Validation of predictive regression models Ewout W. Steyerberg, PhD Clinical epidemiologist Frank E. Harrell, PhD Biostatistician

Personal background  Ewout Steyerberg: Erasmus MC, Rotterdam, the Netherlands  Frank Harrell: Health Evaluation Sciences, Univ of Virginia, Charlottesville, VA, USA “Validation of predictions from regression models is of paramount importance”

Learning objectives: knowledge of  common types of regression models  fundamental assumptions of regression models  performance criteria of predictive models  principles of different types of validation

Performance objectives  To be able to explain why validation is necessary for predictive models  To be able to judge the adequacy of a validation procedure

Predictive models provide quantitative estimates of an outcome, e.g.  Quality of life one year after surgery  Death at 30 days after surgery  Long term survival

Predictive models are often based on regression analysis  y ~ a + sum(b i *x i ) y: outcome variable a: intercept b i : regression coefficient i x i : predictor variable i i in [1,many], usually 2 to 20

3 examples of regression  Quality of life one year after surgery: continuous outcome, linear regression  Death at 30 days after surgery: binary outcome, logistic regression  Long term survival: time-to-outcome, Cox regression

Predictive models make assumptions  Distribution  Linearity of continuous variables  Additivity of effects

Example: a simple logistic regression model  30day mortality ~ a + b 1 *sex + b 2 *age Assumptions:  Distribution of 30day mortality is binomial  Age has a linear effect  The effects of sex and age can be added

Assessing model assumptions  Examine model residuals  Perform specific tests  add nonlinear terms, e.g. age+age 2  add interaction terms, e.g. sex*age

Model assumptions and predictions  Better predictions if assumptions are met  Some violation inherent in empirical data  Evaluate predictions in new data

Evaluation of predictions  Calibration  average of predictions correct?  low and high predictions correct?  Discrimination  distinguish low risk from high risk patients?

Example: predicted probabilities

3 types of validation  Apparent: performance on sample used to develop model  Internal: performance on population underlying the sample  External: performance on related but slightly different population

Apparent validity  Easy to calculate  Results in optimistic performance estimates

Apparent estimates optimistic since same data used for:  Definition of model structure: e.g. selection and coding of variables  Estimation of model parameters: e.g. regression coefficients  Evaluation of model performance: e.g. calibration and discrimination

Internal validity  More difficult to calculate  Test model in new data, random from underlying population

Why internal validation?  Honest estimate of performance should be obtained, at least for a population similar to the development sample  Internal validated performance sets an upper limit to what may be expected in other settings (external validity)

External validity  Moderately easy to calculate when new data are available  Test model in new data, different from development population

Why external validation?  Various factors may differ from development population, including  different selection of patients  different definitions of variables  different diagnostic or therapeutic procedures

Internal validation techniques  Split-sample:  development / validation  Cross-validation:  alternating development / validation  extreme: n-1 develop / 1 validate (‘jack-knife’)  Bootstrap

Bootstrap is the preferred internal validation technique  bootstrap sample for model development: n patients drawn with replacement  original sample for validation: n patients  difference: optimism  efficiency: development and validation on n patients

Example: bootstrap results for logistic regression model  30-day mortality ~ a + b 1 *sex + b 2 *age Apparent area under the ROC curve: 0.77 Mean area of 200 bootstrap samples:0.772 Mean area of 200 tests in original: Optimism in apparent performance: 0.01 Optimism-corrected area: 0.76

External validation techniques  Temporal validation: same investigators, validate in recent years  Spatial validation (other place): same investigators, cross-validate in centers  Fully external: other investigators, other centers

Example: external validity of logistic regression model  30-day mortality ~ a + b 1 *sex + b 2 *age Apparent area in 785 patients: 0.77 Tested in 20,318 other patients: 0.74 Tested by other investigators: ?

Example: external validation

Summary  Apparent validity gives an optimistic estimate of model performance  Internal validity may be estimated by bootstrapping  External validity should be determined in other populations

Key references  tutorial and book on multivariable models (Harrell 1996, Stat Med 15:361-87; Harrell: regression modeling strategies, Springer 2001)  empirical evaluations of strategies (Steyerberg 2000: Stat Med19: )  internal validation (Steyerberg 2001:JCE 54: )  external validation (Justice 1999: Ann Intern Med 130:515-24; Altman 2000: Stat Med 19: )

Links  Interactive text book on predictive modeling  Harrell’s Regression modeling strategies