Measures of association - describe the likeness of each variable (species, column) to each other based on how well the values they have for each object (row) match up.
Typically, association is measured by either correlation or covariance.
Measures of distance - describe the likeness of each object (site, row) to each other based on how well the values they have for each site (column) match up.
There are many different measures of distance.
Measures of association
The following community data represent the abundances of three species of gastropods in five quadrats (ranging from
high shore marsh - Quadrat 1, to low shore marsk - Quadrat 5) in a saltmarsh.
In terms of species abundances at each site (rows), which species are most associated with one another?
Measures of association
Peet & Loucks (1977) examined the abundances of 8 species of trees (Bur oak, Black oak, White oak, Red oak, American elm, Basswood, Ironwood, Sugar maple) at 10 forest sites in southern Wisconsin, USA.
The data (given below) are the mean measurements of canopy cover for eight species of north American trees in 10 samples (quadrats).
For this question we will explore the associations between the different species based on the degree to which their abundances in the quadrats match up (covary or correlate).
Note that the abundances of each species of tree in these data are fairly uniform. Each species has the similar minimum and maximum (and thus means and standard deviations).
Indeed it is just elm and basswood that has slightly lower maximums and standard deviations).
It is therefore just association measures involving either of those two species
that are likely to differ in pattern between covariances and correlations.
If we were to standardize (scale) the raw abundances first (such that each species had a mean of 0 and a standard deviation of 1), the covariance measures
would match the correlation measures of the raw data exactly. Recall that such a standardization effectively evens up the relative abundances of each species.
Try it to prove it to yourself.
We return again to the abundances of three species of gastropods in five quadrats (ranging from
high shore marsh - Quadrat 1, to low shore marsk - Quadrat 5) in a saltmarsh.
Finally, we return to Peet & Loucks (1977) Wisconsin tree data.
For this question we will explore the similarities of quadrats (objects) based on how well the abundances of each species match up.
Note, in the case of Bray-Curtis dissimilarity, it is common practice to first perform some sort of standardization of the data so as to
even up the influence of all species and sites irrespective of whether they are abundant or rate (such as a Wisconsin double standardization).