Boundary-Time Degeneracy in Sparse PK Designs

Why a Feasible Schedule Can Still Be Degenerate

A sparse PK schedule can satisfy every clinic-window and sample-count constraint and still provide little information about one or more parameters. This happens, for example, when several planned observations occur at the same boundary time, when concentrations and their parameter sensitivities are negligible, or when different times produce nearly the same sensitivity pattern. The practical issue is not whether a time is at a boundary; it is whether the observations at the chosen times distinguish the parameters of interest.

The earlier material on identifiability diagnostics from sensitivity collinearity and on reading parameter correlations provides useful context. Here the emphasis is on identifying specific schedule degeneracies and deciding how to repair them. The calculations below use local sensitivities: they describe information at a stated nominal parameter vector, not a guarantee that the design remains equally informative at every possible parameter value.

Definition: Suppose a model predicts concentration $C(t;\theta)$, where $\theta$ is a vector of $p$ parameters. For observation time $t_i$, let $g_i$ be the row vector of derivatives of the predicted concentration with respect to the parameters being estimated, evaluated at the nominal parameter vector. If the residual variance at that time is $\sigma_i^2$, the weighted local information matrix for a schedule of $n$ independent observations is $$ M=\sum_{i=1}^{n}\frac{g_i^{\mathsf T}g_i}{\sigma_i^2}. $$ Its rank indicates how many parameter directions are locally informed; its eigenvalues indicate how strongly those directions are informed.

For comparisons across parameters with different units or scales, it is often helpful to use dimensionless parameters, such as log parameters for positive PK parameters. In that case, derivatives are taken with respect to those transformed parameters. Define the weighted sensitivity row $z_i=g_i/\sigma_i$. Then $M=\sum_i z_i^{\mathsf T}z_i$. This form makes clear that information depends on both the model sensitivity and the residual-error scale. A large raw derivative does not necessarily mean a large weighted contribution if the corresponding residual standard deviation is also large.

The formula assumes the observations have independent residual errors with the stated variances and that the local model derivatives adequately describe changes near the nominal parameter vector. If residual errors are correlated, the appropriate information calculation must account for their covariance rather than simply summing independent contributions. The formula is local and conditional on the selected model and parameterization.

How Degeneracy Appears in the Information Matrix

Stack the weighted sensitivity rows into an $n$-by-$p$ matrix $A$. Then $M=A^{\mathsf T}A$. A parameter direction $v$ receives information $v^{\mathsf T}Mv=\|Av\|^2$. If $Av=0$, changing the parameters in direction $v$ produces no first-order change in the predicted observations, so the schedule cannot distinguish that direction locally.

This gives a direct justification for the central diagnostic. Since $M=A^{\mathsf T}A$, its rank equals the rank of $A$. If the weighted sensitivity rows do not span all $p$ parameter directions, $M$ is singular. For a full-rank design, the determinant is the product of the positive eigenvalues; a very small eigenvalue signals a weakly informed direction even when the determinant is not exactly zero. The condition number, the ratio of largest to smallest eigenvalue for a positive-definite matrix, summarizes how unevenly information is distributed. It should be interpreted after sensible parameter scaling.

v^{\mathsf T}Mv =\sum_{i=1}^{n}(z_i v)^2, \qquad \det(M)=\prod_{j=1}^{p}\lambda_j

For D-optimal design, a singular information matrix has zero determinant and log-determinant tending to negative infinity. A nearly singular matrix can have a very poor determinant because one eigenvalue is tiny. But a single design criterion should not replace diagnosis: a large improvement in one parameter direction can obscure continued weakness in another, and parameter scaling affects numerical comparisons. Inspect the eigenvalues or singular values as well as the objective used to rank candidates.

Three Common Sources of Degeneracy

Repeated boundary times. If observations are scheduled at exactly the same time under the same model and error assumptions, they have the same local sensitivity row. Repeating the time adds information in that row's direction, but cannot create a new direction. A boundary time is not intrinsically uninformative: an early or late sample may be highly useful if its sensitivity complements the rest of the schedule. The concern is repeated or nearly repeated times that contribute only redundant rows.

