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Simple polynomial regression

Polynomial regression fits y as a polynomial in x of a chosen degree (2–6), useful when the relationship is curved but its parametric form is unknown.

What is Simple polynomial regression?

Polynomial regression fits y = β₀ + β₁·x + β₂·x² + … + β_d·x^d via OLS. Quadratic (d = 2) handles a single U or inverted-U; cubic (d = 3) allows an inflection point; higher degrees fit increasingly wiggly curves.

Polynomials are global — they fit one curve to the whole x range. For local flexibility (different shapes in different x regions), use splines or LOESS instead. For dose-response data with a known parametric form, the nonlinear_* curves give interpretable parameters rather than a polynomial's harder-to-read coefficients.

High-degree polynomials overfit and behave wildly at the extremes — keep d ≤ 3 unless you have a specific reason.

When should I use Simple polynomial regression?

Curved x-y relationship without a strong parametric expectation.
Adding a single quadratic term to a linear model to test for curvature.
Quick-and-dirty interpolation when you don't need the parameter interpretability of a nonlinear fit.

What data does it need?

Response y + predictor x + polynomial degree.

What does it report?

Coefficient table, R², adjusted R², F-test, fitted curve overlay on the scatter.

How do I interpret the result?

Each higher-degree coefficient is a *shape* effect, not directly interpretable on its own — interpret the fitted curve visually.