Why Parameterization Matters
A PK model can describe the same concentration-time curve using different parameter coordinates. For example, a one-compartment IV bolus model may use clearance and volume, or volume and the elimination rate constant. The numerical entries of an information matrix depend on the coordinates and units used. Yet, under a smooth one-to-one change of coordinates, a local D-optimal schedule has a useful invariance: every candidate schedule’s information determinant is multiplied by the same positive factor. Its ranking therefore does not change.
This result concerns a fixed model, a fixed nominal parameter point, and a fixed set of candidate designs. It does not say that every design criterion is invariant, or that changing the model or the nominal point leaves the selected schedule unchanged.
The Transformation Rule
Suppose the same model is parameterized by \(\phi=g(\theta)\), where \(g\) is one-to-one and continuously differentiable near \(\theta_0\), and its Jacobian is nonsingular there. Let \(\theta=h(\phi)=g^{-1}(\phi)\), and define \(B=\partial\theta/\partial\phi\), evaluated at the corresponding parameter points. The map is a change of coordinates: it does not change the sampling distribution or the observations being modeled.
For a regular likelihood, the score is the vector of derivatives of the log likelihood with respect to the parameters. The chain rule gives the score in the new coordinates. Taking its expected outer product gives the Fisher information transformation:
Here \(B\) depends on the parameter point and the reparameterization, but not on the sampling design \(\xi\). Taking determinants and using \(\det(B^T)=\det(B)\) gives
Because \(B\) is nonsingular, \(\det(B)^2>0\). Thus for any two candidate designs \(\xi_1\) and \(\xi_2\), \(\det I_\phi(\xi_1;\phi_0)>\det I_\phi(\xi_2;\phi_0)\) exactly when \(\det I_\theta(\xi_1;\theta_0)>\det I_\theta(\xi_2;\theta_0)\). The same factor also adds a design-independent constant to the log determinant: \(\log\det I_\phi=\log\det I_\theta+2\log|\det B|\), whenever the information matrices are positive definite.
If instead the forward Jacobian \(J=\partial\phi/\partial\theta\) is used, then \(B=J^{-1}\) and the equivalent formula is \(I_\phi=J^{-T}I_\theta J^{-1}\). Keeping track of which direction the Jacobian maps is important: using \(J\) where \(J^{-1}\) is needed reverses the scaling factor.
Worked Examples
Worked Example: Rescaling Two Parameters
Suppose two hypothetical candidate schedules have information matrices in coordinates \(\theta=(\theta_1,\theta_2)\): \(I_\theta(\xi_A)=\begin{pmatrix}100&0\\0&4\end{pmatrix}\) and \(I_\theta(\xi_B)=\begin{pmatrix}80&0\\0&6\end{pmatrix}\). Consider coordinates \(\phi=(10\theta_1,\theta_2/2)\). Calculate the determinants before and after reparameterization and check the ranking.
The forward map has Jacobian \(J=\partial\phi/\partial\theta=\operatorname{diag}(10,0.5)\), so \(B=J^{-1}=\operatorname{diag}(0.1,2)\). Its determinant is \(0.2\), and the determinant multiplier for information is \(0.2^2=0.04\).
In the original coordinates, \(\det I_\theta(\xi_A)=100(4)=400\), while \(\det I_\theta(\xi_B)=80(6)=480\). Thus schedule B ranks higher. In the new coordinates, \(I_\phi(\xi_A)=\operatorname{diag}(1,16)\), with determinant \(16=0.04(400)\), and \(I_\phi(\xi_B)=\operatorname{diag}(0.8,24)\), with determinant \(19.2=0.04(480)\).
The numerical determinants change, but schedule B still ranks higher. The example illustrates why determinant values should not be compared across parameterizations without accounting for the coordinate scale.
Worked Example: Clearance and Elimination Rate
For a one-compartment IV bolus model, the concentration may be written as \(C(t)=\mathrm{Dose}\,V^{-1}\exp(-kt)\), with \(k=CL/V\). Let the original coordinates be \(\theta=(CL,V)\), and the new coordinates be \(\phi=(k,V)\). At the hypothetical nominal point \(CL=5\) L/h and \(V=20\) L, \(k=0.25\) h\(^{-1}\). Find the information determinant multiplier.
As a function of \((k,V)\), the original parameters are \(CL=kV\) and \(V=V\). Therefore
The determinant multiplier is \(\det(B)^2=400\). For illustration, suppose a hypothetical schedule has \(I_\theta=\begin{pmatrix}4&1\\1&9\end{pmatrix}\), whose determinant is \(4(9)-1^2=35\). The transformed matrix is
For any other candidate schedule evaluated at this same nominal point, the same multiplier of 400 applies. Thus changing between clearance-volume and rate-volume coordinates cannot by itself change which schedule is locally D-optimal. The numerical scaling depends on the coordinates and the nominal volume, not on the sampling times.
Worked Example: Logarithmic Parameters
Let \(\theta=(CL,V)\) and use log coordinates \(\phi=(\log CL,\log V)\), with positive clearance and volume. At the hypothetical nominal values \(CL=5\) L/h and \(V=20\) L, calculate the determinant change for a schedule with \(I_\theta=\begin{pmatrix}2&0\\0&3\end{pmatrix}\).
