Polynomial Regression and Splines

Representing nonlinear mean relationships with basis functions

Statistics
Regression
Nonlinear Modeling
Splines
Polynomial regression, regression and smoothing splines, basis construction, penalties, uncertainty, diagnostics, and responsible nonlinear interpretation.
Author

Jiahong Sun

Published

October 23, 2026

Modified

October 23, 2026

1 Nonlinear in a Predictor, Linear in Coefficients

A linear regression model need not force a straight-line relationship. It is linear when the conditional mean is linear in its unknown coefficients. For one continuous predictor,

\[ Y_i=f(x_i)+\varepsilon_i, \qquad E(\varepsilon_i\mid x_i)=0, \]

and a chosen collection of basis functions \(B_1,\ldots,B_K\) gives

\[ \boxed{f(x)=\beta_0+\sum_{k=1}^{K}\beta_k B_k(x).} \]

Once the basis is fixed, estimation is ordinary or penalized regression. The scientific target is usually the curve \(f(x)\), a contrast such as \(f(x_b)-f(x_a)\), or a derivative—not an isolated basis coefficient. Different bases can span the same function space and produce the same fitted curve while having entirely different coefficient values.

This is a conditional-mean model. It does not by itself specify causal effects, the error distribution, the variance function, or dependence among repeated observations.

2 Global Polynomials

A degree-\(d\) polynomial uses

\[ f(x)=\beta_0+\beta_1x+\beta_2x^2+\cdots+\beta_dx^d. \]

It is easy to fit and can represent gentle curvature. A test of \(H_0:\beta_2=\cdots=\beta_d=0\) compares the polynomial with a straight line, provided the degree was not chosen after repeatedly inspecting the same test. Centering \(x\) reduces correlation among raw powers; an orthogonal polynomial basis improves numerical conditioning further without changing the fitted degree-\(d\) polynomial on the training data.

Global polynomials have an important weakness: changing one coefficient changes the curve everywhere. High degrees can oscillate, behave erratically near the boundaries, and extrapolate dramatically. They also impose a shape that is often hard to justify scientifically. Low-degree polynomials are therefore useful for prespecified, smooth curvature, not as an automatic strategy of increasing the degree until the residual plot looks flat.

3 Regression Splines

A spline is a piecewise polynomial joined at knots. A cubic spline uses cubic pieces and ordinarily constrains the function and its first two derivatives to be continuous at each knot. These constraints remove visible jumps and kinks while allowing local flexibility.

One truncated-power representation with knots \(\kappa_1,\ldots,\kappa_m\) is

\[ f(x)=\beta_0+\beta_1x+\beta_2x^2+\beta_3x^3 +\sum_{j=1}^{m}\gamma_j(x-\kappa_j)_+^3, \]

where \((u)_+=\max(u,0)\). It makes the piecewise construction explicit, but a B-spline basis is usually better conditioned and locally supported. Basis choice changes computation and coefficient interpretation, not the function space when degree, knots, and boundary constraints agree.

A natural cubic spline adds the constraint that the curve is linear beyond the boundary knots. This usually produces more stable boundary behavior than an unconstrained cubic spline, but extrapolation is still an assumption: data inside the observed range cannot validate the continuation outside it.

3.1 Knots and degrees of freedom

More knots or a larger basis dimension permit more complicated curves. With an unpenalized regression spline, this complexity must be chosen directly, for example by prespecifying degrees of freedom or using validation. Knot locations are often placed at predictor quantiles so that basis flexibility follows data density. Sparse regions cannot support detailed shape regardless of how many knots are supplied.

The effective sample size is determined by independent experimental units, not by the number of distinct technical measurements. A flexible curve fitted to thousands of cells from three donors still has only three independent donor trajectories for population inference unless the dependence is modeled.

4 Smoothing Splines and Penalization

Instead of selecting a small knot set, a smoothing spline balances fit against roughness by minimizing

\[ \boxed{ \sum_{i=1}^{n}\{y_i-f(x_i)\}^2 +\lambda\int \{f''(t)\}^2\,dt. } \]

The first term rewards fidelity to the observations. The second penalizes rapid changes in slope, and \(\lambda\geq0\) is the smoothing parameter. As \(\lambda\rightarrow0\), the curve can approach an interpolant; as \(\lambda\rightarrow\infty\), the second derivative is forced toward zero and the fit approaches a straight line. The solution is a natural cubic spline with knots at the distinct observed predictor values, so an infinite-dimensional optimization reduces to a finite basis problem.

