Column Coordinates in Multiple Correspondence Analysis

Click the Summary: Column coordinates button on either the Quick tab or the Advanced tab of the Multiple Correspondence Analysis Results dialog box to display a spreadsheet with the coordinates for the column points, as well as various additional statistics that are useful for evaluating the adequacy of the number of dimensions in the current solution. The statistics contained in the spreadsheet are described here.

The results of a multiple correspondence analysis are identical to a simple correspondence analysis of the indicator or design matrix, whose inner product is the current Burt table (see the MCA - Introductory Overview). Therefore, the interpretation of the typical statistics reported as the result of a multiple correspondence analysis are analogous to those of the simple correspondence analysis (of an indicator matrix).

Coordinates for dimension 1, 2, ... The first several columns in the spreadsheet report the coordinates for the selected number of dimensions (see the options in the Number of dimensions group box on either the Quick tab or the Options tab). The specific method of computation is described in Computational Details (note that in multiple correspondence analysis, the typical standardization for the column points is based on the column profile matrix, see also the description of the standardization options for the Correspondence Analysis Results dialog box).

Mass. The Mass column contains the respective column totals (and row totals, since the Burt table is symmetrical) for the table of relative frequencies (i.e., for the table where each entry is the respective mass; see the Introductory Overview).

Quality. The Quality column contains information concerning the quality of representation of the respective column point in the coordinate system defined by the respective numbers of dimensions, as chosen by the user. The quality of a point is defined as the ratio of the squared distance of the point from the origin in the chosen number of dimensions, over the squared distance from the origin in the space defined by the maximum number of dimensions (remember that the metric here is Chi-square, as described in the Introductory Overview). A low quality means that the current number of dimensions does not well represent the respective column.

Relative Inertia. The quality of a point (see above) represents the proportion of the contribution of that point to the overall inertia (Chi-square) that can be accounted for by the chosen number of dimensions. However, it does not indicate whether or not, and to what extent, the respective point does in fact contribute to the overall inertia (Chi-square value). The relative inertia represents the proportion of the total inertia accounted for by the respective point, and it is independent of the number of dimensions selected. Note that a particular solution may represent a point very well (high quality), but the same point may not contribute much to the overall inertia (e.g., a column point with a pattern of relative frequencies across the rows that is similar to the average pattern across all columns).

Inertia for each Dimension. This column contains the relative contribution of the respective column point to the inertia "accounted for" by the respective dimension. Thus, this value will be reported for each column point,

Cosine² (quality or squared correlations of each dimension). This column contains the quality for each point, by dimension. The sum of the values in these columns across the dimensions is equal to the total quality value. This value may also be interpreted as the "correlation" of the respective point with the respective dimension. The term Cosine² refers to the fact that this value is also the squared cosine value of the angle the point makes with the respective dimension (refer to Greenacre, 1984, for details concerning the geometric aspects correspondence analysis).

Note: statistical significance. It should be noted at this point that correspondence analysis is a descriptive and/or exploratory technique. Actually, the method was developed based on a philosophical orientation that emphasizes the development of models that fit the data, rather than the rejection of hypotheses based on the lack of fit (Benzecri's "second principle" states that "The model must fit the data, not vice versa;" see Greenacre, 1984, p. 10). Therefore, there are no statistical significance tests that are customarily applied to the results of a correspondence analysis; the primary purpose of the technique is to produce a simplified (low-dimensional) representation of the information in a large frequency table (or tables with similar measures of correspondence).