exponent q - 1 approaches the maximum value, which is the proportional abundance of the most abundant species in the dataset. In practice, increasing the value of q hence increases the effective weight given to the most abundant species. This leads to obtaining a larger mean value and a smaller true diversity (qD) value.

When q = 1, the geometric mean of the values is used, and each species is exactly weighted by its proportional abundance (in the geometric mean, weights are the exponents). When q > 1, the weight given to abundant species is exaggerated, and when q < 1, the weight given to rare species is. At q = 0, the species weights exactly cancel out the species proportional abundances, such that mean equals 1/R even when all species are not equally abundant. At q = 0, the effective number of species (0D) hence equals the actual number of species (R). In the context of diversity, q is generally limited to non-negative values. This is because negative values of q would give rare species so much more weight than abundant ones that qD would exceed R.

The general equation of diversity is often written in the form:

The term inside the parentheses is called the basic sum. Some popular diversity indices correspond to the basic sum as calculated with different values of q.

For diversity of order one, an alternative equation is:

where H' is the Shannon index (see below).