χ2 similarity, metric, and distance
Hellinger distance
The χ2 similarity, metric, and distance measures are based on contingency tables and can thus handle non-monotonic and qualitative variables (see this endpoint for details).
This asymmetric distance is similar to the χ2 metric. While no weights are applied, the square roots of conditional probabilities are used as variance-stabilising data transformations. This distance measure performs well in linear ordination. Variables with few non-zero counts (such as rare species) are given lower weights.
Use the rank-order correlation coefficients below only to analyse variables and never objects (e.g. an R mode analysis on a transposed sites x species table). The relationships between your variables need not be linear, but must be monotonic (i.e. they should either go "up" or "down" together).
Spearman's rho
Kendall's τ
This is a non-parametric measure of correlation which uses ranks rather than the original variable values. Variables should have monotonic relationships: that is, their ranks should either go up or down across objects, but not necessarily in a linear fashion. Like Pearson's r, Spearman's rho is based on the principal of least squares, but is concerned with how strongly the rankings between two variables disagree. The larger the disagreement the lower the rho value. This statistic is sensitive to large disagreements. That is, if one variable ranks an object as "1" and another variable ranks the same object as "100", the correlation reported by Spearman's rho will be strongly affected (relative to Kendall's tau, for example), even if these variables agree on all other ranks. This measure is suitable for raw or standardized abundance data and any monotonically related variables.
Like the Spearman's rho, Kendall's tau uses ranked values to calculate correlation. This measure, however, is not based on the principal of least squares and instead expresses the degree of concordance between two rankings. The tau statistic is the quotient of 1) the difference between concordant and discordant pairs (i.e. ranks that agree and ranks that differ) and 2) the total number of pairs compared. This statistic is not sensitive to the scale of the disagreement. As above, variables should have monotonic relationships: that is, their ranks should either go up or down across objects, but not necessarily in a linear fashion. This measure is suitable for raw or standardized abundance data and any monotonically related variables.