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- 1. Correspondence Analysis with XLStat Guy Lion Financial Modeling April 2005
- 2. Statistical Methods Classification
- 3. The Solar (PCA) System
- 4. Capabilities <ul><li>Correspondence Analysis (CA) handles a Categorical Independent variable with a Categorical Dependent variable. </li></ul><ul><li>CA analyzes the association between two categorical variables by representing categories as points in a 2 or 3 dimensional space graph. </li></ul>
- 5. 4 Steps <ul><li>Testing for Independence of both Variables (in XLStat only). </li></ul><ul><li>Compute Category Profiles (relative frequencies) and Masses (marginal proportions) for both Points-rows & Points-columns. </li></ul><ul><li>Calculate distance (Chi Square distance) between Points-rows, and then Points-columns. </li></ul><ul><li>Develop best-fitting space of n dimensions (relying on PCA). </li></ul>
- 6. An Example: Moviegoers You classify by Age buckets the opinions of 1357 movie viewers on a movie.
- 7. Testing Independence: Chi Square One cell (16-24/Good) accounts for 49.3% (73.1/148.3) of the Chi Square value for all 28 cells. Observed Expected Bad Average Good Very Good Total Bad Average Good Very Good Total 16-24 69 49 48 41 207 16-24 124.2 41.2 14.9 26.7 207 25-34 148 45 14 22 229 25-34 137.4 45.6 16.5 29.5 229 35-44 170 65 12 29 276 35-44 165.6 54.9 19.9 35.6 276 45-54 159 57 12 28 256 45-54 153.6 50.9 18.5 33.0 256 55-64 122 26 6 18 172 55-64 103.2 34.2 12.4 22.2 172 65-74 106 21 5 23 155 65-74 93.0 30.8 11.2 20.0 155 75+ 40 7 1 14 62 75+ 37.2 12.3 4.5 8.0 62 Total 814 270 98 175 1357 Total 814 270 98 175 1357 60% 20% 7% 13% 100% 60% 20% 7% 13% 100% Chi Square Calculations (Observed - Expected) 2 /Expected Bad Average Good Very Good Total (48 - 14.9) 2 /14.9 = 73.1 16-24 24.5 1.5 73.1 7.7 106.7 25-34 0.8 0.0 0.4 1.9 3.1 35-44 0.1 1.9 3.2 1.2 6.3 45-54 0.2 0.7 2.3 0.8 4.0 55-64 3.4 2.0 3.3 0.8 9.5 Chi Squ. 148.3 65-74 1.8 3.1 3.4 0.5 8.8 DF 18 = (7 -1)(4 - 1) 75+ 0.2 2.3 2.7 4.5 9.7 p value 1.613E-22 31.1 11.5 88.3 17.3 148.3
- 8. Row Mass & Profile
- 9. Eigenvalues of Dimensions Dimension F1 Eigenvalue 0.095 explains 86.6% (0.095/0.109) of the Inertia or Variance. F1 Coordinates are derived using PCA.
- 10. Singular Value Singular value = SQRT(Eigenvalue). It is the maximum Canonical Correlation between the categories of the variables in analysis for any given dimension.
- 11. Calculating Chi Square Distance for Points-rows Chi Square Distance defines the distance between a Point-row and the Centroid (Average) at the intersection of the F1 and F2 dimensions. The Point-row 16-24 is most distant from Centroid (0.72).
- 12. Calculating Inertia [or Variance] using Points-rows XLStat calculates this table. It shows what Row category generates the most Inertia (Row 16-24 accounts for 72% of it)
- 13. 2 other ways to calculate Inertia <ul><li>Inertia = Chi Square/Sample size. 148.27/1,357 = 0.109. </li></ul><ul><li>Inertia = Sum of Dimensions Eigenvalues. Shows how much each Dimension explains overall Inertia. </li></ul>
- 14. Contribution of Points-rows to Dimension F1 The contribution of points to dimensions is the proportion of Inertia of a Dimension explained by the Point. The contribution of Points-rows to dimensions help us interpret the dimensions. The sum of contributions for each dimension equals 100%.
- 15. Contribution of Dimension to Points-rows. Squared Correlation . <ul><li>Contribution of Dimensions to Points-rows tells how much of a Point Inertia is explained by a dimension. </li></ul><ul><li>Contributions are called Squared Correlations since they are the same as COS 2 for the angle between the line from the Centroid to the point and the principal axes. </li></ul>
- 16. Squared Correlation = COS 2 If Contribution is high, the angle between the point vector and the axis is small.
- 17. Quality Quality = Sum of the Squared Correlations for dimensions shown (normally F1 and F2). Quality is different for each Point-row (or Point-column). Quality represents whether the Point on a two dimensional graph is accurately represented. Quality is interpreted as proportion of Chi Square accounted for given the respective number of dimensions. A low quality means the current number of dimensions does not represent well the respective row (or column).
- 18. Plot of Points-Rows
- 19. Review of Calculation Flows
- 20. Column Profile & Mass
- 21. Calculating Chi Square Distance for Points-column Distance = SQRT(Sum(Column Profile – Avg. Column Profile 2 /Avg. Column Profile)
- 22. Contribution of Points-column to Dimension F1 Contribution = (Col.Mass)(Coordinate 2 )/Eigenvalue
- 23. Contribution of Dimension F1 to Points-columns
- 24. Plot of Points-Columns
- 25. Plot of all Points
- 26. Observing the Correspondences
- 27. Conclusion <ul><li>The Age Categories and Opinion Categories are dependent. Overall Chi Square P value 0.00%. </li></ul><ul><li>The most different Point-row is 16-24. 0.72 Chi Square distance from Centroid. Accounts for 72.0% of Inertia. </li></ul><ul><li>The most different Point-column is “Good.” 0.90 Chi Square distance from Centroid. Accounts for 59.6% of Inertia. </li></ul>
- 28. Conclusion (continued) We have to remember that we can’t directly compare the Distance across categories (Row vs Column). We see that the 16-24 Point-row makes a greater contribution to Inertia and overall Chi Square vs the Good Point-column. This is because the 16-24 Point-row has a greater mass (207 occurrences vs only 98 for Good).

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