Multidimensional Scaling of Discrete Probability Distributions"
In dad: Three-Way / Multigroup Data Analysis Through Densities

knitr::opts_chunk$set(echo = TRUE)

library(dad)

Introduction: example and objective of the method

The dataset dspg of the dad package is a list of $T = 7$ matrices. For each of the $T$ years 1968, 1975, 1982, 1990, 1999, 2010 and 2015, we have the contingency table of Diploma $\times$ Socioprofessional group in France. Each table has:

4 rows corresponding to a level of diploma (diplome):
bepc: brevet
cap: NCQ (CAP)
bac: baccalaureate
sup: higher education (supérieur)
6 columns corresponding to socio professional groups (csp):
agri: farmer (agriculteur)
cardr: senior manager (cadre supérieur)
pint: middle manager (profession intermédiaire)
empl: employee (employé)
ouvr: worker (ouvrier)

data("dspg")
print(dspg)

After the computation of the distances or divergences between each pair of occasions, that is the distances $(\delta_{ts})$ between their corresponding distributions, the MDS technique looks for a representation of the distributions by $T$ points in a low dimensional space such that the distances between these points are as similar as possible to the $(\delta_{ts})$.

The dad package includes functions for all the calculations required to implement such a method and to interpret its outputs:

The mdsdd function performs MDS and generates scores;
The plot function generates graphics representing the probability distributions on the factorial axes;
The interpret function returns other aids to interpretation based on the marginal distributions.

The `mdsdd` function

MDS of discrete probability distributions can be carried using the mdsdd function. This function applies to

an object of class "folder" (in this case, it is used the same way as fmdsd (see help), except that the columns of each data frame of the folder are not numeric, but factors)
or a list of arrays (or a list of tables).

The following example shows the application of mdsdd on a list of arrays. The mdsdd function is built on the cmdscale function of R. It is carried out on the dataset dspg as follows:

resultmds <- mdsdd(dspg)

In addition to the add argument of cmdscale, the mdsdd function has two sets of optional arguments:

The first, consisting of distance, controls the method used to compute the distances between the distributions.
The second set consists of optional arguments which control the function outputs.

Interpretation of `mdsdd` outputs

The mdsdd function returns an object of S3 class "mdsdd", consisting of a list of 9 elements, including the scores, also called principal coordinates, and the marginal and joint distributions of the variables per occasion.

names(resultmds)

The outputs are displayed with the print function:

print(resultmds)

Graphical representations on the principal planes are generated with the plot function:

plot(resultmds, fontsize.points = 1)

In this example, a single axis is enough to explain the general trends; the first principal coordinate explains 92% of the inertia.

This graph shows an evolution of the value of the first principal score, which gets higher for recent years.

The interpretation of outputs is based on the relationships between the principal scores and the marginal or joint frequencies. These relationships are quantified by correlation coefficients and are represented graphically by plotting the scores against the frequencies. These interpretation tools are provided by the interpret function which has two optional arguments: nscores indicating the indices of the column scores to be interpreted and mma whose default value is "marg1" (the probability distributions of each variable).

interpret(resultmds, nscore = 1)

From the correlations between the principal coordinates (PC) and the distributions of the variables, we deduce that:

The higher $PC1$, the higher the frequencies of the diplomas "diplome.bac" and "diplome.sup", the higher "diplome.cap" tends to be, and the lower the frequencies of "diplome.bepc".
The higher $PC1$, the higher the frequencies of the socio professional groups "csp.cadr", "csp.pint" and "csp.empl", and the lower the frequencies of "csp.agri", "csp.arti" and "csp.ouvr".

So, reminding that $PC1$ gets higher for recent years, these results highlight that in France, since 1968:

the number of brevet graduates have decreased and higher degrees have increased,
the number of farmers, craftsmen and workers have decreased and the number of employees, middle and senior managers have increased.