Relationship square


In statistics, the relationship square is a graphical representation for use in the factorial analysis of a table individuals x variables. This representation completes classical representations provided by principal component analysis or multiple correspondence analysis, namely those of individuals, of quantitative variables and of the categories of qualitative variables. It is especially important in factor analysis of mixed data and in multiple factor analysis.

Definition of ''relationship square'' in the MCA frame

The first interest of the relationship square is to represent the variables themselves, not their categories, which is all the more valuable as there are many variables. For this, we calculate for each qualitative variable and each factor, the square of the correlation ratio between the and the variable, usually denoted :


Thus, to each factorial plane, we can associate a representation of qualitative variables themselves.
Their coordinates being between 0 and 1, the variables appear in the square having as vertices the points,, and.

Example in MCA

Six individuals makes easier the reading of the classic factorial plane. It indicates that:
  • The first factor is related to the three variables but especially and then.
  • The second factor is related only to and and that in a strong and equal manner.
All this is visible on the classic graphic but not so clearly. The role of the relationship square is first to assist in reading a conventional graphic. This is precious when the variables are numerous and possess numerous coordinates.

Extensions

This representation may be supplemented with those of quantitative variables, the coordinates of the latter being the square of correlation coefficients. Thus, the second advantage of the relationship square lies in the ability to represent simultaneously quantitative and qualitative variables.
The relationship square can be constructed from any factorial analysis of a table individuals x variables. In particular, it is used systematically:
An extension of this graphic to groups of variables is used in Multiple Factor Analysis

History

The idea of representing the qualitative variables themselves by a point is due to Brigitte Escofier. The graphic as it is used now has been introduced by Brigitte Escofier and Jérôme Pagès in the framework of multiple factor analysis

Conclusion

In MCA, the relationship square provides a synthetic view of the connections between mixed variables, all the more valuable as there are many variables having many categories. This representation iscan be useful in any factorial analysis when there are numerous mixed variables, active and/or supplementary.