Regression Splines: An Interactive Introduction

António S. Barros — Thu, 07 May 2026 00:00:00 GMT

The problem

Many clinical predictors do not relate linearly to the outcome. A textbook example is age: systolic blood pressure rises gently in young adulthood, accelerates around midlife, and may flatten in the very old.

Let’s simulate a dataset that captures this pattern and compare three modelling strategies - all running inside your browser.

Attempt 1: a straight line

The residuals are systematically positive in the middle and negative at the tails - a clear misspecification signal.

Attempt 2: polynomials (editable)

A common quick fix is to add higher-order terms. Edit the chunk below: change degree from 3 to 7, 10, 15. Watch the curve oscillate at the boundaries - Runge’s phenomenon in action.

The pedagogical point: polynomials are global - a single tail observation can yank the entire curve.

Attempt 3: natural cubic splines (reactive)

Splines are local: piecewise polynomials joined smoothly at knots. Drag the slider to change the degrees of freedom.

viewof df_val = Inputs.range(
  [3, 20],
  { value: 5, step: 1, label: "Spline df:" }
)

António Barros

Regression Splines: An Interactive Introduction

The problem

Attempt 1: a straight line

Attempt 2: polynomials (editable)

Attempt 3: natural cubic splines (reactive)