Dimension Reduction in Workers Compensation

1. Dimension Reduction in Workers Compensation CAS predictive Modeling Seminar Louise Francis, FCAS, MAAA Francis Analytics and Actuarial Data Mining, Inc. Louise_francis@msn.com www.data-mines.com

2. Objectives • Answer questions: What is dimension reduction and why use it? • Introduce key methods of dimension reduction • Illustrate with examples in Workers Compensation • There will be some formulas, but emphasis is on insight into basic mechanisms of the procedures

3. Introduction • “How do mere observations become data for analysis?” • “Specific variable values are never immutable characteristics of the data” • Jacoby, Data Theory and Dimension Analysis, Sage Publications • Many of the dimension reduction/measurement techniques originated in the social sciences and dealt with how to create scales from responses on attitudinal and opinion surveys

4. Unsupervised learning • Dimension reduction methods generally unsupervised learning • Supervised Learning • A dependent or target variable • Unsupervised learning • No target variable • Group like variables or like records together

5. The Data • BLS Economic indexes • Components of inflation • Employment data • Health insurance inflation • Texas Department of Insurance closed claim data for 2002 and 2003 • Employment related injury • Excludes small claims • About 1800 records

6. What is a dimension? • Jacoby – The number of separate and interesting sources of variation • In many studies each variable is a dimension • However, we can also view each record in a database as a dimension

7. Dimensions

8. The Two Major Categories of Dimension Reduction • Variable reduction • Factor Analysis • Principal Components Analysis • Record reduction • Clustering • Other methods tend to be developments on these

9. Principal Components Analysis • A form of dimension (variable) reduction • Suppose we want to combine all the information related to the “inflation” dimension of insurance costs • Medical care costs • Employment (wage) costs • Other • Energy • Transportation • Services

10. Principal Components • These variables are correlated but not perfectly correlated • We replace many variables with a weighted sum of the variables • These are then used as independent variables in a predictive model

11. Factor Analysis: A Latent Factor

12. Factor/Principal Components Analysis • Linear methods – use linear correlation matrix • Correlation matrix decomposed to find smaller number of factors the are related to the same underlying drivers • Highly correlated variables tend to have high load on the same factor

13. Factor/Principal Components Analysis

14. Factor/Principal Components Analysis • Uses eignevectors and eigenvalues • R is correlation matrix, V eigenvectors, lambda eigenvalues

15. Inflation Data

16. Factor Rotation • Find simpler more easily interpretable factors • Use notion of factor complexity

17. Factor Rotation • Quartimax Rotation • Maximize q • Varimax Rotation • Maximizes the variance of squared loadings for each factor rather than for each variable

18. Varimax Rotation

19. Plot of Loadings on Factors

20. How Many Factors to Keep? • Eigenvalues provide information on how much variance is explained • Proportion explained by a given component=corresponding eigenvalue/n • Use Scree Plot • Rule of thumb: keep all factors with eigenvalues>1

22. WC Severity vs Factor 1

23. WC Severity vs Factor 2

24. What About Categorical Data? • Factor analysis is performed on numeric data • You could code data as binary dummy variables • Categorical Variables from Texas data • Injury • Cause of loss • Business Class • Health Insurance (Y/N)

25. Optimal Scaling • A method of dealing with categorical variables • Can be used to model nonlinear relationships • Uses regression to • Assign numbers to categories • Fit regression coefficients • Y*=f(X*) • In each round of fitting, a new Y* and X* is created

26. Variable Correlations

27. Visualizations of Scaled Variables

28. Can we use scaled variables in prediction?

29. Tree Using Optimal Scaling Scores

30. Tree for Subrogation

31. Row Reduction: Cluster Analysis • Records are grouped in categories that have similar values on the variables • Examples • Marketing: People with similar values on demographic variables (i.e., age, gender, income) may be grouped together for marketing • Text analysis: Use words that tend to occur together to classify documents • Fraud modeling • Territory definition • Note: no dependent variable used in analysis

32. Clustering • Common Method: k-means, hierarchical • No dependent variable – records are grouped into classes with similar values on the variable • Start with a measure of similarity or dissimilarity • Maximize dissimilarity between members of different clusters

33. Dissimilarity (Distance) Measure – Continuous Variables • Euclidian Distance • Manhattan Distance

34. Binary Variables

35. Binary Variables • Sample Matching • Rogers and Tanimoto

36. Example: Texas Data • Data from 2002 and 2003 closed claim database by Texas Ins Dept • Only claims over a threshold included • Variables used for clustering: • Report Lag • Settlement Lag • County (ranked by how often in data) • Injury • Cause of Loss • Business class

37. Results Using Only Numeric Variables Used Euclidian distance measure

38. Two Stage Clustering With Categorical Variables • First compute dissimilarity measures • Then get clusters • Find optimum number of clusters

40. Loadings of Injuries on Cluster

41. Age and Cluster

42. County vs Cluster

43. Means of Financial Variables by Cluster

44. Tying Things Together: Multidimensional Scaling • A mathematical way to connect clustering and factor analysis • Data can be decomposed into key row dimensions times a diagonal weight matrix times key column dimensions

45. Modern dimension reduction • Hidden layer in neural networks like a nonlinear principle components • Projection Pursuit Regression – a nonlinear PCA • Kahonen self-organizing maps – a kind of neural network that does clustering • These can be understood as enhancements factor analysis or clustering

47. Kahonen SOM for Fraud

48. Recommended References • Hacher, 1994, A Step-by-Step Approach for Using the SAS System for Factor Ananlysis and Structural Equation Modeling, SAS Publications • Jacoby, 1991, Data Theory and Dimension Analysis, Sage Publications • Kaufman and Rousseeuw,1990, Finding Groups in Data, Wiley • Kim and Mueller, 1978, Factor Analysis: Statistical Methods and Practical Issues, Sage Publications

49. Questions?