Why Sampling Cost Changes the Design
A sampling time can be statistically attractive but operationally expensive. A concentration collected at a routine clinic visit may cost less than one requiring an extra visit, an overnight stay, or a specialized assay. When those costs differ, choosing the times that maximize information for a fixed number of samples is not necessarily the same as choosing the times that maximize information for a fixed budget.
The design question is therefore two-part: which times should be sampled, and how many samples should be assigned to each? In an individual PK design, the answer depends on the model’s local sensitivities, the residual error model, the parameter or parameters of interest, and the costs assigned to candidate measurements. The cost constraint changes which schedules are feasible; it does not replace the statistical design criterion.
The inequality allows unused budget. If the operational rule requires spending the entire budget, the constraint can instead be written as equality, but exact equality may rule out otherwise useful schedules when costs are discrete. Costs may represent money, staff time, participant burden, or a combined resource, provided they are defined consistently.
Information Under a Cost Constraint
For a precise formulation, consider an individual PK model with observation \(Y_{ij}\) at candidate time \(t_j\). Write its model-predicted concentration as \(f(t_j;\theta)\), where \(\theta\) is the vector of individual parameters being estimated. For the examples below, assume independent observations with normally distributed residual error of known variance \(\sigma^2\):
Assume the model and residual error specification are adequate, the sensitivities are evaluated at a stated nominal parameter value, and each measurement has the same variance unless otherwise stated. Define the sensitivity vector \(g_j=\partial f(t_j;\theta)/\partial\theta\). Under these assumptions, one observation at \(t_j\) contributes \(g_jg_j^{\mathsf T}/\sigma^2\) to the Fisher information matrix. Independent observations add, so a schedule has information matrix
If residual variance depends on time or the predicted concentration, the corresponding variance-weighted information contribution must be used instead; the formula above is not valid unchanged for every residual error model. In either case, the cost constraint is applied to the number of observations, while the design criterion is calculated from their information contributions.
For estimating the full parameter vector, a common criterion is D-optimality: among feasible schedules, maximize \(\det(I)\), or equivalently \(\log\det(I)\) when \(I\) is positive definite. Larger determinant corresponds to a smaller generalized volume of the local parameter uncertainty ellipsoid. For a particular linear combination \(c^{\mathsf T}\theta\), c-optimality instead minimizes \(c^{\mathsf T}I^{-1}c\), provided \(I\) is nonsingular. These criteria can favor different schedules even under the same budget.
Formulating the Optimization
List the allowable times \(t_j\) and the cost \(c_j\) of each measurement, including any relevant operational restrictions.
Use the PK model, residual error model, and nominal parameter values to calculate each candidate time’s information contribution.
State whether the aim is, for example, full-vector precision through D-optimality or precision for a target \(c^{\mathsf T}\theta\) through c-optimality.
Optimize the chosen criterion over nonnegative integer counts satisfying the total-budget constraint, then check that the result is not an artifact of omitted constraints or assumptions.
Worked Example: A Scalar Parameter and Three Unequal Costs
Worked Example: A Scalar Parameter and Three Unequal Costs
Suppose three candidate PK sampling times, A, B, and C, have costs of 1, 2, and 3 budget units per sample. Their local information contributions for one scalar parameter are 4, 9, and 6.25 information units, respectively. Assume independent measurements, these contributions remain constant over replicates, and the budget is 6 units. Which schedule maximizes total information?
For a scalar parameter, independent information contributions add. If \(n_A,n_B,n_C\) are the sample counts, the objective and constraint are
First consider schedules with no C sample. For \(n_B=0,1,2,3\), spending the remaining budget on A gives total information \(24,25,26,27\), respectively: the corresponding schedules are six A samples; one B and four A; two B and two A; or three B samples. Thus the best schedule without C is three B samples, costing 6 units and providing \(3(9)=27\) information units.
If there is one C sample, it costs 3 units and leaves 3. The best use of those remaining units is one B sample and one A sample, for total information
If there are two C samples, they use the whole budget and provide \(2(6.25)=12.5\) information units. Therefore, the best schedule containing C provides 19.25 units, still below the 27 units from three B samples. The optimal fixed-budget schedule is three B samples.
