Joint correspondence analysis versus multiple correspondence analysis: A solution to an undetected problem

The problem of the proper dimension of the solution of a Multiple Correspondence Analysis (MCA) is discussed, based on both the re-evaluation of the explained inertia sensu Benźecri (Les Cahiers de l'Analyse des Donńees 4:377-379, 1979) and Greenacre (Multiple correspondence analysis and related methods, Chapman and Hall (Kluwer), Dordrecht, 2006) and a test proposed by Ben Ammou and Saporta (Revue de Statistique Appliqúee 46:21-35, 1998). This leads to the consideration of a better reconstruction of the off-diagonal sub-tables of the Burt's table crossing the nominal characters taken into account. Thus, Greenacre (Biometrika 75:457-467, 1988) Joint CorrespondenceAnalysis (JCA) is introduced, the results obtained on an application are shown, and the quality of reconstruction of both MCA and JCA solutions are compared to that of a series of Simple Correspondence Analyses run on the whole set of two-way tables. It results that JCA's reduced-dimensional reconstruction is much better than the MCA's one, that reveals highly biased and non-monotone, but also than the MCA's re-evaluation, as suggested by Greenacre (Multiple correspondence analysis and related methods, Chapman and Hall (Kluwer), Dordrecht, 2006).