Fisher-Optimal Designs Under Sampling Constraints

Why Sampling Constraints Change the Design Problem

A sampling time can be statistically attractive but operationally unavailable. Clinic hours, participant burden, assay logistics, minimum spacing between draws, and prespecified visit schedules can all restrict when samples may be collected. A design that ignores those restrictions may identify a useful theoretical benchmark, but it is not a usable recommendation.

The task here is to optimize a Fisher-information objective over the set of designs that satisfy the actual timing rules. The information calculation must use the chosen NLME likelihood and the same parameterization and nominal parameter values throughout the comparison. In particular, population information should be calculated using the appropriate population-level likelihood; it should not be replaced automatically by a sum of conditional individual information matrices.

Definition: Let \(\mathcal D\) denote a sampling design, such as a set of times or a schedule assigned to each subject. Let \(\mathcal F\) be the set of designs satisfying all prespecified timing and operational constraints. A constrained Fisher-optimal design maximizes a stated information objective over \(\mathcal D\in\mathcal F\). For a D-optimal objective with nonsingular Fisher information matrix \(\mathcal I(\mathcal D)\), this means maximizing \(\det\{\mathcal I(\mathcal D)\}\), or equivalently \(\log\det\{\mathcal I(\mathcal D)\}\).

A constraint should be expressed in the form that will actually govern sampling. For example, a visit window can be written \(L_j\leq t_j\leq U_j\); a prohibited interval can be excluded from the permitted set; and a minimum gap can be written \(|t_j-t_k|\geq\Delta\). If visit order is fixed, use \(t_1\leq t_2\leq\cdots\) and express the relevant gaps as \(t_{j+1}-t_j\geq\Delta_j\). State whether interval endpoints are permitted: that detail can change the feasible set and the optimum.

The usual local Fisher-information interpretation also requires the model and its likelihood to be sufficiently differentiable at the nominal parameter values, and the matrix used for a full-parameter D-optimal design must be nonsingular. A singular matrix has determinant zero and cannot provide finite local precision in every parameter direction. If the design target is a subset of parameters in the presence of nuisance parameters, use the corresponding adjusted information consistently rather than silently changing the target during optimization.

Optimize Within the Feasible Set

Suppose the design objective is \(J(\mathcal D)=\log\det\{\mathcal I(\mathcal D)\}\). The constrained problem is

\max_{\mathcal D\in\mathcal F} J(\mathcal D).

This is not, in general, the same as finding an unconstrained optimum and then moving its times to nearby permitted values. Such a repair can change the information matrix in a way that is not captured by distance in time. The corrected design must be evaluated under the model, and all feasible alternatives should be compared using the stated objective.

For a continuously varying design, a basic justification for checking boundaries is the derivative identity

d\log\det\{\mathcal I\} = \operatorname{tr}\!\left(\mathcal I^{-1}d\mathcal I\right),

which follows from the determinant differential \(d\det(\mathcal I)=\det(\mathcal I)\operatorname{tr}(\mathcal I^{-1}d\mathcal I)\) for nonsingular \(\mathcal I\), followed by \(d\log u=du/u\). Thus, in an unconstrained interior optimum, the derivative of the objective with respect to each freely varying time must be zero. If a time is at an active bound, or a minimum-gap constraint is active, the allowed changes are restricted; the constrained optimum can therefore occur at a boundary even when the unconstrained derivative is not zero.

If \(\mathcal F\) is closed and bounded and the information objective is continuous and defined throughout it, a maximum is attained. Discrete schedules can be handled by enumerating feasible candidates. Continuous windows may require numerical constrained optimization, with careful checks at the bounds, at excluded-window edges, and where two timing constraints meet.

Worked Example: Choosing Between Discrete Two-Sample Schedules

Hypothetical example. Two samples must be selected from the nominal times \(0,2,4,6\) hours after dose. A clinic rule excludes 2 hours, and the samples must be separated by at least 3 hours. For a two-parameter target, suppose the model-based Fisher information matrices at nominal parameter values are as follows. The entries are illustrative information units; each matrix is recalculated for the complete schedule.

