StatEL : Multiple Correspondence Analysis - MCA

This command launches analysis procedure of a measures table (both qualitative or quantitative variables of a group of observations) :

1 - Principles of Multiple Correspondence Analysis :

Graphical representation of a measures table can be made very easily as soon as we have very few variables : it will be plane if there is only 2 variables (one for abscissa and one for ordinate), it will be in 3 dimensions if we add a third variable. Representation is impossible if number of variables is higher than 3, since it requires an axis to code each variable.

Principle of MCA is to synthesize informations contained in a measures table of both qualitative and quantitative variables, whatever the number of variables and observations, by detecting main tendencies of this table, due to variables as well as observations.

MCA allows to detect axes (= principal components) by which points spread the most. These "synthetic" axes may result of influence of one or several variables or observations.

Thus, by projection of points (observations or variables) on plans defined by these new axes, we have a representative "photograph" of points.

By analogy, imagine someone is trying you to recognize an animal drawn on a paper sheet (that is a camel) but you see only its face shape. You can not conclude if it is a camel or a dromedary, just because angle of analysis is not the most informative. You need a profile image of the animal to conclude with certitude that it is a camel. On the same way, PCA calculates new axes (and then plans) able to advise you at the best about repartition of points.

MCA can be practiced after transformation of the table of observations in a new type of table, containing frequencies of every modality (tabe of Burt).

Eventually, it is possible to add some "observations" points or "variables" points (illustrative) in both representations. These points are not used to define new axes, but they are added in the representations for helping interpretation of new axes.

Nota bene : MCA realized on a same data set with differents softwares may supply different representations, some axes are inverted. There is no mistake, but the result of the way inertia matrix has been diagonalized.

In spite of differences in graphs, you can notice that related positions of points are always the sames (as well as results of correlation, contribution, quality and distance - cf. below). Indeed specificity of MCA is to proceed analysis of data, the ones compared to the others.

2 - Launch of Multiple Correspondence Analysis :

Firts dialog box allows you to select the ddifferent variables (qualitative or quanttative, independantly)

2.1 - Selection of a qualitative variable :

Click on the corect button and select the range of cells containing one of the variables that you want to analyze, with th name of the variable in the first cell.

Modalities of the selected variable appear in a new dialog box. You can delete on or the other of the modalities, or specify this variable as "illustrative".

Once your choice is validated, the selected variable appeear in the list of the first dialog box, with its specificities.

2.2 - Selection of a quantitative variable :

Click on the corect button and select the range of cells containing one of the variables that you want to analyze, with th name of the variable in the first cell.

A new dialog box allows you to specify the way you want to code the numerical variable. You can also decide if this variable must be considered as an illustrative variable.

Once your choice is validated, the selected variable appeear in the list of the first dialog box, with its specificities.

3 - Results of Multiple Correspondence Analysis :

Results are displayed on a new Excel sheet.

Please notice that some cells have comments to explain their content (red triangle).

Details of analysis are displayed upper left of the results sheet :

• nb of analysed variables,
• nb of analysed modalities,
• nb of illustrative variables,
• nb of illustrative modalities),
• nb of new axis extractes by MCA.

Below these recalls are displayed characteristics of new axes (or factors) of MCA, as well as a resume of them on a graph.

In the middle of the results sheet, you can see both representations :

• graph représenting relative positions of modalities of the different variables,
• graph représenting relative positions of modalities, and the weight of every modality (sum of every row or column of the table of Burt).

On the right part of the results sheet are displayed every numerical data necessary for MCA analysis :

• the table of Burt,
• data telated to graphical representations :
• relative weights of every modality,
• coordinates of points in the graphs,
• points' contributions that represent importance of each observation or variable in variance of each factor,
• qualities of representation (or cos˛) that inform on angle between the line linking center of points to studied point, and the considered axis. If cos˛ is close to 1, it implies that this angle is close to 0 ; thus if projection of a point is close to an axis, this point will be, in space, indeed close to the axis,
• distances of observations points and variables points from center of points. Notice that variables points are all located on a sphere whose radius = 1 and the center is the center of points, whereas observations points can be anywere in space,
• inertia that expresses percentage of variance of points explained by the considered observation point or the variable point.

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