Negligible concentrations. Under an additive-error model with approximately constant residual variance, a very small concentration may have small derivatives with respect to the parameters, and therefore contribute little weighted information. However, low concentration alone does not prove low information: the derivatives, the error model, and any quantification limits matter. Under proportional-error assumptions, weighting can change substantially at low concentrations, while observations below quantification limits require treatment consistent with the observation process. The earlier material on designing around below-quantification concentrations addresses that distinct issue.

Nearly redundant sensitivities. Two different times can produce sensitivity rows that point in almost the same direction. Both observations may be precise and have appreciable concentrations, yet they mostly reinforce the same parameter combination. This is a local sensitivity version of collinearity. The schedule then estimates one combination well but leaves another weakly determined.

Worked Example: Repeated Boundary-Time Observations

Consider a hypothetical two-parameter design with unit residual standard deviations. At a feasible boundary time, suppose the weighted sensitivity row is $z=(1,0)$. Three independently measured observations are all scheduled at that same time. What does their information say about the two parameter directions?

Each observation contributes $z^{\mathsf T}z$, so the total information is $$ M=3 \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix} = \begin{pmatrix} 3&0\\ 0&0 \end{pmatrix}. $$ The eigenvalues are 3 and 0, and the determinant is 0. The schedule gives three times the information in the first direction compared with one such observation, but provides no local information in the second direction. Repeating the sample has not made the two-parameter problem identifiable.

Suppose a feasible later time instead has weighted sensitivity $z=(0,1)$. Replacing one repeated observation with that sample gives $$ M= \begin{pmatrix} 2&0\\ 0&1 \end{pmatrix}. $$ The determinant is 2, and both eigenvalues are positive. This replacement supplies information in the previously missing direction. The numbers are hypothetical local sensitivities, not results from a clinical study.

Worked Example: A Very Small Concentration Under Additive Error

Suppose a hypothetical two-parameter PK model is evaluated with additive residual standard deviation $\sigma=0.1$ concentration units. At a very late time, the predicted concentration is $0.002$ concentration units and the derivative row with respect to the chosen parameters is $g=(0.001,0.0005)$ concentration units. Calculate the weighted information contribution.

The weighted row is $z=g/\sigma=(0.01,0.005)$. Its contribution is $$ z^{\mathsf T}z= \begin{pmatrix} 0.0001&0.00005\\ 0.00005&0.000025 \end{pmatrix}. $$ The nonzero eigenvalue of this rank-one contribution is $0.0001+0.000025=0.000125$; the other eigenvalue is zero. Thus this observation adds only a small amount of information, and only along one direction, in the stated scaling.

For comparison, a candidate time with weighted row $(0,1)$ contributes a nonzero eigenvalue of 1, much larger in this illustrative comparison. The comparison supports checking the actual weighted derivatives rather than rejecting all late samples categorically. If a different residual-error model or a different sensitivity pattern applies, the conclusion can change.

Worked Example: Detecting Near-Redundancy and Repairing It

For a hypothetical two-parameter design, suppose two candidate times have unit-variance weighted rows $z_1=(1,0)$ and $z_2=(0.99,0.10)$. Both rows are substantial, but their directions are close. Calculate the information matrix and assess the weakness.

Adding the two outer products gives $$ M= \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix} + \begin{pmatrix} 0.9801&0.099\\ 0.099&0.01 \end{pmatrix} = \begin{pmatrix} 1.9801&0.099\\ 0.099&0.01 \end{pmatrix}. $$ Its determinant is $1.9801(0.01)-(0.099)^2=0.019801-0.009801=0.01000$, and its trace is $1.9901$. The eigenvalues are approximately $1.9851$ and $0.005038$. The condition number is therefore approximately $1.9851/0.005038=394.1$. The design is full rank, but information is highly uneven: one parameter direction is much less informed.

If a feasible alternative time has weighted row $(0,1)$, replacing $z_2$ by that row produces $M=I$, with both eigenvalues equal to 1 and condition number 1. This illustrates the aim of repair: find a time with a sufficiently different sensitivity direction, while retaining useful information and respecting the schedule constraints. The actual best replacement should be assessed using the chosen design objective and parameter scaling.