\mathcal I(0,4)= \begin{pmatrix}5&0\\0&5\end{pmatrix}, \qquad \mathcal I(0,6)= \begin{pmatrix}7&2\\2&3\end{pmatrix}.

After excluding 2 hours, the available times are \(0,4,6\). The candidate pairs are \((0,4)\), \((0,6)\), and \((4,6)\). The first has a 4-hour separation and is feasible. The second has a 6-hour separation and is also feasible. The third has a 2-hour separation, so it violates the minimum gap and must be rejected.

For \((0,4)\), the determinant is \(5(5)-0(0)=25\). For \((0,6)\), it is \(7(3)-2(2)=21-4=17\). Both matrices are positive definite: their leading diagonal entries are positive and their determinants are positive. Among the feasible pairs, \((0,4)\) has the larger determinant, so it is D-optimal for the stated local information objective. The comparison does not claim that the excluded pair is feasible, or that the ranking would be unchanged under a different model or parameter target.

Clinical Windows and Boundary Solutions

A clinical window is often better represented as an interval than as a single nominal time. Additional rules can make the feasible region disconnected: for example, a time may be allowed between 3 and 5 hours except during a period when a clinic is unavailable. A numerical optimizer must not step through a prohibited interval as if it were continuous and permitted.

Worked Example: Optimizing Within Windows With a Prohibited Visit Period

Hypothetical example. Two ordered sampling times, \(t_1\) and \(t_2\), are measured in hours after dose. The first must lie in \([0.5,2]\). The second must lie in \([3,5]\), but the open interval \((3.8,4.2)\) is unavailable; the endpoints 3.8 and 4.2 are allowed. The visits must be at least 2 hours apart. Suppose the illustrative local information matrix is

\mathcal I(t_1,t_2)= \begin{pmatrix} 4\exp\{-(t_1-1.8)^2\}&0\\ 0&9\exp\{-(t_2-3.95)^2\} \end{pmatrix}.

The matrix is positive definite at every permitted pair because both diagonal entries are positive. Its determinant is

\det\{\mathcal I(t_1,t_2)\} =36\exp\{-(t_1-1.8)^2-(t_2-3.95)^2\}.

The first factor is largest at \(t_1=1.8\), which lies within its window. For the second factor, the preferred time 3.95 is prohibited. Among the permitted \(t_2\) values, 3.8 is closest to 3.95: its distance is 0.15, compared with 0.25 at 4.2, and greater distances elsewhere. Thus \(t_2=3.8\) maximizes the second factor. The resulting pair meets the gap rule because \(3.8-1.8=2.0\) hours. Since each factor is individually maximized by this jointly feasible pair, it is the global constrained optimum.

The maximum determinant is \(36\exp(-0.15^2)=36\exp(-0.0225)\approx35.20\). A time of 3.95 hours is not a valid substitute, even though it maximizes the unconstrained second factor: it lies inside the prohibited interval.

Checking Feasibility Before Comparing Information

For discrete schedules, feasibility can be checked explicitly before computing or ranking objectives. When there are many candidate times, a systematic procedure prevents an attractive but invalid schedule from entering the comparison.

Write down the permitted times.
Include the visit windows, any excluded intervals, and whether endpoints are allowed.

Apply schedule rules.
Check visit order, required sample counts, minimum gaps, and any subject-level restrictions for every candidate.

Evaluate the information objective.
Use the same likelihood, nominal parameter values, parameterization, and target for every feasible candidate.

Verify the selected design.
Recheck every constraint and recalculate the objective at the reported times, including numerical boundary values.

Worked Example: Enumerating Feasible Three-Sample Schedules

Hypothetical example. Three samples must be taken from \(0,1,3,5,6\) hours after dose, with at least 2 hours between any consecutive samples. Enumerating the triples gives four feasible schedules: \((0,3,5)\), \((0,3,6)\), \((1,3,5)\), and \((1,3,6)\). For instance, \((0,1,3)\) is rejected because its first gap is only 1 hour; \((0,3,5)\) is feasible because its gaps are 3 and 2 hours.

