Worked Example: Selecting a Continuous Covariate Function

Why the Covariate Function Matters

A continuous covariate such as renal function can be associated with clearance in several plausible ways. A straight-line change in log clearance, a power relationship, and a smooth curve may all describe the observed data reasonably well over part of the covariate range. Choosing among them matters: a function that is too rigid can miss systematic patterns, while an unnecessarily flexible one can follow noise and produce unstable estimates. The aim here is to compare candidate functions while keeping the pharmacokinetic model and likelihood on a common basis.

Let \(x_i\) be a positive continuous covariate for subject \(i\), such as renal function in mL/min. A useful shared structure is to model the typical clearance at a reference covariate value, then allow the function of \(x_i\) to modify that typical value. For illustration, suppose individual clearance is

Here \(\theta_{CL}\) is typical clearance at the reference value, \(f(x_i)\) is the covariate function scaled so that \(f(x_{\mathrm{ref}})=0\), and \(\eta_{CL,i}\) represents between-subject variability on the log scale. The concentration data likelihood also depends on the structural and residual-error models. In a functional-form comparison, those parts of the model, the analyzed subjects, and the likelihood calculation should be held fixed unless there is a specific reason to change them.

Three Candidate Functions

For a linear function on log clearance, define \(z_i=(x_i-x_{\mathrm{ref}})/s_x\), where \(s_x\) is a fixed positive scaling constant with the same units as \(x\). Then

The scaling makes the coefficient correspond to a change over \(s_x\) units of the covariate. For example, with \(s_x=30\) mL/min, \(z=1\) means an increase of 30 mL/min from the reference value. This is linear in log clearance, not in clearance itself.

A power function is

The positivity conditions ensure the ratio is positive, so the expression is real-valued for any real \(\gamma\). Taking logs shows that this is linear in \(\log(x_i/x_{\mathrm{ref}})\), not in \(x_i\). The exponent \(\gamma\) is estimated from the data.

A spline represents a smooth, potentially nonlinear function using basis functions. One common choice is a restricted cubic spline: it is cubic between knots and constrained to be linear beyond the boundary knots. Write a centered spline function with one linear term and two nonlinear basis terms as

The basis functions \(B_2\) and \(B_3\) are constructed from prespecified knots and centered so that the function is zero at the reference value. This particular specification has two additional nonlinear coefficients relative to the linear model. Other knot choices and spline bases are possible; the basis and knot locations must be stated because they determine the fitted function and its flexibility.

Likelihood Comparison and Its Conditions

Let \(\ell\) denote the maximized log likelihood for a candidate model, with all its parameters estimated from the same data. For nested models, the likelihood-ratio statistic is

Under the null hypothesis that the additional parameters in the larger model are zero, and under regularity conditions, \(D\) has approximately a chi-square distribution with degrees of freedom equal to the number of additional freely estimated parameters. The null model must be obtainable by imposing parameter restrictions on the larger model. The models should use the same observations and compatible likelihoods, and the null parameter values should not lie on a problematic boundary. For population PK models, the approximation can also be unreliable with weak identification, small samples, or difficult estimation.

A spline containing the linear function when its nonlinear coefficients are zero is nested within that spline. Thus, testing the linear model against the three-coefficient spline above tests \(H_0:\beta_2=\beta_3=0\), using \(df=2\), provided the model is regular and the spline basis and knots are fixed for the comparison. A linear function of \(x\) and a power function of \(x\) are generally not nested in one another: neither is generally obtained from the other by fixing a subset of its parameters. An ordinary likelihood-ratio test between them is therefore not justified. Their likelihoods can still be compared descriptively, or through criteria such as AIC, or with predictive assessment methods.

Comparing Candidate Shapes and Checking Uncertainty

For nonnested candidates, likelihood values alone reward flexibility: a more flexible model cannot have a lower maximized likelihood when it includes the other model, but independently specified nonnested functions need not have that ordering. AIC makes an explicit complexity adjustment, \(AIC=-2\ell+2k\), where \(k\) is the number of estimated parameters under the chosen likelihood convention. AIC is a relative comparison, not a significance test; lower AIC indicates the preferred trade-off among the specified candidates, not proof that a model is correct. Predictive checks or cross-validation can address prediction-oriented questions, but they answer a different question from a nested likelihood-ratio test.

Uncertainty in a coefficient is not always well summarized by a symmetric standard-error interval. Profile likelihood examines how well the model can fit when a parameter is fixed at successive values and all other parameters are re-estimated. For a scalar parameter \(\gamma\), let \(\ell_{\mathrm{profile}}(\gamma)\) be the maximized log likelihood subject to that fixed value, and let \(\hat{\gamma}\) be its unrestricted maximum-likelihood estimate. The profile deviance is

For a regular scalar parameter, an approximate 95% likelihood-ratio confidence set contains values satisfying \(D_{\mathrm{profile}}(\gamma)\leq 3.841\), the 95th percentile of a chi-square distribution with one degree of freedom. This approximation relies on the same large-sample and regularity considerations as other likelihood-ratio inferences. A profile can reveal asymmetry, flat regions, or boundaries that a local standard error may conceal.

Diagnostics, Limitations, and a Common Pitfall

A likelihood comparison does not replace checking whether the fitted relationship is credible. Plot the estimated typical clearance function over the observed covariate range, with uncertainty where available, and inspect covariate-stratified goodness-of-fit patterns. Examine whether residual or prediction discrepancies show a systematic trend with the covariate. Check the covariate distribution: sparse observations at the extremes provide little information about tail shape, and spline estimates near boundaries may be especially uncertain. Confirm that estimates, gradients, and objective-function values are stable across reasonable initial values and that the added terms are estimable.

Splines can reduce functional-form bias within the data range, but they do not guarantee a biologically plausible relationship. A restricted cubic spline has linear tails by construction, yet predictions beyond the observed range still rely on extrapolation and should be treated cautiously. A power function requires positive covariate values and reference value; it may be inappropriate if zero or negative values are possible. A linear-on-log-clearance model can also predict implausible changes when extrapolated far from the data. No model-selection statistic can compensate for poor covariate measurement, confounding, an unsuitable structural PK model, or inadequate individual information.

Common mistake: “The spline has the largest likelihood, so it must be the right function; I can use a likelihood-ratio p-value to compare it with the power model.” This reasoning is wrong for two reasons. First, a better maximized likelihood does not establish that the function is correct, especially when a model has extra flexibility. Second, the ordinary likelihood-ratio chi-square test requires nested models; a power model and a spline are generally nonnested. Instead, use a valid nested test for linear versus spline when its assumptions hold, and compare nonnested candidates with a clearly stated criterion such as AIC or predictive performance. Then consider diagnostics, uncertainty, and the covariate range before selecting a function.