1. Optimal scaling for a logistic regression model with ordinal covariates
Sanne JW Willems, Marta Fiocco, and Jacqueline J Meulman Leiden University & Stanford University

2. Optimal scaling for generalized linear models with nonlinear covariates
Sanne JW Willems, Marta Fiocco, and Jacqueline J Meulman Leiden University & Stanford University

3. Goal
Reducing linearity in Generalized Linear Models using Optimal Scaling Transformations

4. Generalized Linear Models
Linear predictor:
Link function - (nonlinear) relation between the linear predictor and the outcome:

5. Generalized Linear Models
Nonlinear predictor:
Link function - (nonlinear) relation between the linear predictor and the outcome:

6. Why?

7. Data types

8. Data types – Nominal Categorical
Grouping

9. Data types – Nominal Categorical
Grouping
Dummy Coding

10. Data types – Ordinal Categorical
Grouping
Ordering

11. Data types – Ordinal Categorical
Grouping
Ordering
Dummy Coding

12. Data types – Ordinal Categorical
Grouping
Ordering
Continuous variable via integer Coding

13. Data types – Numeric
Grouping
Ordering
Equal relative spacing

14. Data types – Numeric Grouping Ordering Equal relative spacing
Continuous variable
Grouping
Ordering
Equal relative spacing

15. What if the linear predictor should be nonlinear?

16. What if the linear predictor should be nonlinear?
Keep ordinal property, but do not introduce equal relative spacing

17. What if the linear predictor should be nonlinear?
Keep ordinal property, but do not introduce equal relative spacing
Remove property of equal relative spacing

18. Solution: Optimal Scaling transformations
Transform variables:

19. Solution: Optimal Scaling transformations
Transform variables:
Scaling levels:
Nominal spline
Numeric
Nominal
Ordinal
Ordinal spline

20. How?

21. Optimal Scaling Generalized Linear Models
Nonlinear predictor:
Link function - (nonlinear) relation between the linear predictor and the outcome:

22. Algorithm

23. Algorithm

24. Optimal Scaling step
Apply restrictions according to the chosen scaling level

25. Algorithm

26. Example: logistic regression

27. Example: logistic regression
Inpatient treatment or ?
Day clinic treatment

28. Result nominal scaling level

29. Result ordinal scaling level

30. Predictions for training data nominal vs ordinal
Nominal: Ordinal:
Sensitivity = 0.924
Specificity = 0.829
Efficiency (correct classification rate) = 0.880
Sensitivity = 0.918
Specificity = 0.823
Efficiency (correct classification rate) = 0.874

31. Predictions for training data ordinal vs numeric
Ordinal: Numeric:
Sensitivity = 0.918
Specificity = 0.823
Efficiency (correct classification rate) = 0.874
Sensitivity = 0.864
Specificity = 0.810
Efficiency (correct classification rate) = 0.839

32. Summary Optimal Scaling GLMs
More flexibility by transforming variables
Can be helpful when linear predictor should be nonlinear is nonlinear