In matrix form, with design matrix \(\mathbf B\) and penalty matrix \(\mathbf S\),

\[ \hat{\boldsymbol\beta}_\lambda =(\mathbf B^\mathsf T\mathbf B+\lambda\mathbf S)^{-1} \mathbf B^\mathsf T\mathbf y. \]

The fitted values are \(\hat{\mathbf y}=\mathbf H_\lambda\mathbf y\). A useful complexity summary is the effective degrees of freedom \(\operatorname{tr}(\mathbf H_\lambda)\), which need not be an integer. Cross-validation, generalized cross-validation, or likelihood-based criteria such as REML can select \(\lambda\). That estimated choice contributes uncertainty; treating it as fixed can make intervals too optimistic.

5 Estimation, Contrasts, and Uncertainty

For an unpenalized Gaussian model with design matrix \(\mathbf B\),

\[ \hat{\boldsymbol\beta} =(\mathbf B^\mathsf T\mathbf B)^{-1}\mathbf B^\mathsf T\mathbf y. \]

At a target \(x_0\), let \(\mathbf b(x_0)\) be the basis row. Then

\[ \hat f(x_0)=\mathbf b(x_0)^\mathsf T\hat{\boldsymbol\beta}, \qquad \widehat{\operatorname{Var}}\{\hat f(x_0)\} =\mathbf b(x_0)^\mathsf T \widehat{\operatorname{Var}}(\hat{\boldsymbol\beta}) \mathbf b(x_0). \]

Pointwise intervals concern one prespecified \(x_0\). A simultaneous confidence band is wider because it aims to cover the whole curve. A prediction interval for a new outcome is wider still because it also includes outcome variation. For penalized fits, uncertainty is conditional on a bias-inducing smoothness penalty and often uses approximate Bayesian or frequentist covariance formulas.

Report curves on a scientifically meaningful scale, with intervals and the data density shown. Derivatives can summarize a local rate of change, but their uncertainty is often large and they are more sensitive to smoothing choices than the curve itself.

6 Worked Bioinformatics Example

Suppose methylation proportion is summarized for 80 independently sampled people and related to age. A natural cubic spline can represent a gradual change that plateaus in older ages:

fit <- lm(methylation ~ splines::ns(age, df = 4) + sex + batch,
          data = donor_data)

This ordinary syntax-highlighted example is not executed by the site. The 80 people are experimental units. CpG reads and repeated sequencing lanes are technical observations; they improve measurement but do not create additional independent age profiles. If multiple tissues or time points are observed per person, the dependence requires a subject-aware model.

Statistically, the fitted curve estimates an adjusted conditional mean under the chosen spline space and covariates. Biologically, it does not establish that aging causes the methylation pattern: cohort effects, cell composition, selection, and unmeasured exposures remain possible explanations. The useful summary may be a prespecified contrast between ages 30 and 60 rather than a collection of basis-coefficient p-values.

7 Assumptions, Diagnostics, and Sensitivity

Issue Why it matters Assessment Sensitivity or remedy
Mean structure Too little flexibility leaves pattern; too much follows noise Residuals against predictor and fitted values; held-out error Compare plausible degrees of freedom or smoothness penalties
Independent units Pseudoreplication makes uncertainty too small Inspect sampling, nesting, time, and batch structure Aggregate appropriately or use correlated/mixed models
Variance and outcome family Gaussian constant-variance errors may be inappropriate Residual spread and distribution; measurement process Robust covariance, transformation, or outcome-appropriate model
Boundary support Splines are least constrained near edges Rug plots, leverage, uncertainty bands Restrict claims, use natural constraints, vary boundary knots
Extrapolation Polynomial and spline continuations are model assumptions Compare requested range with observed range Avoid distant extrapolation; report range explicitly
Smoothing selection Data-driven flexibility affects bias and uncertainty Effective degrees of freedom and criterion profile Refit across reasonable settings; use nested validation for prediction
Confounding Flexible adjustment cannot recover missing causal design Scientific design and causal reasoning Better design, justified adjustment, sensitivity analysis

A diagnostic smooth can reveal curvature, but selecting a final shape and then reporting ordinary p-values as though it were prespecified ignores selection. Plot partial effects together with uncertainty and covariate density; inspect influential units; check whether a few sparse boundary observations create the apparent turn; and evaluate conclusions across reasonable bases and penalties.

8 Connections and Take-home Messages

  1. Polynomial and spline regressions are linear in basis coefficients even when the fitted relationship is curved.
  2. Interpret the fitted function, contrasts, and derivatives—not arbitrary basis coefficients.
  3. Splines localize flexibility; penalties trade residual fit against roughness.
  4. Boundary behavior, smoothing selection, and extrapolation require explicit diagnostics and sensitivity analyses.
  5. Biological specimens determine replication; dense technical measurements do not justify an equally flexible population-level curve.

9 References

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