Polynomial and Spline Regression for Dose-Response

The important point is that the model does not require one cubic equation to determine the entire curve. The shape can change locally.

Restricted Cubic Splines

One particularly useful form for clinical regression is the restricted cubic spline, also called a natural cubic spline. The restriction applies linear behavior in the tails beyond the boundary knots.

That restriction is valuable because unrestricted high-order behavior at the extreme ends of the data can be unstable. The model remains flexible in the interior while avoiding unnecessary curvature in the tails.

For the simulated dose-response example, consider internal knots at:

The exact knot locations are not sacred. In practice, they are commonly selected using prespecified quantiles of the observed dose distribution, subject to having enough information in each region.

The Computed Spline Fit

The restricted cubic spline fit to the same simulated data produces:

The spline provides an excellent fit while avoiding the requirement that the entire dose-response curve be described by one global polynomial.

Predicted responses from the spline are:

Notice what the spline does: it rises rapidly at lower doses and gradually flattens as dose increases. There is no requirement for the model to turn downward simply because a quadratic term exists.

Why Knots Matter

Knots determine where the spline is allowed to change its local curvature. Too few knots can produce an overly rigid curve. Too many knots can allow the model to follow random noise.

A common strategy is to place a small number of knots at prespecified quantiles of the dose distribution. For example, with enough observations, a model might use knots near the 5th, 35th, 65th, and 95th percentiles.

The precise number depends on sample size, the distribution of dose, the amount of information available at each dose, and the scientific purpose of the analysis.

Testing for Nonlinearity

A useful distinction is between asking:

1. Is there any association between dose and response?
2. Is the association adequately represented by a straight line?

The second question is the nonlinear-effect question. For a polynomial model, this can be evaluated by testing the coefficients of the higher-order terms jointly. For example, in the quadratic model:

For a spline, the nonlinear component generally contains multiple degrees of freedom, so the appropriate test is a joint test of those nonlinear spline terms.

This is preferable to inspecting one spline coefficient at a time. Individual spline coefficients are usually not directly clinically interpretable because the basis functions are mathematical constructs.

Clinical Interpretation: Do Not Interpret Spline Coefficients Like Ordinary Slopes

One of the most common mistakes with spline models is trying to interpret the individual coefficients as if they represented simple dose effects. They generally do not.

Instead, interpretation should focus on:

• Predicted response at clinically relevant doses
• Differences in predicted response between doses
• Confidence intervals around the fitted dose-response curve
• Tests of overall dose association
• Tests of nonlinearity
• Clinically meaningful features such as plateaus or regions of rapidly increasing response

For example, the scientifically useful statement is not: "the coefficient of spline basis function 3 was 13.74."

A much more useful statement is: "the estimated treatment effect increased rapidly through approximately 40–60 mg and showed progressively smaller incremental gains at higher doses."

Prediction Is Usually More Important Than the Coefficients

Suppose the objective is to compare 40 mg with 80 mg. The clinically relevant quantity is:

Using the spline predictions above:

That is directly interpretable as the estimated increase in response associated with moving from 40 mg to 80 mg. A confidence interval for this contrast should accompany the estimate in a clinical analysis.

Polynomial vs. Spline Regression

Why Not Just Use a Fourth-, Fifth-, or Sixth-Degree Polynomial?

A tempting strategy is to keep increasing the polynomial degree until the fitted curve follows the observed data almost perfectly. That is generally a poor strategy.

High-degree polynomials can oscillate dramatically, particularly near the boundaries of the observed dose range. A model can therefore produce an apparently excellent in-sample fit while producing implausible predictions at doses that were not actually studied.

Splines solve much of this problem by allowing flexibility where the data support it rather than forcing the entire curve to obey one high-degree equation.

Interpolation vs. Extrapolation

This distinction is critical in dose-response analysis.

Interpolation means predicting within the range of observed doses. For example, if the study contains doses from 0 to 160 mg, estimating the response at 70 mg is interpolation.

Extrapolation means predicting outside that range. Estimating the response at 300 mg from a study whose highest dose was 160 mg is extrapolation.

Both polynomial and spline models become much less trustworthy when extrapolated. Even a beautifully fitted spline cannot recover information that was never observed.

Polynomial and Spline Regression in R

A quadratic model can be fit with ordinary lm(). Here the dose is scaled by 80 to keep the polynomial terms numerically manageable:

dat$x <- dat$dose / 80

fit.linear <- lm(response ~ x, data = dat)

fit.quad <- lm(response ~ x + I(x^2),
               data = dat)

fit.cubic <- lm(response ~ x + I(x^2) + I(x^3),
                data = dat)

AIC(fit.linear, fit.quad, fit.cubic)

A restricted cubic/natural cubic spline can be fit using the splines package:

library(splines)

fit.spline <- lm(
  response ~ ns(dose,
                knots = c(20, 60, 120),
                Boundary.knots = c(0, 160)),
  data = dat
)

AIC(fit.linear, fit.quad, fit.cubic, fit.spline)

Predictions across a smooth grid of doses can then be generated with:

newdat <- data.frame(
  dose = seq(0, 160, by = 1)
)

newdat$pred <- predict(
  fit.spline,
  newdata = newdat,
  interval = "confidence"
)

The resulting predictions can be plotted against the observed dose-response data to display the fitted curve and its confidence band. That graphical display is often one of the most informative outputs from a nonlinear dose-response analysis.