Diagnose Before Moving a Sample

A useful diagnostic sequence starts with the planned times and the model predictions at the nominal parameter vector. Check for duplicated times, times that become duplicates after rounding or software constraints, and observations whose predicted concentrations or weighted derivatives are very small. Then inspect the weighted sensitivity matrix and the information matrix. A singular value near zero, a tiny information eigenvalue relative to the largest, or an extreme condition number can reveal a weak direction that may not be obvious from the schedule itself.

Interpret diagnostics with the intended parameterization and objective in mind. Raw parameter scales can make a condition number look poor simply because parameters have different units; using log parameters or another justified scaling can make the comparison more interpretable. Likewise, parameter correlations can be useful, but a correlation summary alone does not show the absolute amount of information. A pair can have moderate correlation while the entire design is weak, or high correlation while total information is otherwise substantial.

When repairing a candidate, compare feasible alternatives by recomputing the weighted sensitivities and information matrix. Prefer a change that increases information in the weak direction, not merely one that adds another observation at a time already represented. Check that the repaired schedule still satisfies timing and spacing constraints, and that the model and residual-error assumptions remain appropriate. If numerical derivatives are used, verify that the apparent weak direction is not an artifact of unstable finite differences.

Locate possible duplicates and weak observations.
Review exact and near-duplicate times, predicted concentrations, and weighted sensitivities under the stated nominal model.

Quantify the weakness.
Inspect rank, eigenvalues or singular values, determinant or log-determinant, and a scaled condition number.

Propose a complementary time.
Seek a feasible observation whose weighted sensitivity points toward a weakly informed parameter direction.

Recalculate and verify.
Recompute diagnostics and confirm the repair improves the relevant objective without violating constraints or relying on a numerical artifact.

Alternatives, Limitations, and a Common Pitfall

D-optimality summarizes overall local information through the determinant, while A-optimality summarizes average parameter-variance information through a function of $M^{-1}$, and c-optimality targets uncertainty in a specified parameter combination. These criteria answer different questions. Eigenvalue inspection is a diagnostic for weak directions, not a substitute for choosing the scientific target. A correlation matrix is another useful view of parameter trade-offs, but does not by itself measure precision. In practice, a defensible assessment can use the selected criterion together with rank and eigenvalue diagnostics.

These calculations have limitations. They depend on the model, parameterization, nominal values, residual-error assumptions, and local derivatives. A locally well-conditioned information matrix does not establish that the model is correct, that parameters are globally identifiable, or that the same schedule will be informative away from the nominal point. Nor does a low concentration automatically imply a useless sample: the derivative and error model determine its contribution. Designs should be checked under scientifically plausible assumptions rather than judged by a time label alone.

Common mistake: “The schedule has four samples, so it must contain enough information for two parameters.” Sample count alone is not enough. Four repeated rows may all point in the same direction, leaving the information matrix rank one. The correction is to inspect the span and scale of the weighted sensitivity rows: a useful schedule needs adequate information in every parameter direction relevant to the analysis.

Key takeaway: Degeneracy is a property of the information contributed by the schedule, not simply of whether a time is at the beginning or end of a window. Diagnose it through weighted sensitivities and information eigenvalues, then repair it with feasible, complementary observations.

Check Your Understanding

Try each question, then check the concise solutions below.

For two parameters, three observations all have weighted sensitivity $(2,0)$. What is the rank of their information matrix, and what does that imply?
Under independent errors, why can a repeated observation improve precision in one direction without resolving a missing parameter direction?
A late sample has a low predicted concentration. What additional quantities must be checked before concluding that it contributes little information?
For two sensitivity rows, $z_1=(1,0)$ and $z_2=(1,0.01)$, would you expect the design to be well-conditioned or nearly degenerate? Explain.

Solutions. (1) The information is $3(2,0)^{\mathsf T}(2,0)=\begin{pmatrix}12&0\\0&0\end{pmatrix}$, which has rank 1. One parameter direction is uninformed locally. (2) Repetition adds another outer product in the same sensitivity direction; it increases information there but does not expand the span of the rows. (3) Check the model derivatives, residual variance or weighting, and how the observation is handled relative to quantification limits. (4) The rows are almost parallel, so one eigenvalue will be small relative to the other; the design is nearly degenerate despite being full rank.