Generalized biplots - Understanding Biplots - page 420

The axis predictivities given in the top left panel of Figure 9.9 for the quantitative
variables are calculated from the output of the call to Genbipl with argument predic-
tions.sample = 1:nrow(X) . Note that these predictions are expressed in terms of the
original quantitative measurements. The correct way of calculating the axis predictivities
is thus to first normalize these predictions exactly as the input quantitative variables were
normalized using the components centred.vec and scaling.vec from the output list
of the call to Genbipl . The sum of the squared normalized predictions for a particular
variable divided by the sum of the squared normalized original observations (which is
unity when normalizing to unit sum of squares) gives the required axis predictivity.

• Finally, we can compare Figure 9.9 with Figure 8.20, which gives a categorical

(ordinal) PCA of the same data. It seems that the optimal ordinal scores determined
by categorical PCA give very similar axes to those found by the generalized biplot
using the normalized original data. In Figure 8.20 these axes have been shifted to
the edges of the figure, and they have not (but could have been) in Figure 9.9. In
this case, it seems that the optimal scores were similar to the original. The treatment
of the fundamentally categorical variables ( Gender , Rank and Faclty ) necessarily
differs in the two figures because the category regions are two-dimensional, whereas
the categorical PCA has represented each variable linearly. In the case of Gender ,
with only two levels, we find that bisecting the line joining the points representing
male and female in Figure 8.20 divides the space into regions that are very similar to
the category regions for gender in Figure 9.9. Rather surprisingly, it seems that the
EMC has given better gender separation than that given by the optimal PCA scores.

9.9 Function for constructing generalized biplots

Our main function for constructing generalized biplots is the function Genbipl .This
function shares the following arguments with PCAbipl :

X c.hull.n marker.size pch.samples.size

G colours max.num predictions.sample

X.new.samples exp.factor n.int specify.bags

e.vects label offset specify.classes

alpha label.size ort.lty Title

ax line.length parplotmar Tukey.median

ax.name.size line.type pch.means zoomval

ax.type line.width pch.means.size

ax.col markers pch.samples

The following arguments are specific to Genbipl and need special consideration.

Arguments

Logical value specifying if CLPs must be shown
on the biplot. Defaults to FALSE.

CLPs.plot

One of "none" , "unitVar" , "unitSS" ,
"unitRange" for specifying normalization of
quantitative variables.

cont.scale