Overview Overview Correspondence analysis is a method of factoring categorical variables and displaying them in a property space which maps their association in two or more dimensions. It is often used where a tabular approach is less effective due to large tables with many rows and/or columns. Though not limited to that arena, correspondence analysis been popular in marketing research, as to display such variables as customer color preference, size preference, and taste preference in relation to preferences for Brands A, B, and C. Correspondence analysis is a special case of canonical correlation, where one set of entities (categories rather than variables as in conventional canonical correlation) is related to another set. Correspondence analysis starts with tabular data, usually two-way cross-classifications, though the technique is generalizable to n-way tables with more than two variables. The variables must be discrete: nominal, ordinal, or continuous variables segmented into ranges. The technique defines a measure of distance between any two points, where points are the values (categories) of the discrete variables. Since distance is a type of measure of association (correlation), the distance matrix can be the input to principal components analysis, just as correlation matrices may be the input for conventional factor analysis. However, where conventional factor analysis determines which variables cluster together, correspondence analysis determines which category values are close together. This is visualized on the correspondence map, which plots points (categories) along the computed factor axes. Because the definition of point distance in correspondence analysis does not support significance testing, it is recommended that some other technique compatible with discrete data, such as log-linear modeling or logistic regression, be used to test alternative models. After selecting a best-fitting model using another technique, then correspondence analysis may be very useful in exploring relationships within that model. [Note: Click here for former PA 765 unit on correspondence analysis (not currently taught).]

Contents Key concepts and terms Assumptions SPSS output Frequently asked questions Bibliography




Key Concepts and Terms

• Correspondence analysis has also been called correspondence mapping, perceptual mapping, social space analysis, correspondence factor analysis, principal components analysis of qualitative data, and dual scaling; these are largely synonymous terms, though there are many variants of the technique.

• Correspondence table. This is the raw crosstabulation of two discrete variables, with marginals. The object of correspondence analysis is to explain the inertia (variance) in this table. In essence, the correspondence map is a graphical tool which helps the researcher easily to notice relationships within this table. When interpreting a correspondence map it is often helpful to refer back to the original correspondence table.

• Points: Also known as "profile points," a point is one of the values (categories) of one of the discrete variables in the analysis. For instance, "male" would be a point for the variable "gender."

• Point distance: Correspondence analysis uses a definition of chi-square distance rather than Euclidean distance between points. This is discussed further below. The point distance matrix is the input to principal components analysis, yielding the dimensions (factors) which correspondence analysis uses to map points.

• Correspondence map. A correspondence map displays two of the dimensions which emerge from principal components analysis of point distances, and points are displayed in relation to these dimensions. For instance, a correspondence analysis may seek to relate political outlook (conservative, liberal, etc.) with region (South, West, etc.), and the correspondence map might show the South is close to conservative, whereas the West is closer to liberal. This is illustrated below using 1993 General Social Survey data:
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  • Interpreting correspondence maps. The scores used as coordinates are based on the analysis of the row profile matrix and are computed to maximize the differences between points with respect to the row profiles (percentages). That is, the scaling of coordinates is usually standardized to the rows (but not in the illustration above, which uses symmetrical standardization). It is also possible to standardize to the columns, symmetrically, or in other ways discussed below. Because these options result in different scaling, some authors prefer to present the correspondence map of the row points in a separate plot from the correspondence map of the column points. However, a combined biplot is much more common, displaying both row and column points in a single map.

  • Standardization/normalization. When a biplot is used, it is important to keep in mind that the distance between one row point and another row point is interpretable if row standardization (the norm) has been used, as are distances between one column point and another column point if column standardization has been used. SPSS offers four forms of standardization, which it calls normalization. Row principal is the traditional type, used to compare row variable points. Column principal is the corresponding normalization for comparing column variable points. Principal normalization is a compromise used for comparing points within either or both variables but not between variables. Symmetrical normalization, called canonical standardization elsewhere, standardizes on both row and column profiles and is suitable for comparing two variables (that is, comparing row points to column points), though it involves a form of averaging which could lead to less meaningful results than row or column standardization employed separately..

    Though symmetrical normalization is designed for this purpose, under any form of standardization one cannot precisely interpret the distance between a row point and a column point. Rather one must make a non-precise general statement, such as noting where particular row points and column points appear in the same map quadrant.

    Example. Let column points be various colleges and let row points be various traits (ex., affordability, location, etc.). The correspondence map distance between a college and a trait is not an indicator of how highly rated that college is on that trait. It will not always be true that the higher the rating, the less the map distance between the college and the trait. That is, the map location of a college will be a multivariate "compromise" position in which the distances are not reliably precise indicators of "closeness" of row points to column points As a result (1) researchers must make general statements, such as whether row and column points are in the same quadrant, not making specific comparisons of exact map distances of row points to column points; and (2) the researcher may find greater understanding of the meaning of map distances by referring back to values in the correspondence table, using the map as an easy graphical guide for where to examine the correspondence table closely.