The inverse coordinate map is \(CL=\exp(\phi_1)\), \(V=\exp(\phi_2)\), so \(B=\partial(CL,V)/\partial(\phi_1,\phi_2)=\operatorname{diag}(5,20)\). Hence \(\det(B)=100\), the multiplier is \(100^2=10000\), and \(\det I_\theta=2(3)=6\).
The transformed information matrix is \(I_\phi=\operatorname{diag}(5,20)I_\theta\operatorname{diag}(5,20) =\operatorname{diag}(50,1200)\). Its determinant is \(50(1200)=60000=10000(6)\). For a second schedule at the same nominal point, its determinant would also be multiplied by 10000; the D-optimal ranking is preserved.
Why the Result Holds and What It Does Not Say
The determinant result follows from the score transformation and the determinant identity for a product of square matrices. A related interpretation comes from the local covariance approximation. Under standard regularity conditions, the covariance matrix of an estimator is approximately the inverse information matrix. Under a change of coordinates, covariance transforms with the forward Jacobian \(J\), so the volume scale of a corresponding confidence ellipsoid changes by \(|\det J|\). That change is common to all schedules at the same nominal point. D-optimality therefore selects the same schedule whether the local ellipsoid’s volume is expressed in \(\theta\)-coordinates or \(\phi\)-coordinates.
This invariance is particular to the determinant criterion. For example, the A-criterion uses the trace of the inverse information matrix. In general, \(\operatorname{tr}(J I_\theta^{-1}J^T)\) is not a fixed multiple of \(\operatorname{tr}(I_\theta^{-1})\); its value can depend on how information is distributed across parameter directions. The E-criterion, based on an extreme eigenvalue, can also change under a general coordinate transformation. Such criteria require their own interpretation of parameter scales.
The information transformation is exact for Fisher information under a regular reparameterization. For an observed Hessian, the second-derivative chain rule can include additional terms involving the score; these terms need not vanish away from a point where the score is zero. Do not assume that every observed-Hessian calculation transforms by the Fisher-information formula in every finite data set.
Assumptions, Diagnostics, and Failure Modes
The invariance claim relies on the following conditions:
- The two parameter vectors describe the same probability model and the same observations.
- The parameter transformation is one-to-one and differentiable near the nominal point, with a nonsingular Jacobian there.
- The same nominal point is used in both coordinate systems, expressed as corresponding values \(\theta_0\) and \(\phi_0=g(\theta_0)\).
- The transformation does not depend on the sampling design. Each candidate schedule is evaluated using the same transformation at the same nominal point.
- The information matrices are calculated for the same observation and error model. A change of coordinates is not a justification for changing the residual-error assumptions or the likelihood.
A direct diagnostic is to compute the determinant ratio for several candidate schedules. If the calculation is correct and the conditions hold, then \(\det I_\phi(\xi)/\det I_\theta(\xi)=\det(B)^2\) for each schedule, up to numerical rounding. If these ratios differ substantially, check that the parameter order and Jacobian direction are consistent, that information was evaluated at corresponding nominal points, and that the same observation model and variance assumptions were used.
A singular Jacobian invalidates the result. For example, a transformation that loses a parameter direction cannot provide a valid local coordinate system for the full model. Near a singular Jacobian, the determinant multiplier may be extremely small or large, and numerical conditioning can make the transformed calculation unstable. A zero information determinant is also a warning: the information matrix is singular, so a finite full-dimensional local covariance approximation is unavailable. A common positive multiplier preserves zero versus positive determinants, but it does not resolve the underlying lack of information.
Common Mistake
Parameterization invariance is useful because it separates a design conclusion from a mere choice of coordinates. It does not make parameter scales irrelevant to every inferential question: units affect the displayed information matrix, and non-D criteria can respond to those scales. It also does not guarantee that a selected design is robust to uncertainty in the nominal parameter values. The result is local: it compares schedules at a specified point under a specified model.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A smooth transformation has \(\det(B)=-3\) at the nominal point. By what factor does it multiply every information determinant, and does the sign of \(\det(B)\) reverse the design ranking?
- Suppose two schedules have information determinants 12 and 15 in \(\theta\)-coordinates. A reparameterization has \(\det(B)=0.5\). What are their determinants in \(\phi\)-coordinates, and which schedule ranks higher?
- Why must the nominal parameter points correspond under the transformation for the invariance comparison to apply?
- Does a general smooth reparameterization necessarily preserve the ranking under the A-criterion? Explain briefly.
Solutions. (1) The multiplier is \((-3)^2=9\). It is positive, so the determinant ranking is not reversed. (2) The multiplier is \(0.5^2=0.25\); the transformed determinants are \(12(0.25)=3\) and \(15(0.25)=3.75\). The second schedule remains higher. (3) The transformation rule relates information matrices at corresponding representations of the same model point. Evaluating at unmatched points changes the local sensitivities being compared, so the same design-independent multiplier need not relate the results. (4) No. The A-criterion depends on the trace of the inverse information matrix, and a general coordinate transformation changes that trace in a way that need not be a common positive multiple across schedules.