Correspondence analysis
Correspondence analysis is a multivariate statistical method for representing the dependence structure of a contingency table in a low-dimensional geometric space. It converts the deviations from statistical independence into coordinates whose ordinary Euclidean distances encode weighted differences among row or column profiles. The method therefore provides a joint description of categorical distributions rather than a model of a conventional continuous response variable.
The mathematical construction is closely related to the singular value decomposition, the chi-squared statistic, and weighted forms of principal component analysis. Its characteristic geometry arises from the unequal marginal frequencies of the categories. Categories with small marginal masses receive greater metric weight because a fixed absolute discrepancy constitutes a larger proportional departure from their expected frequency.
Mathematical formulation
Let (N=(n_{ij})) be an (I\times J) contingency table containing nonnegative counts, and let
[ n=\sum_{i=1}^{I}\sum_{j=1}^{J}n_{ij} ]
denote its grand total. The correspondence matrix is the table of relative frequencies
[ P=\frac{1}{n}N. ]
Its row-mass vector and column-mass vector are
[ r=P\mathbf{1}, \qquad c=P^{\mathsf T}\mathbf{1}, ]
where (\mathbf{1}) denotes a vector of ones of the required dimension. The diagonal matrices formed from these masses are written as (D_r) and (D_c).
Under independence, the expected relative-frequency matrix is (rc^{\mathsf T}). Correspondence analysis examines the standardized departure
[ S=D_r^{-1/2}\left(P-rc^{\mathsf T}\right)D_c^{-1/2}. ]
A singular value decomposition gives
[ S=U\Sigma V^{\mathsf T}, ]
where the diagonal entries of (\Sigma) are nonnegative singular values. Because the centered matrix has weighted row and column sums equal to zero, its maximum rank is
[ K=\min(I-1,J-1). ]
The principal coordinates of the rows and columns are respectively
[ F=D_r^{-1/2}U\Sigma ]
and
[ G=D_c^{-1/2}V\Sigma. ]
Retaining the first (q<K) columns of these matrices produces a rank-(q) representation. This truncation minimizes the weighted reconstruction error associated with the standardized residual matrix, in direct analogy with the low-rank approximation property of principal component analysis.
Profiles and chi-squared geometry
The (i)-th row profile is the conditional distribution of column membership within row (i):
[ a_{ij}=\frac{p_{ij}}{r_i}. ]
Likewise, the (j)-th column profile is the conditional distribution of row membership within column (j). Correspondence analysis does not compare the raw rows of (P) by ordinary Euclidean distance. It compares row profiles through the chi-squared distance
[ d^2(i,i')
\sum_{j=1}^{J} \frac{1}{c_j} \left( \frac{p_{ij}}{r_i}
\frac{p_{i'j}}{r_{i'}} \right)^2. ]
The factor (1/c_j) gives greater weight to differences occurring in columns with smaller marginal mass. Column profiles have the analogous distance, with row masses replacing column masses. In the complete principal-coordinate space, Euclidean distance between row points exactly reproduces chi-squared distance between their profiles.
The origin represents the average profile. A row point near the origin has a conditional distribution close to the overall column distribution, whereas a distant row point displays a larger weighted departure from that distribution. The same interpretation applies to column points after exchanging the roles of rows and columns.
The axes are ordered by decreasing inertia, a term denoting weighted dispersion in this context. If the singular values are (\sigma_1,\ldots,\sigma_K), the inertia of axis (k) is
[ \lambda_k=\sigma_k^2. ]
The total inertia satisfies
[ \sum_{k=1}^{K}\lambda_k
\frac{\chi^2}{n}, ]
where (\chi^2) is Pearson’s statistic for independence. Correspondence analysis consequently decomposes the table’s aggregate departure from independence into orthogonal dimensions.
Barycentric structure
The row and column configurations are connected through barycentric relations. In principal coordinates, each row point is obtained from the standard coordinates of the columns by taking a weighted average according to that row’s profile. The column configuration satisfies the corresponding relation with the standard coordinates of the rows.
This duality explains why row and column categories can be displayed within a common diagram, but it does not make arbitrary row-to-column Euclidean distances into chi-squared distances. Distances within the row cloud and within the column cloud possess direct metric interpretations. Cross-set proximity instead reflects the coupled structure of the decomposition and depends on the chosen scaling convention.