• Contribution of points to dimensions. Just 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. The contribution of points to dimensions shows the percent of inertia (variance) of a particular dimension which is explained by a point. By looking at the more heavily loaded points, one may induce the meaning of a dimension. Contribution of points to dimensions will sum to 1.0 across the categories of any one variable. SPSS labels this "Contribution of Row Points to the Inertia of Each Dimension."

• Contribution of dimensions to points, also known as squared correlations. Also known as cr values or the quality of representation of the description of a point, these reflect how well the principal components model is explaining any given point. That is, the contribution of dimensions to points is the percent of variance in a point explained by a given dimension. One would like the points on which one's analysis focuses to have a high contribution of dimensions to points value. Less analytic focus must be placed on points which are not well described by the model.

  The sum of contributions of dimensions to points will add to 1.0 across all dimensions for a given point in the full solution where all possible dimensions are computed. However, the interpreted dimensions usually will sum to less than 1.0.

  Note that high contribution of points to dimensions implies a high squared correlation, but the reverse is not true. That is, if a point explains a lot of the variance in a dimension, usually that dimension will also describe the point very well (high squared correlation). However, just because a dimensions describes a point well does not mean the point will necessarily be important in explaining the dimension.

  • Scree test. Since only the first few dimensions will have a clear interpretation, the researcher needs some criterion for determining how many dimensions to interpret. The first dimension will have the highest eigenvalue, explaining the largest percentage of variance. Later dimensions will explain successively less. If the eigenvalues are plotted, they form a curve heading toward almost 0% explained by the last dimension. The "scree test" is to stop interpreting dimensions when this curve makes an "elbow," often around the 3rd, 4th, or 5th dimension. The scree test is a bit subjective but is widely used in correspondence analysis. Other criteria are the Kaiser criterion (stop when the eigenvalue falls below 1.0) and the variance explained criterion (stop when 90%, others use 80%, is explained). The scree plot and other criteria are discussed in the section on factor analysis.

• Eigenvalues. These are the "characteristic roots" of the principal components solution. There is one eigenvalue for each dimension, sometimes labeled the inertia for that dimension. Each eigenvalue is the amount of inertia (variance) a given factor explains in the correspondence table. It is called "Inertia" in SPSS output. Eigenvalues reflect the relative importance of the dimensions. The first dimension always explains the most inertia (variance) and has the largest eigenvalue, the next the second-most, and so on. The sum of eigenvalues is total inertia, discussed above.
  • Total inertia. Inertia means variance in the context of correspondence analysis. Total inertia is the sum of eigenvalues and reflects the spread of points around the centroid. Total inertia may be interpreted as the percent of inertia(variance) in the original correspondence table explained by all the computed dimensions in the correspondence analysis. However, usually only the first two dimensions are used in the correspondence map, so the effective model will explain a percent of inertia in the original table equal to the sum of eigenvalues for the first two dimensions only.

  • Chi-square significance of total inertia. SPSS computes a chi-square test for total inertia, along with the corresponding probability level. If this level is <= .05, the conventional cutoff, the researcher concludes the dimensions are associated with the values of the variables in the original correspondence table. Note this does not demonstrate that the two variables which are the source of the points in the original correspondence table are significantly associated. A significant chi-square for total inertia merely shows that the total inertia is not so low as to be insignificantly different from zero.

  • Proportion of inertia accounted for by a given dimension is its eigenvalue divided by total inertia. Thus, if the proportion of inertia accounted for by dimension 1 is .549, then dimension 1 explains 54.9% of the variance which is explained (measured by total inertia) in the original correspondence table. Thus if the total inertia is .172, meaning all the factors explain 17.2% of the variance in the original correspondence table, then factor 1 explains 54.9% of this 17.2%. It does not explain 54.9% of the variance in the original table. as is sometimes mis-reported.

  • Singular value. A singular value is the square root of an eigenvalue. It is interpreted as the maximum canonical correlation between the categories of the variables in analysis for any given dimension. In SPSS output it is "Singular Value."

• Row and column profiles: These are the relative frequencies of the row or column discrete variable. Profile elements are the entries in each row or column profile. Note that the row variable is normally the variable to be explained, and the column variable(s) is/are the explanatory variable(s).

• Category masses: These are the marginal proportions of a discrete variable, used to weight the point profiles when computing point distance. This weighting has the effect of compensating for unequal numbers of cases in the columns, as discussed below.

