Transforming Random-Effect Covariances Between Parameterizations

Why Transforming Random-Effect Covariances Matters

A pharmacokinetic model may be written using one set of structural parameters but interpreted or reported using another. For example, a two-compartment model can use microconstants such as \(k_{10}\), \(k_{12}\), and \(k_{21}\), while its concentration-time curve is described by macroconstants \(\alpha\) and \(\beta\). If subjects vary jointly in the microconstants, their macroconstants also vary jointly. The question is how to approximate the variances and covariance after this nonlinear transformation.

The answer depends on the scale of the random effects. In particular, transforming natural-scale parameters and transforming log-scale random effects are related but are not the same calculation. The delta method provides a local, first-order approximation: it uses the derivative of the transformation near the typical parameter values to carry the covariance forward.

The Covariance Transformation

For a vector of natural-scale parameters \(\boldsymbol{\theta}\) with mean or expansion point \(\boldsymbol{\theta}_0\), and a differentiable transformation \(\mathbf{z}=g(\boldsymbol{\theta})\), the corresponding approximation is

This follows by a first-order Taylor expansion:

The constant term has zero covariance with everything, and for any fixed matrix \(A\), \(\operatorname{Cov}(A\mathbf{X})=A\operatorname{Cov}(\mathbf{X})A^T\). Applying that identity to the linear term gives the stated approximation. The approximation neglects second- and higher-order terms in the Taylor expansion; it is not generally an exact transformation of the full random-effects distribution.

In a common lognormal parameterization, \(\theta_j=\theta_{j,\mathrm{pop}}\exp(\eta_j)\), where \(\eta_j\) is a mean-zero random effect. For positive transformed parameters \(z_i=g_i(\boldsymbol{\theta})\), define the transformed log effect as \(\xi_i=\log\{g_i(\boldsymbol{\theta})/g_i(\boldsymbol{\theta}_{\mathrm{pop}})\}\). The Jacobian with respect to the original random effects is then

The resulting \(G\Omega G^T\) is a covariance matrix for the transformed log effects. To approximate the covariance matrix of the transformed natural-scale parameters themselves, the Jacobian must instead be taken for \(g(\boldsymbol{\theta})\), or the local log-effect covariance must be rescaled by the typical transformed values.

From Microconstants to Macroconstants

For a standard two-compartment model with elimination from the central compartment, the two positive macroconstants are the roots of \(\lambda^2-S\lambda+P=0\), where \(S=k_{10}+k_{12}+k_{21}\) and \(P=k_{10}k_{21}\). When \(S^2-4P>0\), they are

These formulas define a nonlinear transformation. If the microconstants have log random effects, the needed Jacobian is the derivative of \((\log\alpha,\log\beta)\) with respect to those log effects, evaluated at the typical microconstants. One convenient derivative follows by differentiating the root equation \(\lambda^2-S\lambda+P=0\):

Here \(k_j=k_{j,\mathrm{pop}}\exp(\eta_j)\), so \(\partial k_j/\partial\eta_j=k_j\) at the expansion point. The denominator \(2\lambda-S\) is \(+\sqrt{S^2-4P}\) for \(\alpha\) and \(-\sqrt{S^2-4P}\) for \(\beta\).

Scale, Curvature, and Approximation Quality

A covariance has meaning only together with the scale of the variables. A variance of a log effect is dimensionless; a natural-scale variance has squared parameter units. Likewise, a natural-scale covariance between \(\alpha\) and \(\beta\) has units of \(\mathrm{h}^{-2}\), while their log-effect covariance is dimensionless. Do not transfer a numerical covariance from one scale to another without the appropriate Jacobian or local rescaling.

The delta method is most useful when random effects are sufficiently concentrated near the expansion point and the transformation is smooth there. Check whether the Jacobian is finite and whether plausible random-effect values remain in the transformation’s valid domain. In the two-compartment example, as \(S^2-4P\) approaches zero, the macroconstants approach one another and derivatives involving \(1/\sqrt{S^2-4P}\) can become large. The local approximation may then be unstable, even if the typical values still satisfy the root formulas.

A transformed covariance matrix from \(G\Omega G^T\) is positive semidefinite when \(\Omega\) is positive semidefinite, but that algebraic property does not establish that the approximation describes the full transformed distribution well. Large transformed variances, strong curvature, or simulated parameter draws near invalid regions are reasons to be cautious. If the transformation is linear, covariance propagation is exact. For a clearly nonlinear transformation, one alternative is to transform representative draws from the specified random-effects distribution and calculate their empirical covariance; this retains curvature but depends on the distributional assumptions and the adequacy of the simulation.

Common Pitfall

A second frequent error is evaluating a derivative at an arbitrary value rather than at the point defining the random effects. For lognormal effects centered at \(\boldsymbol{\eta}=\mathbf{0}\), evaluate at the typical parameters \(\boldsymbol{\theta}_{\mathrm{pop}}\). The transformation of typical values need not equal the mean of the transformed random parameters, especially under substantial variability and curvature.