The conclusion is not that C is uninformative: it provides 6.25 information units per sample. B provides the most information per sample, while C’s cost is high relative to its information contribution in this particular example. The comparison uses total information under the budget, not information per sample alone.
Worked Example: Fixed Sample Count Versus Fixed Budget
Worked Example: Fixed Sample Count Versus Fixed Budget
Consider a two-parameter local model with independent unit-variance observations at three candidate times. Let the sensitivity vectors be \(g_E=(1,0)^{\mathsf T}\), \(g_M=(1,1)^{\mathsf T}\), and \(g_L=(0,1)^{\mathsf T}\). The costs are 1, 2, and 1 budget units for E, M, and L. Which schedules are D-optimal with exactly two samples, and which are D-optimal under a budget of 4?
Because the residual variance is 1, each observation contributes \(g_jg_j^{\mathsf T}\). With exactly two samples, taking one E and one L gives \(I\) equal to the identity matrix and \(\det(I)=1\). One E and one M gives \(I=\begin{pmatrix}2&1\\1&1\end{pmatrix}\), whose determinant is \(2(1)-1(1)=1\). One M and one L likewise has determinant 1. Two observations at the same time give rank-one information matrices and determinant 0. Thus the best two-sample schedules have determinant 1.
Now let \(e,m,l\) denote the counts at E, M, and L. The information matrix and determinant are
If \(m=0\), then \(e+l\leq4\), and \(\det(I)=el\). The largest product is 4, achieved by \(e=l=2\). If \(m=1\), the remaining budget allows \(e+l\leq2\), so \(\det(I)=e+l+el\); its maximum is 3 at \(e=l=1\). If \(m=2\), no budget remains for E or L and the determinant is 0. The budget-optimal schedule is therefore two E and two L samples, with determinant 4. It uses four samples rather than two, but respects the same total budget of 4 units.
This example shows why equal-count and fixed-budget questions are different. The equal-count rule compares schedules with the same number of observations, while the budget rule allows the number to vary according to cost. It also shows why a “most informative time per cost” shortcut can fail for a matrix criterion: the determinant depends on complementary information across parameter directions.
Worked Example: The Target Parameter Changes the Choice
Worked Example: The Target Parameter Changes the Choice
Use the same candidate sensitivities and costs as in the previous example, but suppose the target is the first parameter, so \(c=(1,0)^{\mathsf T}\). Under the budget of 4, which schedule minimizes the local variance approximation \(c^{\mathsf T}I^{-1}c\)?
For \(I=\begin{pmatrix}e+m&m\\m&m+l\end{pmatrix}\), the first diagonal entry of \(I^{-1}\) is \((m+l)/\det(I)\), when \(I\) is nonsingular. If \(m=0\), both E and L are needed for full rank, and the target variance is \(1/e\). With \(e+l\leq4\) and \(l\geq1\), the largest possible \(e\) is 3, giving \(e=3,l=1\) and variance \(1/3\).
If \(m=1\), the remaining budget permits \(e+l\leq2\). The nonsingular possibilities \((e,l)=(2,0),(1,1),(0,2)\) give variances \(1/2,2/3,3/2\), respectively. Cases with fewer E or L samples do not improve on these values. If \(m=2\), the budget is exhausted and the information matrix is singular, so the two parameters cannot both be estimated with finite local variance. The c-optimal schedule is three E samples and one L sample, with target variance approximation \(1/3\).
The D-optimal schedule in the previous example was two E and two L samples. It is not c-optimal for this target: reallocating one L sample to E reduces the approximate variance of the first parameter while retaining enough information to estimate the second. A design criterion must match the scientific estimation objective.
Derivation and Practical Checks
The information formula used above follows directly from the normal observation model. For one observation with mean \(f(t;\theta)\) and known variance \(\sigma^2\), the log likelihood, apart from a constant, is \(-[Y-f(t;\theta)]^2/(2\sigma^2)\). Its score vector is \([Y-f(t;\theta)]g/\sigma^2\). Taking the expected outer product of the score and using \(E\{[Y-f(t;\theta)]^2\}=\sigma^2\) gives \(gg^{\mathsf T}/\sigma^2\). Independent observations have additive log likelihoods and hence additive information. Repeating a candidate measurement \(n_j\) times multiplies its contribution by \(n_j\).