• Centroid: The weighted mean of the row and column profiles, it is the origin in a correspondence map.

• Score in dimension. These are the scores used as coordinates for points when plotting the correspondence map. Each point has a score on each dimension.
  • Confidence row points and confidence column points. These show the standard deviations of the row or column scores and are used to assess their precision.

Assumptions

• Significance testing is not supported, so model comparison and selection of a best-fit model should be done using another technique compatible with discrete variables, such as log-linear modeling or logistic regression. That is, correspondence analysis assumes that the researcher has already specified the appropriate variables and value categories using some other technique prior to correspondence analysis. Correspondence analysis is an exploratory, not a confirmatory technique.

• Correspondence analysis assumes that its measure of chi-square distance between points (category values) can be treated the same as correlation among variables. To justify this assumption, it is recommended that the two variables be shown to be associated using the conventional chi-square significance test for tabular data. Correspondence is not association, and the correspondence map may show points to be close even though the model (reflected in total inertia) explains only a small percentage of the inertia (variance) in the correspondence table.

• Homogeneity of column variables across categories of row variables is assumed, otherwise the measure of distance between points of the row variable is misleading.

• As in other forms of factor analysis, the meaning of correspondence dimensions is induced from loadings (from contributions of points to dimensions) and the ensuing labeling of dimensions is subject to human discretion, judgment, and error.

• Although in principle able to handle n-way tables, like other forms of factor analysis, the dimensions in correspondence analysis fall off sharply in interpretability. Typically, correspondence analysis handles only two or three variables well.

• Correspondence analysis is a nonparametric technique which makes no distributional assumptions, unlike factor analysis.

• Correspondence analysis is usually used with discrete variables which have many categories (that is, with large tables). With only two or three categories, the dimensions computed in correspondence analysis usually are not more informative than the original small table itself. For variables with few categories, log-linear analysis may be preferable to correspondence analysis.

• While correspondence analysis may be used with any level of data, if continuous data are used, they must be categorized into ranges. This involves a loss of information which may attenuate the level of computed association of variables. Differences in ranging continuous data also may have a significant effect on later interpretation of results. For this reason, some researchers prefer other techniques when key variables are continuous.

• Case values cannot be negative.

SPSS Correspondence Analysis Output

Click here to view commented SPSS 10 output (Analysis, Data Reduction, Correspondence) for the variables region of the country and political outlook, from the 1993 General Social Survey.

Frequently Asked Questions

• What software supports correspondence analysis?

  Correspondence analysis is now supported by several programs, some of which are:

  • SPAD, a French program preferred by many researchers who use correspondence analysis
  • SPSS (CORRESPONDENCE in the CATEGORIES module; when installed, look under Analysis, Data Reduction, Correspondence in the menus)
  • SAS (PROC CORRESP)
  • BMDP
  • SIMSTAT
  • EDA (Exploratory Data Analysis), whose ANACOR module supports correspondence analysis
  • ViSta from VisualStats, a free program.
  • XL-Stat, an Excel spreadsheet add-in to do correspondence analysis

• What are some good correspondence analysis sites?
  Correspondence analysis, including detrended correspondence analysis (DCA), is widely used in the study of ecology. See the site at http://www.okstate.edu/artsci/botany/ordinate/.

• How is the distance between points computed in correspondence analysis?
  Correspondence analysis uses chi-square distances, d. These are measures of distance betwee the row and column profiles for a set of points. A large d means the two profiles are very different.
  1. One starts with a crosstabulation of two discrete variables, such as party id (Republican, Democratic, Libertarian, Other, None) and primary news source (newspaper, television, radio, magazine, other, none).
  2. One computes row profiles (cell entries as a percent of the row marginal), row masses (row marginals as a percent of n, the sample size), and average row profiles (column marginals as a percent of n).
  3. One computes column profiles (cell entries as a percent of their column marginal), column masses (column marginals as a percent of n), and average column profiles (row marginals as a percent of n). Note that row masses will equal average column profiles, and column masses will equal average row masses by definition.
  4. Compute the chi-square distances between points. Chi-square distance is the Euclidean distance weighted inversely according to the average profile element. Let d(ii') be the chi-square distance from point i to point i' on the row variable. Let a(ij) be the cell elements of the row profile. Let a(.j) be the elements in the row of average row profiles. Then d(ii') = SQRT( SUM (((a(ij) - a(i'j))²)/a(.j))).

     Note that since the average row profile element is used inversely (1/a(.j)), this makes categories with few observations (as reflected in lower average row profiles) contribute more to interpoint distances (because the divisor is smaller). For instance, if party id is columns and media type is rows, and if Libertarian is a small group, their small row profile elements are compensated by dividing by their small average row profile. The effect is to equalize the importance of the column categories, with Libertarians being as important as Democrats when comparing distances among media types.

