Sunday, March 31, 2013

Factor Analysis for Mixed Data

Usually, as a factor analysis approach, we use the principal component analysis (PCA) when the active variables are quantitative; the multiple correspondence analysis (MCA) when they are all categorical. But what to do when we have a mix of these two types of variables?

A possible strategy is to discretize the quantitative variables and use the MCA. But this procedure is not recommended if we have a small dataset (a few number of instances), or if the number of qualitative variables is low in comparison with the number of quantitative ones. In addition, the discretization implies a loss of information. The choice of the number of intervals and the calculation of the cut points are not obvious.
Another possible strategy is to replace each qualitative variable by a set of dummy variables (a 0/1 indicator for each category of the variable to recode). Then we use the PCA. This strategy has a drawback. Indeed, because the dispersions of the variables (the quantitative variables and the indicator variables) are not comparable, we will obtain biased results.

The Jérôme Pages' "Multiple Factor Analysis for Mixed Data" (2004) [AFDM in French] relies on this second idea. But it introduces an additional refinement. It uses dummy variables, but instead of the 0/1, it uses the 0/x values, where 'x' is computed from the frequency of the concerned category of the qualitative variable. We can therefore use a standard program for PCA to lead the analysis (Pages, 2004; page 102). The calculation process is thus well controlled. But the interpretation of the results requires a little extra effort since it will be different depending on whether we study the role of a quantitative or qualitative variable.

In this tutorial, we show how to perform an AFDM with Tanagra 1.4.46 and R 1.15.1 (FactoMinerR package). We emphasize the reading of the results. We must study simultaneously the influence of quantitative and qualitative variables for the interpretation of the factors.

Keywords: PCA, principal component analysis, MCA, multiple correspondence analysis, AFDM, correlation, correlation ratio, FactoMineR package, R osftware
Tutorial: en_Tanagra_AFDM.pdf
Dataset: AUTOS2005AFDM.txt
References :
Jerome Pages, « Analyse Factorielle de Données Mixtes », Revue de Statistique Appliquee, tome 52, n°4, 2004 ; pages 93-111.