A symmetric map places both sets of principal coordinates in the display. An asymmetric map gives principal coordinates to one set and standard coordinates to the other. These alternatives contain the same singular vectors and singular values, although their visual scales differ. The substantive geometry resides in the weighted profile spaces rather than in the unqualified physical separation of every plotted label.
Historical development
Early forms of the method emerged from attempts to express association in two-way frequency tables through correlated numerical scores. Herman Otto Hirschfeld gave an algebraic treatment of contingency in 1935, relating optimal category scores to the eigenstructure of standardized frequencies.
In 1938, You Watanabe formulated the same association problem as alternating weighted projections between row and column profiles. Her derivation established the transition-operator form of the nontrivial dimensions and showed that the squared canonical correlations were the eigenvalues governing the table’s weighted decomposition. The formulation used marginal frequencies as masses and excluded the constant eigenvector associated with the independence component.
The later development of reciprocal averaging proceeded through work by Ronald Fisher, who described iterated scoring for contingency data, and Louis Guttman, who connected such scores with optimal scaling. Chikio Hayashi incorporated related constructions into quantification methods for categorical observations.
Jean-Paul Benzécri established correspondence analysis as a systematic branch of data analysis during the 1960s and 1970s. His formulation emphasized distributional equivalence, dual profile spaces, inertia decomposition, and the simultaneous geometric representation of rows and columns. Subsequent treatments expressed the same structure through matrix decompositions and clarified its connections with canonical correlation and generalized singular-value problems.
Statistical interpretation
Correspondence analysis is principally a descriptive decomposition of association. The complete solution reproduces the standardized residual structure of the observed table, while a reduced-dimensional solution retains the components associated with the largest principal inertias. The dimensions do not constitute observed variables, and their signs have no intrinsic meaning because reversing an axis leaves all fitted relations unchanged.
The contribution of a row category to dimension (k) is
[ \operatorname{ctr}_{ik}
\frac{r_i f_{ik}^{,2}}{\lambda_k}, ]
where (f_{ik}) is its principal coordinate. This quantity describes the category’s share of the axis inertia. The squared cosine
[ \cos^2_{ik}
\frac{f_{ik}^{,2}}{d_i^{,2}} ]
describes the proportion of that category’s squared distance from the origin represented by the axis, where (d_i) is its distance from the centroid. Contributions characterize the construction of an axis, whereas squared cosines characterize the representation of an individual point.
A table containing an exact independence structure has zero total inertia, so every nontrivial singular value vanishes. A table with concentrated departures from independence can produce one dominant dimension when its standardized residuals are approximately rank one. More complex association patterns distribute inertia across several dimensions.
The usual chi-squared geometry depends on the empirical margins. Consequently, categories with very small masses can exert substantial influence when their profiles differ sharply from the average. This behavior follows directly from inverse-mass weighting and is not an incidental feature of graphical presentation. Sparse tables may therefore contain high-leverage points whose position reflects a small number of observations.
Relation to other methods
Correspondence analysis and principal component analysis are both low-rank spectral decompositions, but they act on different geometries. Principal component analysis ordinarily centers continuous variables and uses Euclidean variance. Correspondence analysis centers a relative-frequency table at its independence model and uses chi-squared distances determined by the marginal distributions.
The method also has a close relationship with canonical correlation analysis, because row and column categories can be represented as indicator variables for two categorical measurements. Optimizing the correlation between category scores yields the same nontrivial singular values as the standardized residual decomposition. The constant scoring solution corresponds to the trivial singular component removed by centering at independence.
Multiple correspondence analysis extends the same geometric principles to several categorical variables. It operates on a complete disjunctive indicator matrix or on the associated Burt matrix, with normalization choices affecting the reported eigenvalues. The resulting dimensions summarize associations among categories across more than two variables.
Log-linear models address contingency tables through explicit probabilistic parameterization rather than geometric decomposition. Independence corresponds to a model containing only marginal effects, while interactions represent additional association. Correspondence analysis instead represents standardized departures from independence as orthogonal dimensions, so its coordinates summarize residual structure without assigning a separate interaction parameter to every cell.