  5. The computed matrix of interpoint distances is treated like a correlation matrix for purposes of input to principal components analysis (PCA). As in conventional PCA, it is necessary to rotate the axes to achieve good interpretability of the dimensions.
  6. The dimensions emerging from PCA are used as axes in plotting correspondence maps.

• What are supplementary points?
  Active points are the category values of the variables used to compute the dimensions used to plot the correspondence map. However, once the dimensions have been computed, it is possible to compute the squared correlation of additional variables not used in the original analysis. These can be plotted on the correspondence map also. One use of supplementary points is to validate the correspondence model: active points should fall on the map as one would expect similar supplementary points to do. Another use of supplementary points is to handle outliers. If one point has an extremely divergent profile, it may unduly affect the computation of dimensions. In this case the researcher may wish to demote that point from being an active one to being a supplementary one. As of Version 10, SPSS started supporting supplementary points (look in Analyze, Data Reduction, Correspondence Analysis, Define Range, Category Constraints, Category is supplemental.

• What is detrended correspondence analysis (DCA)?
  Correspondence analysis can suffer from two problems - the arch effect and compression. The arch effect occurs when one variable has a unimodal distribution with respect to a second (ex., fish population is highest at a given pH level but decreases above or below that level. This will cause the distribution of points in the correspondence map to form an arch shape. Compression occurs when points at the ends of the distribution appear on the map very close together, such that their spacing along the primary map axis is not well related to the amount of change along that gradient. Detrended correspondence analysis (DCA) was invented to correct these problems.

  Detrending removes the arch effect. This is done by dividing the map into a series of vertical partitions, thus dividing the map along the primary (horizontal) axis. Within each partition, that cluster of points is relocated to center on the second (vertical) axis's 0 point. This arbitrary adjustment of the data has been the subject of methodological criticism.

  Rescaling is a second step in DCA. Where detrending realigned the points with respect to the secondary (vertical) axis, rescaling realigns the points along the primary (horizontal) axis as well as the vertical axis. Both axes are rescaled such that units represent standard deviations, seeking to make distance in ordination space mean the same thing along the axes of the map. Note that rescaling requires numeric (not nominal) measurement of points associated with the primary axis.

  The effects of detrending and rescaling may remove the arch effect, remove compression at the ends of axes, and distances separating points are more easily interpreted. Detrending is common in ecological uses of correspondence analysis.

• What is the largest number of dimensions which may be computed in correspondence analysis?
  The full PCA solution (PCA) will yield (I - 1) or (J - 1) dimensions, whichever is smaller, where I and J are the number of categories of the two variables in the table. However, only the first few dimensions will be interpretable and some cut-off criteria is used to look at a smaller number of dimensions. The scree "elbow" test is one such common criterion, as is the Kaiser criterion of using only dimensions with eigenvalues of 1.0 or higher.

• How does correspondence analysis of three variables work (MCA, multiple correspondence analysis)?
  In three-way correspondence analysis, a common approach is to combine the two variables of least interest. For instance, in an analysis of gender, age range, and media preference, the variable of most interest (media preference) would be the rows. The columns would be age ranges for men and age ranges for women. The computation would be the same as for two-way correspondence analysis, but in plotting the correspondence map, different symbols would be used for the points representing men and those representing women.

  Multiple correspondence analysis (MCA) is the generalized extension of correspondence analysis to handle more than two variables. The input to MCA is a design matrix in which cases are rows and categories of variables are columns. Cell values in this matrix are 1's or 0's, depending on whether the case does or does not belong to the category. Interpretation of correspondence maps is similar to that for simple correspondence analysis.

Bibliography

• Benzecri, J. P. (1992). Correspondence analysis handbook. Paris: Dunod.
• Bourdieu, Pierre (1984). Distinction: A social critique of the judgment of taste. Cambridge, MA: Harvard University Press. A seminal sociological example of correspondence analysis.
• Clausen, Sten-Erik (1998). Applied correspondence analysis. Quantitative Applications in the Social Sciences Series No. 121. Thousand Oaks, CA: Sage Publications.
• Greenacre, M. J. (1984). Theory and applications of correspondence analysis. NY: Academic Press.
• Greenacre, M. J. (1993). Correspondence analysis in practice, London: Academic Press.
• Weller, S. C. and A. K. Romney (1990). Metric scaling: Correspondence analysis. Quantitative Applications in the Social Sciences Series No. 75. Thousand Oaks, CA: Sage Publications.

Copyright 1998, 2008 by G. David Garson.
Last update, 3.34/2008.