1. **Stating the problem:** We want to derive the underlying relation between Correspondence Analysis (CA) and Chi-Square ($\chi^2$) analysis of association. 2. **Background:** - Correspondence Analysis is a multivariate graphical technique used to analyze contingency tables. - Chi-Square test measures the association between categorical variables by comparing observed and expected frequencies. 3. **Chi-Square statistic formula:** $$\chi^2 = \sum_{i,j} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$$ where $O_{ij}$ is the observed frequency and $E_{ij}$ is the expected frequency under independence. 4. **Expected frequencies under independence:** $$E_{ij} = \frac{(\text{row total}_i)(\text{column total}_j)}{\text{grand total}}$$ 5. **Correspondence Analysis setup:** - Define the matrix of relative frequencies: $$P = \frac{O}{n}$$ where $n$ is the grand total. - Define row and column marginal proportions: $$r_i = \sum_j P_{ij}, \quad c_j = \sum_i P_{ij}$$ 6. **Standardized residuals matrix in CA:** $$S_{ij} = \frac{P_{ij} - r_i c_j}{\sqrt{r_i c_j}}$$ This matrix measures deviations from independence normalized by expected values. 7. **Relation to Chi-Square:** The Chi-Square statistic can be expressed as: $$\chi^2 = n \sum_{i,j} \frac{(P_{ij} - r_i c_j)^2}{r_i c_j} = n \sum_{i,j} S_{ij}^2$$ 8. **Interpretation:** - CA performs a singular value decomposition (SVD) on the matrix $S$ to find principal axes that explain the association structure. - The total inertia in CA equals $\frac{\chi^2}{n}$, linking the two methods. **Final answer:** Correspondence Analysis decomposes the Chi-Square statistic by analyzing the matrix of standardized residuals $S$, where the total inertia in CA equals $\frac{\chi^2}{n}$, thus providing a geometric interpretation of the Chi-Square test of association.