Tipos de Comunicación Organizacional

quality when the PCA biplot is constructed from the standardised measurements

When the PCA biplot is constructed from the standardised measurements, the sam- ple variance of the

˜

xk is equal to one for all k ∈[1∶p], and hence each of the p axis

predictivities will be assigned a weight equal to 1

p. Consequently, when the PCA

PCA biplot will be equal to the arithmetic average of thep axis predictivities, tr(̂X′X̂) tr(X′X) = p ∑ k=1 1 pπk

and the relative magnitude of a biplot axis’ contribution to the overall quality of the biplot will be equal to the relative magnitude of that biplot axis’ predictivity,

1 pπk ∑p j=1 1 pπj = πk ∑p j=1πj .

The magnitude of a biplot axis’ predictivity indicates the quality of the axis’ con- tribution to the overall quality of the biplot.

When the PCA biplot is constructed from the standardised measurements,

wk=

1

p

and hence the increase in the overall quality of the biplot resulting from an increase in the dimension of the biplot is equal to the arithmetic average of the increases in the paxis predictivities resulting from the increase in dimension. It follows that when the PCA biplot is constructed from the standardised measurements, a plot showing the overall quality of the biplot as well as each of the p axis predictivities against the possible dimensionalities of the biplot is useful in that it allows for the visual assessment of both the relative contributions of the biplot axes to the overall quality as well as the qualities of those contributions. When the dimension of a PCA biplot is increased from r to r+1, the contribution of the kth variable to the resulting increase in the overall quality is

1

pπk,(r,r+1)

while its relative contribution is

πk,₍r,r+1₎

∑p

k=1πk,(r,r+1)

Hence, when the PCA biplot is constructed from the standardised measurements, the relative magnitude of the kth variable’s contribution to α₍r,r+1) is equal to the

relative magnitude of πk,₍r,r+1). Notice that the gradient of the line connecting the

points corresponding to πk,r to πk,r+1 in the plot showing the overall quality and p

axis predictivities against the dimension of the PCA biplot is equal toπk,(r,r+1). The

plot of the overall quality and axis predictivities against the dimension of the biplot therefore also allows for the visual assessment of the relative contributions of the biplot axes to α₍r,r+1) for r ∈[1∶p−1] and the qualities of these contributions. A

plot such as the one just described is illustrated in Figure 3.6 for theUniversity data set. It is evident from Figure 3.6 that the variable whose relative contribution to the overall quality of the one-dimensional PCA biplot of the University data set is the greatest, is the variable, SAT, while the variable whose relative contribution to the overall quality of the one-dimensional PCA biplot is the smallest, is the variable,

Expenses. 1 2 3 4 5 6 0.6 0.7 0.8 0.9 1.0

Dimension of the PCA biplot

Quality Overall Quality SAT Top10 Accept SFRatio Expenses Grad

Figure 3.6: The overall quality and axis predictivities of the PCA biplot constructed from the standardised measurements of the University data set.

When the construction of the PCA biplot is based on the unstandardised data matrix, the interpreter of the biplot needs to take both the axis predictivity as well as the weight assigned to that axis predictivity into account when assessing the axis’ relative contribution to the overall quality of the biplot. A plot showing the overall quality and each of the p axis predictivities against the possible dimensions of the PCA biplot constructed from the unstandardised measurements of a data

set therefore does not allow for the visualisation of the relative contributions of the various biplot axes to the overall quality of the biplot or the relative contributions of the biplot axes to α₍r,r+1) - it only allows for the visualisation of the qualities of

these contributions. When the PCA biplot is constructed from the unstandardised measurements, a plot of the overall quality together with the contributions of the p biplot axes to the overall quality can be made, but this plot would be uninformative as to what the qualities of the contributions are concerned.

Consider the slopes of the six straight lines in Figure 3.6 illustrating the increase in the six axis predictivities resulting from increasing the dimension of the PCA biplot from one to two. It is evident that the increase in the overall quality of the PCA biplot resulting from the increased dimension is mainly due to the increased accuracy with which the variables, Expenses, SFRatio and Grad are predicted, or put differently, the addition of the second dimension to the PCA biplot contributed mainly to the increased accuracy with which the variables, Expenses, SFRatio and

Grad are predicted. This means that the second principal component accounts for a large proportion of the sample variances of these three variables. The relative contribution of the variable, Expenses, to the increase in the overall quality is the greatest of all the variables. When the dimension of the biplot is increased from two to three, it is the variableSFRatio whose relative contribution to the increase in the overall quality of the biplot which is the greatest.