Optimal scaling for a logistic regression model with ordinal covariates Sanne JW Willems, Marta Fiocco, and Jacqueline J Meulman Leiden University & Stanford University s.j.w.willems@math.leidenuniv.nl www.SanneJWWillems.nl

Optimal scaling for generalized linear models with nonlinear covariates Sanne JW Willems, Marta Fiocco, and Jacqueline J Meulman Leiden University & Stanford University s.j.w.willems@math.leidenuniv.nl www.SanneJWWillems.nl

Goal Reducing linearity in Generalized Linear Models using Optimal Scaling Transformations

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

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

Why?

Data types

Data types – Nominal Categorical Grouping

Data types – Nominal Categorical Grouping Dummy Coding

Data types – Ordinal Categorical Grouping Ordering

Data types – Ordinal Categorical Grouping Ordering Dummy Coding

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

Data types – Numeric Grouping Ordering Equal relative spacing

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

What if the linear predictor should be nonlinear?

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

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

Solution: Optimal Scaling transformations Transform variables:

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

How?

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

Algorithm

Algorithm

Optimal Scaling step Apply restrictions according to the chosen scaling level

Algorithm

Example: logistic regression

Example: logistic regression Inpatient treatment or ? Day clinic treatment

Result nominal scaling level

Result ordinal scaling level

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

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

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