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# Appendix 4: Graphical Description of Mantels Test and ANOSIM

Mantel tests and ANOSIM work in similar ways but test slightly different things and provide slightly different output. This figure describes graphically how they work. Imagine five sites where biological data have been collected. The axes in biological space (on the left) represent the abundance of different species. This example uses only three species so that our space is three-dimensional and can be drawn. We have generally been working with a large number of species and therefore multi-dimensional space, however, the principles are exactly the same. Each point is plotted in a biological space using the abundance of each species as the coordinates. From this plot, we can measure the distance between each pair of points. These distances are recorded in a matrix called the biological (pair-wise) distance matrix.

The same process is carried out with the environmental data, in an environmental space. Here the axes represent different environmental variables, e.g. depth, mean SST, tidal current. The environmental space can be changed by adding, weighting and transforming variables. Each axis represents a variable so if we can add or remove a variable we add or remove an axis, if we transform a variable we change the scaling along that axis, and if we want to weight a variable, we duplicate its axis. Once the variables, transformations and weighting have been decided, we plot the points in environmental space using the values of the environmental variables as coordinates and record the results in a matrix called the environmental (pair-wise) distance matrix. This environmental distance matrix can then be compared with the biological distance matrix.

In a Mantels test (Appendix 4, Figure A 4.1) the correlation between the two matrices gives an objective measure of the match between biological and environmental space. The correlation between the two matrices is a measure of how well the combination of our environmental variables represents the biological pattern. Different environmental spaces (defined by adding, weighting and transforming variables) can be tested to examine how correlation can be increased.

An ANOSIM tests how well an imposed grouping of the sites explains variation in actual biological data and is used to test the strength of the environmentally based classification. An ANOSIM works in the biological space with two values:

• the average distances between sites within each imposed class and
• the average distances between classes

Figure A4.1: Graphical representation of Mantel test

The abundance of three different species in the space defined by five sample sites is shown graphically. A biological (pair wise) distance matrix then shows the numerical distance between sites as determined by the abundance of the three species at each site. A similar graph and matrix then depict the relationship between environmental variables in this space. The correlation between the two matrices is a measure of how well the environmental model created fits the biological pattern.

These distances are illustrated graphically in (Appendix 4, Figure A4.2) below. The statistic that ANOSIM reports is the between class distance minus the within class distance. This value is high (i.e. the classification is strong) when the within-class distances are small (i.e. the sites are biologically similar) and the between-class distances are large.

Figure A4.2: Graphical representation of ANOSIM test

A graph shows comparison of within-class variability in biological data with between-class variability in biological data. It tests the biological significance of an environmentally based classification by comparing distances within each imposed class with average distances between classes.