Why the Allocation Must Be Made Feasible
An optimal-design calculation may assign a fraction of the samples to each candidate time. That representation is mathematically convenient: it allows continuous optimization over allocation weights. A person, however, cannot provide 0.6 of a blood sample at one time and 1.4 at another. The schedule for one individual must use whole-number sample counts, and those counts must add to the planned total. The distinction matters when a proposed fractional allocation is translated into an actual individual PK schedule.
This tutorial compares approximate designs, which use fractional weights, with exact designs, which use integer sample counts. The goal is not to assume that rounding automatically preserves the design's properties. It is to understand the information being approximated, make a feasible schedule, and check what was gained or lost in the conversion.
Information Under an Exact or Approximate Design
Suppose there are \(m\) candidate sampling times \(t_1,\ldots,t_m\), and exactly \(n\) samples are to be collected from each individual. For a local PK design, calculate sensitivities at a specified nominal parameter vector \(\theta\). Let \(s(t_j;\theta)\) be the column vector of derivatives of the predicted concentration with respect to the model parameters at time \(t_j\). To make the information calculation explicit, first assume independent observations with normally distributed errors, constant known variance \(\sigma^2\), and a differentiable mean concentration model \(g(t;\theta)\). Thus \(Y_j\sim N(g(t_j;\theta),\sigma^2)\), and \(s(t_j;\theta)=\partial g(t_j;\theta)/\partial\theta\).
Under these assumptions, the information contribution from one observation at \(t_j\) is \(M(t_j)=s(t_j;\theta)s(t_j;\theta)^\mathsf{T}/\sigma^2\). If a different known error variance is used at each time, the corresponding variance belongs in that time's information contribution instead. The allocation definitions below do not depend on choosing constant variance, but the information matrices and comparisons must use the error model that is actually assumed.
The exact information formula adds the contribution from every observation. If a time is sampled more than once, its information contribution is multiplied by its count, provided the observations meet the stated independence and error assumptions. The approximate formula represents the same sum using proportions: if the counts happened to satisfy \(n_j=nw_j\) for every \(j\), the two information matrices would be identical. Approximate designs are useful precisely because their weights need not make those products integers.
For a model with \(p\) parameters, a common comparison is the D-criterion, \(\det(I)\), or its numerically more stable equivalent, \(\log\det(I)\). A larger determinant corresponds to a smaller generalized volume for the local parameter uncertainty approximation, when the information matrix is positive definite. A singular matrix has determinant zero and \(\log\det(I)=-\infty\); it cannot provide finite local information in every parameter direction. For schedules with the same \(n\), multiplying a per-observation information matrix by \(n\) scales its determinant by \(n^p\), so this common scale does not change which allocation has the larger D-criterion.
Why Integer Counts Can Differ From Fractional Weights
The approximate design enlarges the set of allowed allocations: it includes every feasible integer schedule, but also allocations that cannot be implemented for one individual. Therefore, if the approximate design is optimized over the same candidate times and under the same assumptions, its best criterion value cannot be worse than the best exact value. This follows because every exact design is also an approximate design with weights \(w_j=n_j/n\). The converse does not follow: an optimal set of weights may not correspond to integer counts at the planned sample size.
This relationship gives a useful benchmark, not a guarantee that rounding a particular weight vector will preserve its criterion. The claim that the optimized approximate criterion bounds the optimized exact criterion applies only when both problems use the same candidate times, sample total, model, nominal parameter values, and objective. A fractional allocation chosen for practical or scientific reasons need not be the optimized approximate design, and its criterion need not bound that of a feasible schedule.
Worked Example: Measuring the Gap at Four Samples
Consider a hypothetical two-parameter local model with three candidate times. For illustration, suppose the standardized sensitivity vectors \(s(t_j;\theta)/\sigma\) are \((1,0)^\mathsf{T}\), \((1,1)^\mathsf{T}\), and \((0,1)^\mathsf{T}\). Each time's information contribution is the outer product of its standardized sensitivity vector with itself. Plan \(n=4\) observations.
