SPSS 10 Correspondence Analysis Output

This is commented output, with the comments in blue font like this.

Notes
Output Created 21-MAR-2001 20:16:31

Comments

Input Data C:\Program Files\SPSS\GSS93 subset.sav

Filter <none>

Weight <none>

Split File <none>

N of Rows in Working Data File 1500

Syntax CORRESPONDENCE
TABLE = region4(1 4) BY politics(1 5)
/DIMENSIONS = 2
/MEASURE = CHISQ
/STANDARDIZE = RCMEAN
/NORMALIZATION = SYMMETRICAL
/PRINT = TABLE RPOINTS CPOINTS RPROFILES CPROFILES RCONF CCONF
/PLOT = NDIM(1,MAX) BIPLOT(20) RPOINTS(20) CPOINTS(20) TRROWS(20) TRCOLUMNS
(20) .

Resources Elapsed Time 0:00:00.11

**Notes**
Output Created	21-MAR-2001 20:16:31
Comments
Input	Data	C:\Program Files\SPSS\GSS93 subset.sav
Filter	<none>
Weight	<none>
Split File	<none>
N of Rows in Working Data File	1500
Syntax	CORRESPONDENCE TABLE = region4(1 4) BY politics(1 5) /DIMENSIONS = 2 /MEASURE = CHISQ /STANDARDIZE = RCMEAN /NORMALIZATION = SYMMETRICAL /PRINT = TABLE RPOINTS CPOINTS RPROFILES CPROFILES RCONF CCONF /PLOT = NDIM(1,MAX) BIPLOT(20) RPOINTS(20) CPOINTS(20) TRROWS(20) TRCOLUMNS (20) .
Resources	Elapsed Time	0:00:00.11

**Credit**
CORRESPONDENCE
Version 1.0
by
Data Theory Scaling System Group (DTSS)
Faculty of Social and Behavioral Sciences
Leiden University, The Netherlands

The "Correspondence Table" below is simply the crosstabulation of the row and column variables, including the row and column marginal totals, serving as input.

**Correspondence Table**
	Political Outlook
Region	Liberal	Tend Lib	Moderate	Tend Cons	Conservative	Active Margin
Northeast	19	23	58	16	15	131
Midwest	26	31	71	47	35	210
South	18	27	75	46	70	236
West	30	19	40	26	33	148
Active Margin	93	100	244	135	153	725

The "Row Profiles" are the cell contents divided by their corresponding row total (ex., 19/131 = .145 for the first cell).

**Row Profiles**
	Political Outlook
Region	Liberal	Tend Lib	Moderate	Tend Cons	Conservative	Active Margin
Northeast	.145	.176	.443	.122	.115	1.000
Midwest	.124	.148	.338	.224	.167	1.000
South	.076	.114	.318	.195	.297	1.000
West	.203	.128	.270	.176	.223	1.000
Mass	.128	.138	.337	.186	.211

Likewise, the Column Profiles are the cell elements divided by the column marginals (ex., 19/103 = .204). This table also shows the row masses (row marginals as a percent of n) (ex., 131/725= .181). These are intermediate calculations on the way toward computing distances between points.

**Column Profiles**
	Political Outlook
Region	Liberal	Tend Lib	Moderate	Tend Cons	Conservative	Mass
Northeast	.204	.230	.238	.119	.098	.181
Midwest	.280	.310	.291	.348	.229	.290
South	.194	.270	.307	.341	.458	.326
West	.323	.190	.164	.193	.216	.204
Active Margin	1.000	1.000	1.000	1.000	1.000

In the Summary table below, we first look at the table chi-square value and see that it is significant, justifying the assumption that the two variables are related. SPSS has computed the interpoint distances and subjected the distance matrix to principal components analysis, yielding in this case three dimensions. Only the interpretable dimensions are reported, not the full solution, which is why the eigenvalues (labeled Inertia below; these are the percent of variance explained by each dimension) add to something less than 100% -- in this case only .057 = 5.7%. This reflects the fact that the correlation between region and political outlook, while significant, is weak. The eigenvalues reflect the relative importance of each dimension, with the first always being the most important, the next second most important, etc.

The singular values are simply the square roots of the eigenvalues. They are interpreted as the maximum canonical correlation between the categories of the variables in analysis for any given dimension.

Note that the "Proportion of Inertia" columns are the dimension eigenvalues divided by the total (table) eigenvalue. That is, they are the percent of variance each dimension explains of the variance explained: thus the first dimension explains 62.7% of the 5.7% of the variance explained by the model.

The standard deviation columns refer back to the singular values and help the researcher assess the relative precision of each dimension.

