Quiz 5: Correspondence Analysis 📊

Test your knowledge of the material on correspondence analysis in the following quiz to see how much you learned. This is entirely private for you---no records are kept of your performance.

Questions

1. Correspondence analysis (CA) is best described as:

The analog of principal component analysis (PCA) for frequency data A chi-square test for independence in contingency tables A regression method for categorical predictors A clustering algorithm for categorical data

Explanation

CA is the analog of PCA for frequency data. Like PCA, it aims to account for the maximum amount of variation (χ² in this case) in a few dimensions and provides optimal scaling of row and column categories.

2. Correspondence analysis uses which mathematical technique to find optimal row and column scores?

Maximum likelihood estimation Singular value decomposition (SVD) of residuals from independence Iterative proportional fitting Principal component analysis

Explanation

CA uses singular value decomposition (SVD) of the matrix of residuals from independence. This decomposes the residuals as D = XΛY^T, where X and Y are the row and column scores, and Λ contains the singular values.

3. The singular values (λ_i) in correspondence analysis represent:

The variances of the principal components The correlations between row and column category scores on each dimension The chi-square statistics for each dimension The eigenvalues of the contingency table

Explanation

The singular values λ_i are the (canonical) correlations between the row and column category scores on each dimension. For dimension 1, the scores are chosen to have the maximum possible correlation λ₁, and so on for subsequent dimensions.

4. For a two-way contingency table with r rows and c columns, the CA solution has at most how many dimensions?

r + c r × c min(r - 1, c - 1) max(r, c)

Explanation

The CA solution has at most min(r - 1, c - 1) dimensions. This is the rank of the matrix of residuals from independence, similar to how PCA has at most min(n - 1, p) dimensions for n observations and p variables.

5. How does the Pearson chi-square statistic (χ²) relate to the inertia in CA?

χ² = n × Σλ²ᵢ, where n is the sample size and λᵢ are the singular values χ² = Σλᵢ, the sum of singular values χ² = n × λ₁, where λ₁ is the first singular value There is no relationship between χ² and inertia

Explanation

The Pearson χ² statistic equals n times the sum of squared singular values: χ² = n × Σλ²_i. The total inertia is Σλ²_i = χ²/n, so the percentage of inertia explained by each dimension reflects the percentage of χ² accounted for.

6. What does the term "inertia" mean in correspondence analysis?

The total sample size in the contingency table The weighted variation or dispersion of row/column profiles from the centroid The number of dimensions needed for the analysis The correlation between rows and columns

Explanation

Inertia refers to the weighted variation or dispersion of the profile points from their centroid (weighted average). The physical analogy is to mass × distance², hence the term 'inertia.' Higher inertia indicates greater association between rows and columns.

7. In the default "symmetric" CA map, what does it mean when a row point and column point are plotted near each other?

They have a positive association (residual from independence > 0) They are independent They have a negative association They have equal marginal frequencies

Explanation

In a symmetric CA map, when a row point and column point are near each other, it indicates a positive association between them—the residual from independence (d_ij) is positive. Proximity reflects similarity or attraction in the data.

8. Which of the following is TRUE about CA solutions?

CA solutions are nested, similar to PCA (the first two dimensions of a 3D solution are identical to the 2D solution) The centroid of row and column profiles is located at a corner of the plot Distances in a CA plot use Euclidean distance, not chi-square distance Different dimensions from a CA solution can be freely rotated like in factor analysis

Explanation

CA solutions are nested, just like PCA. The first two dimensions of a 3D solution are identical to the 2D solution. Additionally, the centroid (weighted average) of row and column profiles is at the origin, and distances reflect chi-square distances.

9. Multiple correspondence analysis (MCA) differs from simple CA in that MCA:

Analyzes all pairwise bivariate associations among multiple categorical variables Can only be used for binary variables Uses a different mathematical technique than SVD Cannot display all variables in a single plot

Explanation

MCA extends CA to n-way tables and analyzes all pairwise bivariate associations among the categorical variables. It can plot all factors in a single display and provides an optimal scaling of category scores across all variables simultaneously.

10. For MCA with Q categorical variables, the Burt matrix is:

A matrix of dummy variables with 0s and 1s The product of the indicator matrix Z and its transpose, B = Z^T Z The original contingency table A diagonal matrix of singular values

Explanation

The Burt matrix is B = Z^TZ, the product of the indicator matrix Z and its transpose. The diagonal blocks contain marginal frequencies for each variable, and the off-diagonal blocks contain the bivariate contingency tables for each pair of variables.