Correspondence Analysis of Square Tables

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Correspondence Analysis of Square Tables
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Correspondence Analysis of Square Tables is conducted using the Maps dialog box.

Required data

Correspondence analysis of square tables requires a table where the names and ordering of the rows and columns are identical. Most commonly, this will be a crosstab of brand switching or brand repertoire data.

A square table is a table where the rows and columns have the same labels. There are two types of square matrixes. A symmetric square table is one where the same question is selected in both the blue and brown drop-downs. For example, insight into market structure can often be obtained by selecting a brand repertoire question in both the blue and brown drop-downs. An asymmetric square table is a table where the rows and columns have the same labels, but they have different meanings. The most common example in market research is a switching matrix, showing purchasing or consumption of brands on two separate occasions.

Statistic used in the analysis

Q computes the analysis using whichever of the following statistics is available on the table (where multiple are available, the one that appears first is used):

Interpretation

An example of a switching matrix is shown below. (This table was formed with Q7. Company currently with and Company for previous contract - if on contract from Tutorial 1.Q; the small categories have been combined into the Other category, and the Don’t Know and No previous contract categories have been set as Missing Data. The statistic displayed has been changed to Total %.

SwitchingMatrix.png

Correspondence analysis with a square table is different to traditional Correspondence Analysis in that:

  • We now only have one set of coordinates. In traditional correspondence analysis we obtain both rows and column scores because their respective labels are charted in different positions on the map.
  • If the table is asymmetric, an output will indicate the proportion of the patterns that is symmetrical. For example, the output below indicates 89.2% symmetrical, which means that 89.2% of the patterns identified in the table are symmetrical. If say, a relatively large proportion of Optus customers switch to Telstra and a relatively large proportion switch from Telstra back to Optus, this is an example of a symmetric pattern. An asymmetric pattern is a one-way flow (e.g., relatively many switching from Optus to Telstra but relatively few switching back). When using a symmetric table (i.e., the same question in both the blue and brown drop-downs), this result is not shown.

With an asymmetric table, different dimensions reveal either asymmetric or symmetric relationships. In this example, symmetric relationships are revealed by dimensions 1, 2, 5, 8 and 9, which have unique eigenvalues/canonical correlation.

The asymmetric coordinates are not unique. The coordinates reported using the Maps dialogue, or as an R Output may differ by a rotation. Symmetric and Asymmetric dimensions should be considered separately.

The asymmetric patterns revealed in the remaining dimensions are extremely difficult to interpret and it is generally advisable to focus directly on the table for understanding asymmetries; row percentages are usually most easily interpretable (for a detailed discussion of the interpretation of correspondence analysis of asymmetric matrices see Greenacre, M. (2000). "Correspondence analysis of square asymmetric matrices." Applied Statistics 49(3): 297-310).

Total sample
Unweighted
base n = 361; total n = 713; 352 missing

Correspondence analysis of a square table
 
Inertia(s):
              Canonical Correlation  Inertia  Proportion
Dimension 1                   .581     .337        .563
Dimension 2                   .415     .172        .287
Dimension 3                   .170     .029        .048
Dimension 4                   .170     .029        .048
Dimension 5                   .153     .023        .039
Dimension 6                   .060     .004        .006
Dimension 7                   .060     .004        .006
Dimension 8                   .042     .002        .003
Dimension 9                   .000     .000        .000

Standard coordinates:
                       Dimension 1  Dimension 2  Dimension 3  Dimension 4  Dimension 5  Dimension 6  Dimension 7  Dimension 8  Dimension 9
Other                        -.12         -.21        -1.67         -.68         3.70        -1.18         3.86        -2.31         1.20
Optus                         .54        -1.07         1.06        -1.23          .11         -.36          .19          .68          .83
Orange (Hutchison)            .86         -.99        -2.84        -2.50        -2.38        -1.03        -2.02        -3.34          .99
Telstra (Mobile Net)        -1.49          .19         -.12         -.66         -.27         1.59          .21          .06         1.13
Vodafone                      .87         1.46          .46          .06         -.07         -.53         -.30          .14         1.59

Principal coordinates:
                       Dimension 1  Dimension 2  Dimension 3  Dimension 4  Dimension 5  Dimension 6  Dimension 7  Dimension 8  Dimension 9
Other                        -.07         -.09         -.12          .28          .57         -.23         -.07         -.10          .00
Optus                         .31         -.44         -.21         -.18          .02         -.01         -.02          .03          .00
Orange (Hutchison)            .50         -.41         -.42          .48         -.36          .12         -.06         -.14          .00
Telstra (Mobile Net)         -.87          .08         -.11          .02         -.04         -.01          .10          .00          .00
Vodafone                      .51          .60          .01         -.08         -.01          .02         -.03          .01          .00

89.2% symmetrical

Scores of symmetric dimensions:
                       Dimension 1  Dimension 2  Dimension 5  Dimension 8  Dimension 9
Other                        -.07         -.09          .57         -.10          .00
Optus                         .31         -.44          .02          .03          .00
Orange (Hutchison)            .50         -.41         -.36         -.14          .00
Telstra (Mobile Net)         -.87          .08         -.04          .00          .00
Vodafone                      .51          .60         -.01          .01          .00

The chart below shows the first two dimensions (with an asymmetric table, the first two symmetrical dimensions are charted. This reveals that Optus and Orange are relatively strong competitors (i.e., given their market shares, there is a high level of switching between these brands), while all the others are comparatively isolated (this finding is also evidence in the percentages data shown in the earlier table).

CorrespondenceAnalysisTable.png

Technical details

For technical details, please refer to Greenacre, M. (2000). "Correspondence analysis of square asymmetric matrices." Applied Statistics 49(3): 297-310.