**Summary**
	Singular Value	Inertia	Chi Square	Sig.	Proportion of Inertia		Confidence Singular Value
					Accounted for	Cumulative	Standard Deviation	Correlation
Dimension					Accounted for	Cumulative	Standard Deviation	2
1	.189	.036			.627	.627	.035	-.043
2	.124	.015			.268	.895	.040
3	.078	.006			.105	1.000
Total		.057	41.489	.000(a)	1.000	1.000
a 12 degrees of freedom

The Overview Row Points table below, for each row point in the correspondence table, displays the mass, scores in dimension, inertia, contribution of the point to the inertia of the dimension, and contribution of the dimension to the inertia of the point. To recall:

Mass: the marginal proportions of the row variable, used to weight the point profiles when computing point distance. This weighting has the effect of compensating for unequal numbers of cases
Scores in dimension: scores used as coordinates for points when plotting the correspondence map. Each point has a score on each dimension.
Inertia: Variance
Contribution of points to dimensions: as factor loadings are used in conventional factor analysis to ascribe meaning to dimensions, so "contribution of points to dimensions" is used to intuit the meaning of correspondence dimensions.
Contribution of dimensions to points: these are multiple correlations which reflect how well the principal components model is explaining any given point (category).

**Overview Row Points(a)**
	Mass	Score in Dimension		Inertia	Contribution
		1	2		Of Point to Inertia of Dimension		Of Dimension to Inertia of Point
Region		1	2		1	2	1	2	Total
Northeast	.181	-.702	.309	.020	.470	.139	.832	.105	.938
Midwest	.290	-.130	.065	.005	.026	.010	.181	.030	.210
South	.326	.540	.194	.020	.501	.099	.901	.076	.977
West	.204	-.055	-.675	.012	.003	.752	.010	.970	.979
Active Total	1.000			.057	1.000	1.000
a Symmetrical normalization

The Overview Column Points table below is similar to the previous one, except for the column variable in the correspondence table.

**Overview Column Points(a)**
	Mass	Score in Dimension		Inertia	Contribution
		1	2		Of Point to Inertia of Dimension		Of Dimension to Inertia of Point
Political Outlook		1	2		1	2	1	2	Total
Liberal	.128	-.491	-.800	.016	.163	.663	.363	.630	.993
Tend Lib	.138	-.351	.124	.003	.090	.017	.921	.075	.995
Moderate	.337	-.252	.334	.009	.113	.303	.448	.512	.960
Tend Cons	.186	.237	-.037	.006	.055	.002	.308	.005	.313
Conservative	.211	.721	-.094	.022	.579	.015	.940	.010	.950
Active Total	1.000			.057	1.000	1.000
a Symmetrical normalization

The Confidence Row Points and Confidence Column Points tables below display the standard deviations of the row or column scores (the values used as coordinates to plot the correspondence map) and are used to assess their precision.

**Confidence Row Points**
	Standard Deviation in Dimension		Correlation
Region	1	2	1-2
Northeast	.190	.307	.528
Midwest	.169	.323	.066
South	.122	.206	-.685
West	.339	.148	-.026

**Confidence Column Points**
	Standard Deviation in Dimension		Correlation
Political Outlook	1	2	1-2
Liberal	.387	.221	-.694
Tend Lib	.072	.117	.801
Moderate	.171	.122	.575
Tend Cons	.215	.406	.095
Conservative	.127	.302	.304

Next come the requested plots.

The plots of transformed categories for dimensions below display a plot of the transformation of the row category values and of column category values into scores in dimension, with one plot per dimension. The x axis has the category values and the y axis has the corresponding dimension scores. Thus the category "Northeast" in the Overview Row Points table above had a score in dimension of -.702, as shown on the plot below. Note that there are various types of normalization, a.k.a standardization, not just the symmetrical option used in this example. Comparing how different types of normalization affect transformation of category values into dimension scores can be insightful, but that requires re-running the analysis using different normalization options, not illustrated here.

Dimension 1 transformed region categories

Dimension 2 transformed region categories

Dimension 1 transformed political outlook categories

Dimension 2 transformed political outlook categories

The next two plots below the uniplots for the row and column variables. Note that the origin of the axes is slightly different in the two plots. Not also that both plots are based on symmetrical normalization. Usually uniplots are based on row normalization or column normalization, but that requires re-running the analysis using these normalization options, not illustrated here.

Finally the biplot correspondence map is shown. Note the axes now encompass the most extreme values of both of the uniplots. Note that while some generalizations can be made about the association of categories (South more conservative, West more liberal), the researcher must keep firmly in mind that correspondence is not association. That is, the researcher should not allow the map's display of inter-category distances obscure the fact that, for this example, the model only explains 5.7% of the variance in the correspondence table.