Unipolar axes

The interpretation of biplots is made easier by the construction of axes in it. These axes are used in the same way as in a bivariate Cartesian diagram. Perpendicular projection of the points in the diagrams upon a coordinate axis allows us to determine (or reconstruct) the values in the table. 

We consider the special biplot in which both rows and columns are represented in a single display of latent variables subjected to the constraint that a + P equals 1. As we have seen above, this constraint allows us to reconstruct the original data X from which the latent variables U, V and the latent values A have been computed (eq. (31.22)). 

Algebraically, the reconstruction of the values of X has been defined by the matrix product of the scores S with the transpose of the loadings L (eq. (31.22)). Geometrically, one reconstructs the value Xy by perpendicular projection of the point represented by upon the axis represented by s, as shown in Fig. 31.3c  

The same result can be obtained by perpendicular projection of a point s, upon an axis 1 as shown in Fig. 31.3d. 

The distance from the origin or dj is a measure of the information contained in the corresponding row or column. If the distance is relatively large, in comparison to others, then the corresponding row or column can be seen to contribute more information (inertia, variance) to the result of the analysis. In the case when the distance is zero, then the row or column carries no information (inertia, variance) since all its elements are zero. 

---

Figure 31.4 shows the biplot of the trace elements and wind directions for the case when a = p = 0.5. Since here we have that a + P equals 1, we can reconstruct the values in the columns of the data table X by means of perpendicular projections upon unipolar axes. In Fig. 31.4a we have drawn a unipolar axis through Cl. Perpendicular projection of the four wind directions upon this axis reconstructs the order of the concentrations of Cl at the four wind directions as listed in Table 31.1. Now we have established a way which leads back from the graphic display to the tabulated data. This interpretation of the biplot emphasizes the one-to-one relationship between the data and the plot. Such a relationship is also inherent in the ordinary bivariate (or Cartesian) diagram. 

Fig. 31.6. Biplot of chromatographic retention times in Table 31.2, after column-centering of the data. Two unipolar axes and one bipolar axis have been drawn through the representations of the methods DMSO and methylenedichloride (CH2CI2). The projections of three selected compounds are indicated by dashed lines. TTie values read off from the unipolar axes reproduce the retention times in the corresponding columns. The values on the bipolar axis reproduce the differences between retention times.

There are two outstanding poles on this biplot. DMSO and dimethylchloride are at a large distance from the origin and from one another. These poles are the most likely candidates for the construction of unipolar axes. As has been explained in the previous section, perpendicular projections of points (representing compounds) upon a unipolar axis (representing a method) leads to a reproduction of the data in Table 31.3. In this case we have to substitute the untransformed value in eq. (31.35) by Zy of eq. (31.42) ... 

Finally, we have constmcted a bipolar axis through DMSO and methylenedichloride. By perpendicular projection of the centers of the circles upon this bipolar axis we obtain the differences in retention times obtained respectively with DMSO and with methylenedichloride. Using a similar reasoning as developed above for unipolar axes we can perform a substitution of eq. (31.42) in eq. (31.38) which leads to ... 

The distances between compounds in Fig. 31.7 are not notably affected by the transformation in comparison with the previous Fig. 31.6. This biplot allows more easily to perceive the correlations between measurements. Three clusters are now put in evidence, namely (1) DMSO and DMF, (2) ethanol and propanol, (3) octanol, dioxane, THF and methylenedichloride. The line segments drawn from the origin have been added to emphasize these groupings. Unipolar axes could have been defined here in the same way as in Fig. 31.6. Bipolar axes on the column-standardized biplot, however, cannot be interpreted directly in terms of the original data in X. 

With log column-centering we obtain unipolar axes by substituting eq. (31.46) in eq. (31.35) ... 

Unipolar and bipolar axes have been discussed in Section 31.2. Briefly, a unipolar axis is defined by the origin and the representation of a row or column. A bipolar axis is drawn through the representations of two rows or through the representations of two columns. Projections upon unipolar axes reproduce the values in the transformed data table. Projections upon bipolar axes reproduce the contrasts (i.e. differences) between values in the data table. 

Drug-drug interactions Carbamazepine and valproate The mood stabilizers carbamazepine and valproate are often used in combination with antidepressants in patients with unipolar or bipolar affective disorder. The effects of valproate and carbamazepine on the steady-state pharmacokinetics of moclobemide and two metabolites have been studied in a non-randomized crossover study in 21 patients with unipolar depression [3 ]. Valproate had no effect, but carbamazepine was associated with a 35% reduction in moclobemide AUC, a 28% reduction in C ,ax, and a 41% reduction in clearance after 4 weeks of co-administration. These changes were interpreted as being due to induction carbamazepine of the metabolism of moclobemide and its main metabolite. However, there was no concurrent loss of efficacy, throwing into doubt the clinical significance of these significant kinetic effects. 

---