Comparing Power, Exponential, and Spline Effects

Choosing a Shape for a Continuous Covariate Effect

A continuous covariate may relate to a typical pharmacokinetic parameter in ways that are not well represented by a straight line. For example, clearance may increase with body size but do so proportionally, or a parameter may change more rapidly in one part of the covariate range than another. The functional form determines how the model describes that relationship, how many coefficients must be estimated, and what the fitted relationship implies beyond the observations.

The earlier material on Testing Linear Versus Nonlinear Covariate Functions introduced the question of whether a nonlinear function is warranted. Here the focus is on constructing and comparing three alternatives: power, exponential, and restricted cubic spline effects. Each is a covariate relationship for a typical parameter; it does not, by itself, specify the model for between-subject variability or residual error.

Power and Exponential Effects

A power effect is appropriate when a multiplicative change in the covariate is the natural scale for describing the relationship. For positive \(x\), define

The ratio \(x/x_{\mathrm{ref}}\) is dimensionless, so the power is well defined and the parameter retains its original units. At the reference value, the ratio is one, giving \(\theta_{\mathrm{typ}}(x_{\mathrm{ref}})=\theta_{\mathrm{ref}}\). Taking logarithms gives \(\log\{\theta_{\mathrm{typ}}(x)/\theta_{\mathrm{ref}}\}=\beta\log(x/x_{\mathrm{ref}})\). Thus \(\beta\) is the change in log parameter per unit change in log covariate: doubling \(x\) multiplies the typical parameter by \(2^\beta\).

A power effect requires positive covariate values. It is often interpretable for quantities such as body weight, but is not directly suitable for a covariate that can be zero or negative. An offset or a different functional form may be considered only when justified by the variable’s meaning; an arbitrary shift changes the model’s interpretation.

An exponential effect instead uses an additive change in the covariate. To make the coefficient’s scale explicit, choose a positive scaling constant \(s\) in the covariate’s units and set \(z=(x-x_{\mathrm{ref}})/s\). Then

Here \(\beta\) is the change in log typical parameter for an increase of \(s\) covariate units. A change of \(s\) units multiplies the typical parameter by \(\exp(\beta)\). Unlike the power model, the exponential form is defined for covariate values on either side of the reference value, including zero and negative values. It imposes a constant slope in the log parameter per covariate unit.

Restricted Cubic Spline Effects

Power and exponential effects each use one shape coefficient and impose a specific global pattern. A restricted cubic spline (also called a natural cubic spline) allows the log parameter to bend across the observed covariate range while constraining the tails to be linear. It uses knots, which are specified covariate values where the spline’s polynomial pieces join smoothly.

Let there be \(K\) ordered knots \(k_1<k_2<\cdots<k_K\). Define \((u)_+=\max(u,0)\). For \(j=1,\ldots,K-2\), one restricted-cubic-spline basis function is

With \(z=(x-x_{\mathrm{ref}})/s\), a reference-centered model is

Subtracting each basis value at \(x_{\mathrm{ref}}\) ensures that the spline contribution is zero at the reference value. Dividing by \(s^3\) makes the basis term dimensionless; the coefficients \(\beta\) and \(\gamma_j\) then describe changes in log parameter. The spline has \(K-2\) nonlinear basis coefficients plus one linear coefficient, or \(K-1\) covariate-shape coefficients in total. The reference parameter \(\theta_{\mathrm{ref}}\) is estimated as part of the PK model, as it is for the other forms.

The restriction on the tails is important. Below the first knot, each truncated term in the basis is zero, leaving a linear log effect. Above the last knot, the cubic basis terms combine to a linear function. To see why the cubic term cancels, write \(A=(k_K-k_j)/(k_K-k_{K-1})\) and \(B=(k_{K-1}-k_j)/(k_K-k_{K-1})\). The coefficient of \(x^3\) in the upper-tail expression is \(1-A+B=0\). The coefficient of \(x^2\) is also zero because \(A k_{K-1}-B k_K=k_j\). Thus the basis is at most linear in the upper tail. The whole model is linear in \(x\) on the log scale outside the boundary knots, not cubic indefinitely.

Comparing Flexibility, Parameter Count, and Extrapolation

The table describes model structure, not a guarantee that a particular form will fit well. A spline with more knots can represent more curvature, but it also estimates more coefficients and may be poorly determined if observations are sparse in some regions. Knot placement should be chosen with the covariate distribution and scientific context in mind; adding knots does not create information where there are few data.

The forms also make different extrapolations. In a power model, the log parameter changes linearly with \(\log(x/x_{\mathrm{ref}})\). In an exponential model, it changes linearly with \(x\). A restricted cubic spline becomes linear in \(x\) on the log scale beyond its boundary knots, while retaining the slope implied by its fitted tail. None of these continuation rules makes far-out predictions reliable: extrapolation is an assumption about the function beyond the data, not an observation supported by those data.

Diagnostics, Assumptions, and Common Pitfalls

All three forms assume that the chosen relationship is a reasonable description of the covariate effect over the range where predictions are needed. They also assume that the covariate has been recorded and scaled appropriately, that the reference value is meaningful, and that the form is considered conditional on the other terms in the PK model. A covariate coefficient describes a modeled association; it is not automatically a causal effect.

Useful checks include plotting the fitted typical-parameter relationship against the covariate and marking where observations are concentrated. For a spline, inspect the fitted curve and the distribution of observations around each knot, and check whether the curve is driven by a small number of observations near a boundary. For all forms, look for implausible predictions, unexpectedly steep changes, numerical instability, or substantial sensitivity to reasonable changes in knots or scaling. In a population PK model, these checks should be considered alongside the adequacy and stability of the overall model fit.

• Power-domain failure: A power model is not directly defined for \(x\leq0\). Do not apply it to such values without a scientifically justified redefinition of the covariate.
• Confusing coefficients: The exponential coefficient is tied to the chosen scale \(s\), while a power coefficient is an elasticity. State the scale and units when interpreting either coefficient.
• Over-interpreting a spline: A flexible fitted curve is not proof of a complex biological mechanism. With sparse covariate coverage, curvature may be uncertain and tail behavior remains model-imposed.
• Wrong reasoning: “A spline fits better, so it must be the correct relationship.” Correction: A more flexible form can fit the observed data more closely simply because it has more coefficients. Consider parameter count, stability, scientific plausibility, and the covariate range relevant to prediction.

The practical choice is a balance. Prefer the simpler power or exponential form when its scale and implied pattern are scientifically plausible and adequately describe the data. Consider a restricted spline when there is a reason to allow smooth curvature and sufficient covariate information to estimate it. The comparison is about functional form; a fitted curve should not be interpreted beyond its assumptions or data support.