Suppose the hypothetical model-based information matrices for the four feasible schedules are:

\begin{aligned} \mathcal I(0,3,5)&=\operatorname{diag}(8,4,2),\\ \mathcal I(0,3,6)&= \begin{pmatrix}6&1&0\\1&5&0\\0&0&2\end{pmatrix},\\ \mathcal I(1,3,5)&=\operatorname{diag}(5,6,2),\\ \mathcal I(1,3,6)&=\operatorname{diag}(5,5,2). \end{aligned}

Their determinants are respectively \(8(4)(2)=64\), \(2\{6(5)-1(1)\}=58\), \(5(6)(2)=60\), and \(5(5)(2)=50\). The second matrix is positive definite because its upper-left block has positive leading entry 6 and determinant 29, and its remaining diagonal entry is 2. The diagonal matrices are positive definite because all their diagonal entries are positive. Thus the feasible schedule with the largest determinant is \((0,3,5)\), with determinant 64. This conclusion follows from comparing all four feasible candidates, not from comparing only a convenient subset.

Diagnostics, Alternatives, and Limitations

Several checks help identify a constrained design calculation that has gone wrong:

Recheck feasibility independently. List the candidate times or test every inequality. Do not assume that a pair is invalid merely because another nearby pair violates a gap rule.
Inspect active constraints. Report whether a selected time is at a window edge, a prohibited-period edge, or a minimum-gap boundary. Small numerical tolerances can otherwise produce a reported time just inside a forbidden region.
Check matrix properties and numerical stability. Confirm that the information matrix is symmetric to numerical precision and is positive definite when a full-parameter determinant is being used. A near-zero determinant or a failed factorization may indicate weak identification, numerical error, or a singular design.
Check the optimizer against alternatives. For discrete sets, enumerate all feasible schedules when practicable. For continuous windows, compare multiple starting values and explicitly evaluate feasible boundaries and disconnected regions.
Keep the criterion fixed. Do not compare determinant values calculated under different parameter scalings, target parameters, likelihoods, or nominal values as if they were directly interchangeable.

Enumeration is exact for a modest prespecified schedule and gives a transparent audit trail. Continuous constrained optimization can explore a much larger timing space, but it depends on numerical optimization and constraint handling. A fine grid is useful for visualization or as a starting point, but may miss a narrow optimum or misrepresent a boundary. Projecting an unconstrained optimum onto a permitted time is a quick heuristic, not a general optimality result; it is acceptable only if the projected design is re-evaluated and compared with other feasible candidates.

A Fisher-optimal design is conditional on the assumed model, nominal parameter values, error structure, and information objective. It does not guarantee good finite-sample estimation, robustness to model misspecification, or operational acceptability beyond the constraints explicitly encoded. Constraints should therefore be defined with clinical and logistical input, and design performance should be interpreted within those assumptions.

Common pitfall: “The 4-hour visit is too close to 6 hours, so the only feasible pair is 0 and 4 hours.” This reasoning overlooks the pair \((0,6)\), whose separation is 6 hours and therefore satisfies a minimum gap of 3 hours. Check every candidate pair against the rule; then compare the information objective only among the feasible pairs.

Check Your Understanding

Answer each question, then compare with the solutions below.

Candidate times are 0, 2, 4, and 6 hours. If 2 hours is unavailable and the minimum separation is 3 hours, which pairs are feasible?
A time must lie in \([1,5]\), except that the closed interval \([3,4]\) is prohibited. Is 3 hours feasible? Is 4 hours feasible?
For two feasible designs, \(\det\{\mathcal I_A\}=18\) and \(\det\{\mathcal I_B\}=12\). Which is D-optimal for the stated target, and what does this comparison not establish?
Why is rounding an unconstrained optimum to the nearest available time not, by itself, a proof of constrained optimality?

Solutions. (1) The available times are 0, 4, and 6. The feasible pairs are \((0,4)\) and \((0,6)\); \((4,6)\) has a 2-hour gap and is infeasible. (2) Neither endpoint of the prohibited closed interval is allowed, so both 3 and 4 hours are infeasible. (3) Design A is D-optimal among the two stated feasible candidates because 18 exceeds 12; this does not establish that it is best under a different target, model, or parameterization. (4) Rounding can change the information objective and may not produce the best feasible design; the rounded schedule must be checked for feasibility and compared with other feasible candidates.