Canonical variate analysis biplots - Understanding Biplots - page 180

Resrch research output expressed on a continuous scale;

Rank academic position or rank;

Age in years;

Gender male/female;

PQual professional qualification (not used in the CVA discussed here);

AQual academic qualification expressed on a continuous scale;

Faclty integer value representing one of nine faculties included in the study;

Ye a r 2002 or 2005.

The mean values of the samples in 2002 and two-dimensional CVA biplots, both
grouped according to gender, are given in Table 4.12 and Figure 4.14, respectively.
The function CVAbipl was called four times with argument e.vects specified as
c(1,2) , c(1,3) , c(1,4) and c(1,5) respectively to produce Figure 4.14. Arguments
predictions.mean = 1:2 and ort.lty = 2 request the perpendicular dotted lines
for predicting the mean values for the males and the females. The reader can verify
that the predicted values obtained from any of the four biplots in Figure 4.14 correspond
exactly to the values given in Table 4.12. This is so in spite of the large differences in the
appearances of these biplots. Furthermore, a call to the function CVA.predictivities
reveals that all biplot axes in the different biplots have axis predictivity of unity. Similarly,
the class predictivities are all equal to unity (this is already true in the first dimension).
Since we have only two groups, we know that the canonical means differ only in their first
components and the call to PCA.predictivities returns the values for the canonical
means given in Table 4.13.
The biplots in Figure 4.14 illustrate that although each of them provides exact predic-
tions for all the mean values, incorrect conclusions can be drawn by judging from their
appearance only. The reason for this is found in the form of the matrix returned by
the call to CVAbipl : all its diagonal elements, except the first one of 0.2906, are zero.
Therefore, there is no uniquely defined second axis for constructing the scaffolding for
the CVA biplot - only a one-dimensional CVA biplot is uniquely defined when dealing
with two groups. In Figures 4.15 and 4.16 we give one-dimensional CVA biplots for the
remuneration data.
Since in a one-dimensional biplot all the sample points and all the biplot axes are
on the line extending through the means, we have overlaid the biplot in Figure 4.15
with separate density estimates for the two groups and also translated the axes arbitrarily

Ta b l e 4 . 1 2 Mean values of males
and females for the variables in the
remuneration data for 2002.

Variable Female Male

Remun 19.3128 26.1441

Resrch 0.2604 0.6319

Rank 2.4367 3.6046

Age 41.9755 47.9959

Aqual 6.7184 7.6304