Parameter Transformations for Positive and Bounded Parameters

Why Transform Parameters?

Many pharmacometric model parameters have a restricted range by scientific definition: a clearance or variance must be positive, while a fraction or correlation may be confined to a finite interval. An optimizer that proposes values outside the permitted range can produce undefined model predictions or an objective function that cannot be evaluated. One way to preserve the restriction at every finite optimization iterate is to optimize an unconstrained coordinate and map it to the natural parameter scale.

This tutorial develops two common maps. The logarithm maps the real line to strictly positive values; the logit maps the real line to values strictly between two fixed bounds. These are smooth, one-to-one transformations in the interior of their ranges. Their derivatives therefore give a direct way to translate gradients and local uncertainty between scales. The transformations do not remove the need to choose a useful parameterization or starting value: near a boundary, their derivatives can become very small, which has practical consequences.

The reference value makes the logarithm's argument dimensionless. In practice, software often writes \(\eta=\log(\theta)\) while implicitly expressing \(\theta\) in a chosen unit; that convention is equivalent to taking the reference value to be one unit. Changing units changes the numerical log coordinate by a constant, so state the units and reference convention when reporting transformed values. The bounded transformation is also dimensionless: both \(\theta-L\) and \(U-\theta\) have the same units.

Derivatives on the Transformed Scale

Let \(f(\theta)\) be the objective function being minimized, and write \(\theta=g(\eta)\) for either inverse transformation. The optimizer evaluates the same objective at the mapped parameter value, \(f\{g(\eta)\}\). By the chain rule, the gradient used in transformed coordinates is the natural-scale gradient multiplied by the transformation's local rate of change. For a vector parameter, let \(J=\partial\theta/\partial\eta\) be the Jacobian matrix. Then

For a scalar positive parameter, \(\theta=\theta_{\mathrm{ref}}\exp(\eta)\), so \(d\theta/d\eta=\theta\). For a bounded parameter, set \(q=\{1+\exp(-\eta)\}^{-1}\), giving \(\theta=L+(U-L)q\). Since \(dq/d\eta=q(1-q)\), its derivative is \((U-L)q(1-q)\), which is positive in the interior and approaches zero near either bound. Thus the same natural-scale gradient can correspond to a much smaller transformed-scale gradient close to a boundary.

The additional second-derivative term matters away from a stationary point. For the log map, \(g'(\eta)=g''(\eta)=\theta\), so the scalar Hessian is \(\theta^2 f''(\theta)+\theta f'(\theta)\). At an interior optimum, where \(f'(\theta)=0\), the second term vanishes; away from the optimum it does not. For the bounded map, \(g'(\eta)=(U-L)q(1-q)\) and \(g''(\eta)=(U-L)q(1-q)(1-2q)\). These formulas explain why curvature reported in one coordinate system cannot generally be treated as though it were curvature in the other.

Mapping Local Uncertainty and Intervals

Suppose an estimator is obtained on the transformed scale, with estimated covariance matrix \(V_{\eta}\). If the transformation is differentiable and the uncertainty is sufficiently concentrated around the estimate, a first-order Taylor expansion gives the delta-method approximation

For a single parameter, this reduces to \(\operatorname{Var}(\hat{\theta})\approx\{g'(\hat{\eta})\}^2\operatorname{Var}(\hat{\eta})\), or \(SE_{\theta}\approx |g'(\hat{\eta})|SE_{\eta}\). This is a local approximation, not an exact distributional identity. For several transformed parameters, the Jacobian also maps covariances between parameters; dropping off-diagonal entries of \(V_{\eta}\) would generally lose information about their joint uncertainty.

Why the Transformation Works—and Where It Can Struggle

The maps preserve the restrictions because exponentials are positive and the logistic function lies strictly between zero and one. For the bounded map, \(q=1/(1+\exp(-\eta))\) satisfies \(0<q<1\) for every finite \(\eta\). Hence \(L<L+(U-L)q<U\) whenever \(L<U\). Conversely, any interior value \(L<\theta<U\) has a finite coordinate \(\eta=\log\{(\theta-L)/(U-\theta)\}\). The log map similarly gives \(\theta_{\mathrm{ref}}\exp(\eta)>0\) for every finite \(\eta\), and every positive \(\theta\) has a finite log coordinate. These facts establish a smooth one-to-one correspondence between the real line and each parameter's interior range.

The correspondence does not include the endpoints at finite transformed values. As \(\eta\) tends toward negative or positive infinity, the logit-mapped parameter approaches \(L\) or \(U\), respectively, while \(d\theta/d\eta\) tends to zero. The log-mapped positive parameter approaches zero only as \(\eta\) tends to negative infinity. Accordingly, an estimate exactly on a boundary cannot be represented by a finite transformed coordinate. Optimization may also become poorly scaled when a parameter is very close to a boundary: a large change in \(\eta\) can produce only a tiny change in \(\theta\), and gradients in \(\eta\) may be small even when natural-scale gradients are not.

A transformation is a change of optimization coordinates, not automatically a change to the statistical model or objective. If the goal is to minimize \(f(\theta)\), the transformed objective is \(f\{g(\eta)\}\); one does not add a Jacobian term merely because the parameter was re-expressed. A Jacobian term is relevant in a different operation, such as transforming a probability density and expressing that density with respect to a new variable. Confusing these operations changes the target being optimized.

When using transformed-scale standard errors, remember that the delta method is local. It can be inaccurate when uncertainty is wide, when the estimate is near a boundary, or when the inverse transformation is strongly curved over the uncertainty range. Transforming the endpoints of a suitable transformed-scale interval often gives a more visibly asymmetric interval on the natural scale, but it does not guarantee good coverage in small samples or under a poor likelihood approximation. In pharmacometric estimation, inspect the fitted parameter values, gradients, and uncertainty on the scales relevant to the model; a plausible-looking transformed-scale result alone is not a diagnostic of a well-determined natural-scale parameter.

Choosing and Checking a Parameterization

A direct constrained optimization approach instead works with \(\theta\) and explicitly restricts proposals to the permitted range. It can represent an attained boundary, unlike a finite logit coordinate, but requires the optimizer to manage feasibility and boundary behavior. A transformed approach guarantees interior feasibility for finite iterates and allows unconstrained update methods, but can be poorly scaled near an endpoint. Neither is universally preferable: the scientific range, expected parameter location, and behavior of the estimation algorithm all matter. Detailed consequences of active parameter bounds are a separate issue from the chain-rule calculations developed here.

Useful diagnostics follow directly from the maps. Check that the inverse transformation returns the intended natural-scale starting values; verify the gradient chain rule numerically at an interior point; and compare the estimated natural-scale uncertainty with the transformed-scale uncertainty after applying the Jacobian. If the estimate is extremely close to a bound, report that fact and interpret local standard errors cautiously. These checks help separate a genuine feature of the model fit from a scale mismatch or a nearly flat transformation.