Normalised abundance data without object-standardisation
If variables with high abundances and few zeros are to be treated equivalently to variables with low abundances and many zeros, consider the following measures:
If variables with high abundances and few zeros (bearing on the objects under comparison) are to contribute more to the similarity (and less to the dissimilarity) relative to variables with low abundances and many zeros, consider the following measures:
Canberra metric (D10)
Coefficient of divergence (D11)
This metric is the sum of the quotient of absolute differences in abundance between two objects and the sum of the abundances of these objects. Note that the Canberra metric has no upper limit and double zeros must be excluded prior to calculation.
This metric is the square root of the average squared quotient between the differences and sums of corresponding abundance values across two objects. As above, double zeros must be excluded from the calculation.
Legendre and Legendre (1998) note that the measures above are not suited to detect differences at fine granularities and should be avoided in cluster analysis.
More complex measures are available for scenarios where variables with high abundances and few zeros (bearing on the entire data set) are to contribute more to the similarity (and thus less to the dissimilarity) relative to variables with low abundances and many zeros. The asymmetrical Gower coefficient (S19), extends the Gower coefficient (S15; described here) by including a term which is set to zero should double zeros or an absence of information arise. The coefficient of Legendre and Chodorowski (S20), which uses a modified partial similarity function, is another option recommended by Legendre and Legendre (1998).