# CDT for Multiple Correspondence Analysis (MCA): what if there are multiple modalities selected for one variable?

I have the following set of qualitative data:

 ______________________________________________
| Observations | City    | Transportation mean |
|______________|_________|_____________________|
| Individual#1 | Paris   | Bicycle             |
|--------------|---------|---------------------|
| Individual#2 | London  | Car                 |
|--------------|---------|---------------------|
| Individual#3 | Paris   | (Bicycle, Car)      |
|______________|_________|_____________________|


To analyze them, I thus want to run a Multiple Correspondence Analysis (MCA). However, for some variables (here Transportation mean), modalities are non-exclusive (i.e. Individual#3 use both Bicycle and Car).

Questions:

• Is it ok to do a MCA on such data?
• If yes, how to construct the complete disjunctive table (CDT)?

Issue

If yes, I'm not sure what is the best CDT to use between these following two:

 _______________________________________________________
| Observations | C_Paris | C_London | T_Bicycle | T_Car |
|______________|_________|__________|___________|_______|
| Individual#1 | 1       | 0        | 1         | 0     |
|--------------|---------|----------|-----------|-------|
| Individual#2 | 0       | 1        | 0         | 1     |
|--------------|---------|----------|-----------|-------|
| Individual#3 | 1       | 0        | 1         | 1     |
|______________|_________|__________|___________|_______|

_______________________________________________________
| Observations | C_Paris | C_London | T_Bicycle | T_Car |
|______________|_________|__________|___________|_______|
| Individual#1 | 1       | 0        | 1         | 0     |
|--------------|---------|----------|-----------|-------|
| Individual#2 | 0       | 1        | 0         | 1     |
|--------------|---------|----------|-----------|-------|
| Individual#3 | 1       | 0        | .5        | .5    |
|______________|_________|__________|___________|_______|


The difference is that, in the first table, weights are only binary (0 or 1); when in the second, the weight in a cell is: weight = bool_selected? / (number of modalities selected for this category and this observation) (where bin_selected? is a boolean that equals 1 if the conisidered modality has been selected, and 0 otherwise).
E.g. since Individual#3 selected two modalities (Bicycle and Car) of the variable Transportation mean, the weight for a modality of the variable Transportation mean for the observation Individual#3 equals 1/2 = .5 if the modality is selected and 0/2 = 0 if not.

Hy @ebo, first of all, "yes", it is Okay to analyse such data.

I am reading the book Exploratory Multivariate Analysis by Example Using R by the creators from FactoMineR package and watching the playlist of https://www.youtube.com/playlist?list=PLnZgp6epRBbTsZEFXi_p6W48HhNyqwxIu regarding MCA.

It seems that you can analyze it such as another category of your variable. (You should run the example of tea dataset from the function MCA of FactoMineR package. There is a variable such as yours: “What kind of tea do you buy (tea bags, loose tea, both)?”, and it is coded this way in the data.

Your construction is wrong, since the table should be disjunctive and complete: One individual must only be in one and only one category of a variable.

The CDT table will look like this:

_________________________________________________________________________
| Observations | C_Paris | C_London | T_Bicycle | T_Car | T_Bicycle_car |
|______________|_________|__________|___________|_______|_______________|
| Individual#1 | 1       | 0        | 1         | 0     | 0             |
|--------------|---------|----------|-----------|-------|---------------|
| Individual#2 | 0       | 1        | 0         | 1     | 0             |
|--------------|---------|----------|-----------|-------|---------------|
| Individual#3 | 1       | 0        | 0         | 0     | 1             |
|______________|_________|__________|___________|_______|_______________|