Testing Linear Versus Nonlinear Covariate Functions

Why Test Functional Form?

A continuous covariate can be relevant to a population PK parameter without having a linear relationship with that parameter. For example, body weight may predict typical clearance, but a straight-line change in log clearance across the observed weight range may not describe the data adequately. A formal functional-form test asks a focused question: does a prespecified nonlinear extension improve fit enough to justify its additional parameter or parameters?

The key is to construct the comparison as nested models. The simpler model must be obtainable from the more flexible model by fixing specified parameters to known values. Then the likelihood-ratio test evaluates those restrictions. It does not automatically compare any two curves that happen to be called “linear” and “nonlinear.”

Writing the Nested Hypotheses

Let \(u_i\) be a prespecified numerical coordinate for a continuous covariate measured on individual \(i\), and let \(x_i=u_i-1\). Suppose the covariate effect is on typical clearance and is represented on the log scale:

This is linear in \(x_i\) on the log-clearance scale. A quadratic extension is

The linear model is recovered by setting \(\beta_2=0\). Thus the hypotheses are \(H_0:\beta_2=0\) against \(H_a:\beta_2\ne 0\). The linear coefficient remains in both models; the test concerns the additional curvature term, not whether the covariate has any effect at all. A joint extension with quadratic and cubic terms instead tests \(H_0:\beta_2=\beta_3=0\), adding two parameters.

More generally, write the effect as a linear term plus prespecified nonlinear basis functions:

Here each \(h_j\) is fixed before fitting the comparison. The linear model is the restriction \(\gamma_1=\cdots=\gamma_q=0\). A candidate power or exponential relationship can also be tested this way only if its parameterization genuinely contains the linear model as a restriction. If one model cannot be obtained from the other by fixing parameters, the ordinary nested-model likelihood-ratio test does not apply.

The Likelihood-Ratio Test and Its Conditions

Fit both models to the same observations using the same data likelihood and the same treatment of the structural, random-effects, residual-error, and other nuisance parameters. Let \(\ell_0\) be the maximized log likelihood for the restricted linear model and \(\ell_1\) the maximized log likelihood for the nonlinear model. The likelihood-ratio statistic is

when the objective function is defined as minus twice the log likelihood. A correctly optimized nested alternative cannot have a lower maximized likelihood than its restricted model, so \(\Lambda\) should be nonnegative apart from small numerical discrepancies. Under the null hypothesis and standard regularity conditions, \(\Lambda\) has approximately a chi-square distribution with \(q\) degrees of freedom, where \(q\) is the number of independent restrictions being tested. The p-value is the upper-tail probability \(P(\chi^2_q\geq\Lambda)\).

The approximation relies on more than the models being nested. The null value should be an interior point of the alternative parameter space; the added parameters must be identifiable under the null; the likelihood must be sufficiently well behaved; and estimation must have reached the relevant maxima. In nonlinear mixed-effects models, finite samples, weak information, sparse covariate coverage, or complicated random-effects structures can make the chi-square approximation inaccurate. A parameter on a boundary or an added parameter that is undefined under the null can require different calibration.

Worked Comparisons

Diagnostics, Alternatives, and Limitations

A small p-value is meaningful only if the comparison and its numerical fits are credible. Before interpreting it, check that the models use the same records and likelihood, that the alternative truly contains the null model, and that the counted degrees of freedom match the restrictions tested. Verify that the objective-function difference is positive and stable to reasonable changes in starting values or estimation settings. A failed convergence, implausible parameter estimate, or large sensitivity to starting values can signal that the nominal likelihood difference is not trustworthy.

Also inspect whether the covariate data can distinguish the proposed shape. A quadratic and a linear term can be difficult to separate when observations cover only a narrow range. Sparse observations near the extremes may leave curvature largely determined by a few individuals. Examine the fitted covariate relationship over the observed range and relevant residual or predictive diagnostics; a significant test does not make extrapolation beyond that range reliable.

The likelihood-ratio test is not appropriate for two nonnested functions merely because both describe the same covariate. For example, if neither a power curve nor an exponential curve can be recovered by restrictions from the other, their separate maximized likelihoods do not yield the ordinary chi-square test above. One possible strategy is an encompassing model that includes prespecified basis terms and tests a clearly defined joint restriction, provided the nesting and regularity conditions hold. Other questions may call for information criteria or predictive validation, which compare fit with complexity or predictive performance rather than testing the same null hypothesis.

The test is conditional on the functional form being specified. If many transformations are tried and only the most significant one is reported, the nominal p-value does not account for that search. Search procedures and multiplicity adjustments address that broader selection issue; the single-comparison p-value should not be presented as though it covers unreported exploration. Finally, statistical evidence for curvature is not proof of a causal mechanism. It describes the fitted covariate relationship under the model and data assumptions.