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- 1. Eﬀect of Number of Categories and Category Boundaries on Recovery of Latent Linear Correlations from Optimally Weighted Categorical Data Johnny Lin Advisor: Peter Bentler November 19, 2008
- 2. Outline Introduction LINEALS Forming a Hypothesis Method Description Simulation Analysis Results Main Eﬀects Interactions
- 3. Outline Introduction LINEALS Forming a Hypothesis Method Description Simulation Analysis Results Main Eﬀects Interactions
- 4. Introducing LINEALS A Method of Optimal Scaling Algorithm An iterative process that minimizes m m 2 2 2 l=1 (ηjl − rjl ) where ηjl j=1 is a measure of nonlinearity. Developed by Jan de Leeuw and implemented by Patrick Mair. Assumption That bi-linearization is possible. No assumption of normality.
- 5. Plot of LINEALS Transformation Criterion: Linearize both X on Y and Y on X simultaneously. Figure: Red: X on Y , Blue: Y on X
- 6. Outline Introduction LINEALS Forming a Hypothesis Method Description Simulation Analysis Results Main Eﬀects Interactions
- 7. Questions to ask First, deﬁne good recovery as small deviation from true score. 1. Does LINEALS recover true population correlations better than Pearson for categorical data? 2. Is the performance of LINEALS robust? 3. What factors inﬂuence good recovery?
- 8. Outline Introduction LINEALS Forming a Hypothesis Method Description Simulation Analysis Results Main Eﬀects Interactions
- 9. Conditions tested Correlation Type, True Population Correlation, Number of Categories, and Homogeneity Condition Parameters {0=LINEALS, 1=Pearson} 1. Correlation Type (r) {0.3,0.5,0.7,0.9} 2. True Population Correlation (P) {2,3,5,7,10} 3. Number of Categories (V) {0=Non-Homogeneous, 1=Homogeneous} 4. Homogeneity (h) Total of 80 combinations (2x4x5x2).
- 10. Outline Introduction LINEALS Forming a Hypothesis Method Description Simulation Analysis Results Main Eﬀects Interactions
- 11. Creating functions in R For each combination (total of 80): 1. Generate 1000 sets of bivariate normal data. 2. Make “cuts” (homogeneous vs. non-homogeneous). 3. Run through LINEALS / Pearson. 4. Calculate deviation of result and true population correlation. 5. Repeat Steps 1 - 4 twenty-ﬁve times. Result: Total of 2000 deviations (80x25).
- 12. Outline Introduction LINEALS Forming a Hypothesis Method Description Simulation Analysis Results Main Eﬀects Interactions
- 13. Hierarchical Regression Description DV: deviation of sample correlation from true population correlation |ρ12 | − |ˆ12 | ρ IVs: main eﬀect and interactions of four conditions (total of 15) Four main eﬀects (h,r,P,V) Six 2-way interactions (hr, hP, hV, . . . ) Four 3-way interactions (hrP, hrV, . . . ) One 4-way interaction (hrPV)
- 14. Hierarchical Regression Model Selection Tested full model against nested models. Conﬁrmed with Best Subset Regression. Optimal Adj. R 2 and Mallow’s CP found with 7-8 parameters. (a) Adj. R 2 (b) Mallow’s CP
- 15. Final Model SPSS Output Coefficients(a) Unstandardized Standardized Model Coefficients Coefficients t Sig. B Std. Error Beta 1 (Constant) .189 .006 31.240 .000 h -.113 .012 -.620 -9.299 .000 r .007 .002 .041 3.054 .002 V -.024 .001 -.773 -40.558 .000 P .098 .008 .241 12.655 .000 hV .013 .002 .487 7.164 .000 hP .117 .018 .435 6.392 .000 hPV -.017 .003 -.422 -6.326 .000 a Dependent Variable: difference Diﬀerence between LINEALS and Pearson deviations is .007 controlling for other factors.
- 16. Outline Introduction LINEALS Forming a Hypothesis Method Description Simulation Analysis Results Main Eﬀects Interactions
- 17. Plot of Main Eﬀects I Figure: Main Eﬀect of Number of Figure: Main Eﬀect of Population Categories V Correlation P
- 18. Plot of Main Eﬀects II Figure: Main Eﬀect of Homogeneity h Figure: Main Eﬀect of Correlation Type r
- 19. Outline Introduction LINEALS Forming a Hypothesis Method Description Simulation Analysis Results Main Eﬀects Interactions
- 20. Plot of Signiﬁcant Interactions Note: The signiﬁcant 3-way interaction hPV is not plotted. Figure: Population Correlation by Levels Figure: Number of Categories by Levels of Homogeneity hP of Homogeneity hV
- 21. Interaction of Correlation Type and Number of Categories When rV added into regression model, the main eﬀect of Correlation Type r goes away. Suggests that number of categories may contribute to the LINEALS vs. Pearson diﬀerence. Figure: Number of Categories by Correlation Type (rV, marginally sig.)
- 22. Summary 1. LINEALS performs slightly better than Pearson under bivariate normal categorizations. 2. The non-signiﬁcant interactions with Correlation Type suggest that LINEALS is robust. 3. Recovery of true population correlations is highly inﬂuenced by homogeneity (i.e., the underlying equality of interval widths). Future Studies How does it compare against polychoric correlations? Is the resulting matrix positive deﬁnite?

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