The inverse information matrix is a local approximation to parameter covariance under regularity conditions, a correctly specified model, and an identifiable design. It is not a guarantee that a nonlinear estimation algorithm will converge or that finite-sample uncertainty will exactly match the approximation. In particular, the information matrix must have full rank to support finite uncertainty for all parameters; a high objective value from a nearly singular matrix can still indicate unstable estimation.
Practical checks include verifying each candidate’s cost and information contribution, checking the budget arithmetic, inspecting rank and parameter correlations, and repeating the optimization under plausible changes to nominal parameter values and costs. If visit costs include a fixed setup charge, if costs depend on the schedule, or if measurements share correlated errors, the simple additive-cost or additive-information formulation must be adapted. Feasibility constraints such as minimum spacing or clinic windows must also be applied alongside the budget.
Alternatives, Limitations, and a Common Pitfall
An equal-count design is appropriate when the number of samples is fixed and costs are irrelevant or effectively equal. A fixed-budget design is more appropriate when resource use is the binding constraint. A cost-per-information ranking can be a useful screening device for a single scalar target with additive information, but it is not generally an exact solution for D-optimal or c-optimal matrix problems. In those problems, one observation can be valuable because it complements measurements in another parameter direction, not simply because it supplies a large amount of information by itself.
For a small candidate set and modest budget, direct enumeration of integer schedules is transparent and reliable. Larger problems may require integer optimization or carefully checked numerical search. A continuous relaxation, which allows fractional counts, can provide a bound or starting point, but the final design must have integer counts and satisfy the operational constraints. The locally optimal schedule can also change when nominal PK parameters or the assumed residual error model change.
- Pitfall: “Choose the time with the most information per cost, then spend the budget there.” This ignores whether the design remains identifiable and whether measurements complement one another.
- Correction: Optimize the stated design criterion over the complete set of feasible integer schedules. For a D-optimal objective, compare determinants (or log determinants); for a target parameter, compare the relevant \(c^{\mathsf T}I^{-1}c\).
- Pitfall: “The schedule with the largest information per sample must be best under a budget.” A costly measurement can displace a better combination of cheaper measurements, as the first worked example demonstrates.
Check Your Understanding
Work through each question, then check the solution immediately below it.
- A candidate sample costs 2 units and contributes 5 information units for a scalar parameter. What information does one sample provide per cost unit, and why is that ratio not enough to establish its value in a multivariate D-optimal design?
Solution: The ratio is \(5/2=2.5\) information units per cost unit. In a multivariate design, the value also depends on which parameter directions the observation informs and on how it complements the existing information matrix; the ratio alone does not determine the determinant. - In the two-parameter example, calculate the determinant for one E sample and one L sample. What does its positive value imply here?
Solution: The information matrix is the identity, so its determinant is \(1(1)-0(0)=1\). It is positive, indicating full rank for these two parameters under the example’s model and assumptions. - With the three scalar options from the first worked example and budget 6, what is the best schedule that contains exactly one C sample?
Solution: C uses 3 units, leaving 3. One B and one A use those units and provide \(6.25+9+4=19.25\) information units. This exceeds the alternatives of three A samples (12) or one B plus one A only among those remaining units; it is the best completion after one C. - Why can a D-optimal schedule and a c-optimal schedule differ even when they use the same candidate times and budget?
Solution: D-optimality targets joint precision for the full parameter vector by maximizing \(\det(I)\). c-optimality targets the variance of a particular linear combination by minimizing \(c^{\mathsf T}I^{-1}c\). Allocating samples to improve one parameter direction can improve the target variance without maximizing overall information volume. - Suppose a proposed schedule has a singular information matrix. Can it be called D-optimal for estimating the full parameter vector under the usual determinant criterion?
Solution: Not if a feasible full-rank schedule exists: the singular matrix has determinant zero and cannot provide finite local uncertainty for every parameter. If every feasible design is singular, the candidate set or budget is inadequate for full-vector estimation, and the design objective needs reconsideration rather than interpreting the singular schedule as adequate.