Choosing the Parameter Scale for Covariate Effects

Why the Parameter Scale Matters

A covariate relationship can be linear on the parameter scale or linear on the logarithm of that parameter. These are different model assumptions, even when they use the same covariate and the same reference value. They imply different changes in the parameter as the covariate changes, and they impose different constraints on possible parameter values.

For example, an increase of one unit in a covariate produces a constant absolute change under an additive relationship. Under a log-linear relationship, it produces a constant multiplicative change. The distinction matters in a nonlinear mixed-effects model because parameters such as clearance, volume, and rate constants are often constrained to be positive. The scale should reflect both the scientific interpretation and the plausible range of the parameter, not just which formula is easier to write.

These equations describe the covariate component of the typical parameter. They do not, by themselves, specify the random-effects distribution, residual error model, or the rest of the structural model. In an NLME analysis, those components must still be defined and assessed separately.

Deriving the Different Changes

Write \(\Delta x=x-x_0\). For the additive model, subtracting the parameter at the reference value gives

Thus, for a fixed change \(\Delta x\), the absolute parameter change is constant wherever the model is used. The units of \(\beta_A\) are parameter units per covariate unit. For example, if \(\theta\) is clearance in L/h and \(x\) is weight in kg, then \(\beta_A\) has units L/h/kg.

For the log-linear model, exponentiating both sides of its defining equation gives

Dividing by \(\theta_0\) shows the multiplicative change:

The corresponding relative change is \(\exp(\beta_L\Delta x)-1\), or \(100\{\exp(\beta_L\Delta x)-1\}\%\). The slope \(\beta_L\) has units reciprocal to the covariate units: for a covariate measured in kg, it is per kg. It is not generally correct to call \(100\beta_L\%\) the exact change for a one-unit increase; that is an approximation when \(\beta_L\) is small. The exact percentage change for one unit is \(100\{\exp(\beta_L)-1\}\%\).

Worked Examples

When the Two Scales Can Look Similar

Near the reference value, the models may make similar predictions if their slopes are chosen appropriately. The local rate of change for the additive model is \(\beta_A\). Differentiating the log-linear expression gives

At the reference value this derivative is \(\beta_L\theta_0\). Thus, matching the local slopes at \(x_0\) requires \(\beta_A=\beta_L\theta_0\). Away from \(x_0\), however, the log-linear slope on the parameter scale changes with \(\theta_{\mathrm{typ}}(x)\), while the additive slope remains constant. Local agreement does not imply agreement across a broad covariate range.

This also clarifies the interpretation. An additive slope describes an absolute change, while a log-linear slope describes a constant proportional change. Which is more appropriate depends on the scientific question and on how the parameter is expected to behave. A parameter that changes by roughly the same amount per unit of a covariate may be described naturally on its original scale. A parameter expected to change by roughly the same proportion may be described naturally on the log scale.

Assumptions, Diagnostics, and Alternatives

Either formula assumes the specified relationship is a reasonable approximation over the covariate range being modeled, with other covariates held fixed. The additive form assumes a constant slope on the parameter scale. The log-linear form assumes a constant slope on the log-parameter scale and requires a positive reference value. Neither assumption should be extended automatically beyond the observed range.

Useful checks include plotting model predictions against the covariate over the observed range, inspecting residual diagnostics for systematic patterns, and checking whether predicted parameter values remain scientifically plausible. A pattern in residuals may signal that the covariate relationship is not adequately represented, but does not by itself identify which alternative scale is correct. Sparse observations at the extremes make assessments there less reliable.

When comparing candidate scales, keep the data, covariate range, other model components, and fitting conditions comparable. Assess the resulting fit and parameter behavior together; a small improvement in fit does not alone establish a scientifically meaningful relationship. Consider whether slopes have interpretable units, whether predictions respect known bounds, and whether the model behaves sensibly at the covariate values relevant to use.

The log-linear form is not a universal solution to positivity: it cannot represent a parameter that is zero or negative, and it may predict values that are implausibly small or large. Conversely, the additive form may be reasonable over a restricted range even for a parameter that must remain positive, provided its predictions stay positive throughout that range. If neither form captures the scientific pattern, a more flexible relationship may be considered, but flexibility does not remove the need to check interpretability and extrapolation.

Common Pitfall

A frequent error is to say, “The log-scale slope is 0.02, so the parameter rises by exactly 2% per covariate unit.” The coefficient is a change in the log parameter. For one unit, the exact ratio is \(\exp(0.02)\), so the exact percentage increase is \(100\{\exp(0.02)-1\}=2.02\%\), not exactly 2%. The approximation is close for a small coefficient, but the formula matters, especially across several units. Another error is to choose the log-linear form simply because the parameter is positive: positivity is a useful constraint, not proof that proportional change is the right scientific assumption.