What does the circle in the left-hand plot mean? See below...
Now it is time to interpret the two biplots. First, the proportion of variance
accounted for by the first two axes is 0.751, or 75.1%. This high value makes us
confident that our interpretation of the first pair of axes will extract most of the
relevant information from the data. Here is an example of how such a biplot can be
interpreted.
First, the scaling 1 biplot displays a feature that must be explained. The circle is
called a circle of equilibrium contribution. Its radius is equal to
ffiffiffiffiffiffiffi ffi
d=p
p
, where d is
the number of axes represented in the biplot (usually d ¼ 2) and p is the number of
dimensions of the PCA space (i.e., usually the number of variables of the data
matrix)
2 . The radius of this circle represents the length of the vector representing a
variable that would contribute equally to all dimensions of the PCA space. Therefore, for any given pair of axes, the variables that have vectors longer than this radius
make a higher contribution than average and can be interpreted with confidence. In
scaling 2 it is not possible to draw a circle of equilibrium contribution, because
scaling 2 is a projection into a Mahalanobis space, not a Euclidean space. Actually, a
circle could only be drawn if all variables were standardized and strictly orthogonal
to one another, which is never the case in practice.
The scaling 1 biplot shows a gradient from left to right, starting with a group
formed by sites 1–10 which display the highest values of elevation (ele) and slope
(slo), and the lowest values in river discharge (dis), distance from the source
(dfs) and hardness (har). The second group of sites (11–16) has the highest values
in oxygen content (oxy) and the lowest in nitrate concentration (nit). A third
group of very similar sites (17–22) show intermediate values in almost all the
measured variables; they are not spread out by the variables contributing to axes
1 and 2. Phosphate (pho) and ammonium (amm) concentrations, as well as biological oxygen demand (bod) show their maximum values around sites 23–25; the
values decrease afterwards. Overall, the progression from oligotrophic, oxygen-rich
to eutrophic, oxygen-deprived water is clear.
The scaling 2 biplot shows that the variables are organized in groups. The lower
left part of the biplot shows that elevation and slope are very highly, positively
correlated, and that these two variables are very highly, negatively correlated with
another group comprising distance from the source, river discharge and hardness.
Oxygen content is positively correlated with slope and elevation, but very negatively
with phosphate and ammonium concentration and, of course, with biological oxygen
demand. The right part of the diagram shows the variables associated with the lower
2 Note, however, that vegan uses an internal constant to rescale its results, so that the vectors and
the circle represented here are not equal but proportional to their original values. See the code of the
cleanplot.pca() function.
160
5 Unconstrained Ordination
Now it is time to interpret the two biplots. First, the proportion of variance
accounted for by the first two axes is 0.751, or 75.1%. This high value makes us
confident that our interpretation of the first pair of axes will extract most of the
relevant information from the data. Here is an example of how such a biplot can be
interpreted.
First, the scaling 1 biplot displays a feature that must be explained. The circle is
called a circle of equilibrium contribution. Its radius is equal to
ffiffiffiffiffiffiffi ffi
d=p
p
, where d is
the number of axes represented in the biplot (usually d ¼ 2) and p is the number of
dimensions of the PCA space (i.e., usually the number of variables of the data
matrix)
2 . The radius of this circle represents the length of the vector representing a
variable that would contribute equally to all dimensions of the PCA space. Therefore, for any given pair of axes, the variables that have vectors longer than this radius
make a higher contribution than average and can be interpreted with confidence. In
scaling 2 it is not possible to draw a circle of equilibrium contribution, because
scaling 2 is a projection into a Mahalanobis space, not a Euclidean space. Actually, a
circle could only be drawn if all variables were standardized and strictly orthogonal
to one another, which is never the case in practice.
The scaling 1 biplot shows a gradient from left to right, starting with a group
formed by sites 1–10 which display the highest values of elevation (ele) and slope
(slo), and the lowest values in river discharge (dis), distance from the source
(dfs) and hardness (har). The second group of sites (11–16) has the highest values
in oxygen content (oxy) and the lowest in nitrate concentration (nit). A third
group of very similar sites (17–22) show intermediate values in almost all the
measured variables; they are not spread out by the variables contributing to axes
1 and 2. Phosphate (pho) and ammonium (amm) concentrations, as well as biological oxygen demand (bod) show their maximum values around sites 23–25; the
values decrease afterwards. Overall, the progression from oligotrophic, oxygen-rich
to eutrophic, oxygen-deprived water is clear.
The scaling 2 biplot shows that the variables are organized in groups. The lower
left part of the biplot shows that elevation and slope are very highly, positively
correlated, and that these two variables are very highly, negatively correlated with
another group comprising distance from the source, river discharge and hardness.
Oxygen content is positively correlated with slope and elevation, but very negatively
with phosphate and ammonium concentration and, of course, with biological oxygen
demand. The right part of the diagram shows the variables associated with the lower
2 Note, however, that vegan uses an internal constant to rescale its results, so that the vectors and
the circle represented here are not equal but proportional to their original values. See the code of the
cleanplot.pca() function.
160
5 Unconstrained Ordination