A More Clinical Modeling Strategy

A practical dose-response analysis does not necessarily require choosing between "linear" and "spline" based solely on which has the smallest AIC. The model should reflect the scientific question.

A useful workflow is:

1. Plot the observed response against dose.
2. Fit a prespecified linear model.
3. Fit a clinically reasonable nonlinear model.
4. Assess whether there is evidence of nonlinearity.
5. Inspect the fitted curve and confidence interval.
6. Estimate contrasts between clinically relevant doses.
7. Perform sensitivity analyses for reasonable knot specifications.
8. Avoid extrapolation beyond the dose range without strong scientific justification.

What About Log-Dose?

Dose-response relationships sometimes become more naturally represented on the logarithmic dose scale. For example:

can be useful when the relevant biological change is approximately multiplicative rather than additive. However, dose zero cannot be directly log-transformed, so placebo or zero-dose observations require special handling.

A log-dose transformation is therefore not automatically superior to a spline. It is another modeling assumption that should be supported by pharmacologic knowledge and the observed data.

Connection to Emax Models

Polynomial and spline regression are flexible statistical models, but they are not the only way to describe dose-response. Pharmacology often motivates mechanistic or semi-mechanistic models such as the Emax model:

where:

• \(E_0\) is the baseline response,
• \(E_{\max}\) is the maximum additional treatment effect, and
• \(ED_{50}\) is the dose producing half of the maximum effect.

The Emax model has a strong pharmacologic interpretation. A spline, by contrast, makes fewer assumptions about the exact biological shape.

This creates an important modeling tradeoff: mechanistic models provide interpretable parameters but impose stronger structural assumptions; splines provide flexibility but sacrifice simple mechanistic interpretation.

When a Spline Is Especially Useful

• The dose-response relationship is clearly nonlinear.
• There is no compelling reason to assume a specific mechanistic curve.
• The study contains several dose levels across a meaningful range.
• The scientific objective is estimating the shape of the dose-response relationship.
• You want to avoid the global behavior of a high-order polynomial.
• You need clinically relevant predictions at doses between observed dose levels.

Important Limitations

• A spline does not create information. A smooth curve can make sparse data look convincing. The confidence interval should always be considered alongside the fitted curve.
• Knot selection matters. Different reasonable knot specifications can produce somewhat different estimates, particularly when the sample size is modest.
• Too much flexibility can overfit. More knots are not automatically better.
• Polynomial coefficients are difficult to interpret clinically. The higher-order coefficients describe mathematical curvature rather than directly meaningful treatment effects.
• Extrapolation remains dangerous. Neither splines nor polynomials can reliably determine what happens at doses that were not studied.
• Continuous endpoints are not the only possibility. If the endpoint is binary, ordinal, count-based, or time-to-event, the appropriate regression framework changes even though the underlying dose-response concept remains.
• Randomized dose-response trials require careful design. The regression model cannot compensate for a dose allocation scheme that provides little information about the relevant portion of the dose-response curve.

Summary

Dose-response relationships are rarely obligated to be linear. A linear regression model assumes that every incremental increase in dose produces the same incremental change in response, which can be a poor approximation when treatment effects saturate.

Quadratic and cubic polynomial regression introduce curvature, but higher-order polynomials can become unstable because their behavior is globally constrained by a single equation. The computed example demonstrated this clearly: the linear model predicted a continued increase to approximately 67 response units at 160 mg, whereas the observed pattern was approaching a plateau.

Restricted cubic splines provide a more flexible alternative. They allow the shape of the dose-response relationship to change locally while maintaining smoothness and controlling tail behavior. In the simulated example, the spline achieved an excellent fit while producing clinically plausible predictions across the observed dose range.

The most important practical lesson is not that splines should always replace polynomials. Rather, the model should be chosen according to the scientific question, the amount and distribution of information in the dose range, the plausibility of the assumed functional form, and the need for interpretable predictions.

References

ICH E4 (1994). Dose-Response Information to Support Drug Registration.
FDA (2003). Exposure-Response Relationships — Study Design, Data Analysis, and Regulatory Applications. Guidance for Industry.
Harrell, F.E. (2015). Regression Modeling Strategies. Springer.
Ruppert, D., Wand, M.P. & Carroll, R.J. (2003). Semiparametric Regression. Cambridge University Press.
Stone, C.J. (1985). Additive regression and other nonparametric models. The Annals of Statistics.
Pinheiro, J.C., Bretz, F. & Branson, M. (2006). Design and Analysis of Dose-Finding Studies. Springer.