For the quantification of the variables see the Homogeneity Analysis of this dataset.
A two-dimensional PRINCALS-solution for all the variables shows a fit of .63, which is slightly worse than the fit of the corresponding two-dimensional HOMALS-solution.
In Graph 1, the plots give the original vs the optimally transformed category quantifications in both dimensions for the IQ, Teacher's Advice and Father's Profession variables. The transformations are monotone for all variables in both directions (a property of the PRINCALS solution for single ordinal variables; as these three variables are treated in this case). It can be seen immediately that the first dimension discriminates much better than the second one. To get a better picture we can look at the single category quantifications of the Gender, IQ, Teacher's Advice and Father's Profession variables which are presented in Graph 2. As already seen from Graph 1 the IQ, the Teacher's Advice and the Father's Profession variables discriminate really well along the first dimension, while Father's Advice discriminates also quite well along the second dimension. Notice that the quantifications of each variable are ordered and lie on a line that goes through the origin (another property of the PRINCALS solution for single ordinal variables).
The component loadings are given in Graph 3. The IQ and Teacher's Advice variables are highly correlated (long arrows close together in the plot). Relative high correlations occur also among these two variables and the Father's Profession variable. This plot also shows that the first dimension is determined by the IQ and the Teacher's Advice variables (and by Father's Profession to some extent), while the School and the Gender variables manily determine the second dimension.
Finally, the object scores are given in Graph 4. As in the Homals solution the object scores of Boys and Girls continue to form two somewhat separate clusters. However, the horseshoe has disappeared (another property of the PRINCALS solution).
Graphs 2 and 3 indicate that the second dimension is mainly determined by the Gender and the Father's Profession analysis. Therefore, we can create an interactive variable (INT) comprised of 8 categories (2x4) and repeat the PRINCALS analysis.