An approximate design with equal weights \(w_1=w_2=w_3=1/3\) has per-observation matrix \(M(w)=\begin{pmatrix}2/3&1/3\\1/3&2/3\end{pmatrix}\). Thus its total information determinant is \(\det(4M(w))=4^2\det(M(w))=16(4/9-1/9)=16/3\approx5.333\).
The corresponding fractional counts are \(4/3\) at each time, which are not feasible. One nearby exact schedule is \((1,2,1)\). Its total information is \(I_{\mathrm{exact}}=\begin{pmatrix}1&0\\0&0\end{pmatrix}+2\begin{pmatrix}1&1\\1&1\end{pmatrix}+\begin{pmatrix}0&0\\0&1\end{pmatrix}=\begin{pmatrix}3&2\\2&3\end{pmatrix}\). Its determinant is \(3(3)-2(2)=5\).
The determinant ratio for this schedule relative to the fractional design is \(5/(16/3)=0.9375\), a 6.25% decrease in determinant. For two parameters, D-efficiency is the square root of this ratio: \(\sqrt{0.9375}=0.9682\), or 96.82%. This is a close, but not identical, conversion. In this example, schedules \((2,1,1)\) and \((1,1,2)\) also have determinant 5. The fractional optimum is a benchmark; integer schedules cannot attain its determinant at \(n=4\).
Converting Weights Into a Feasible Schedule
A simple conversion starts by calculating the target counts \(nw_j\), taking their integer parts, then assigning the unallocated samples to the largest fractional remainders. This largest-remainder rule ensures that the final counts add to \(n\), and usually keeps each count near its fractional target. It is a bookkeeping method, not an optimality theorem: the schedule it produces need not maximize the chosen information criterion among feasible schedules.
A more defensible conversion treats the rounded counts as a starting point. Check that the sample total is correct, calculate the exact information matrix for the resulting schedule, and compare it with other plausible integer allocations. In a small candidate set, enumeration of feasible counts may be practical. Otherwise, one can consider nearby schedules formed by moving a sample from one time to another, retaining only changes that respect the sampling constraints. The objective is to choose a feasible schedule with good information, not to preserve every fractional weight exactly.
Worked Example: Rounding and Checking a Three-Sample Allocation
For a hypothetical two-parameter PK design, suppose three candidate times have standardized sensitivities \((1,0)^\mathsf{T}\), \((1,1)^\mathsf{T}\), and \((0,2)^\mathsf{T}\), respectively. Consider the proposed approximate weights \((0.20,0.60,0.20)\) with \(n=3\). These weights may reflect a desired allocation across phases; they are not asserted to be the approximate D-optimum.
The target counts are \(3w=(0.60,1.80,0.60)\). Taking integer parts gives \((0,1,0)\), leaving two samples to allocate. The two largest fractional remainders are tied at 0.60, so one largest-remainder schedule is \((1,1,1)\). Its information matrix is \(I=\begin{pmatrix}1&0\\0&0\end{pmatrix}+\begin{pmatrix}1&1\\1&1\end{pmatrix}+\begin{pmatrix}0&0\\0&4\end{pmatrix}=\begin{pmatrix}2&1\\1&5\end{pmatrix}\), with determinant \(2(5)-1=9\).
For comparison, the fractional per-observation matrix is \(M(w)=0.20\begin{pmatrix}1&0\\0&0\end{pmatrix}+0.60\begin{pmatrix}1&1\\1&1\end{pmatrix}+0.20\begin{pmatrix}0&0\\0&4\end{pmatrix}=\begin{pmatrix}0.80&0.60\\0.60&1.40\end{pmatrix}\). Its determinant is \(0.80(1.40)-(0.60)^2=0.76\), so the total approximate determinant is \(3^2(0.76)=6.84\). The integer schedule's determinant is higher than this particular fractional design's value. That is possible because the proposed weights were not specified as the optimized approximate design. The arithmetic demonstrates why the criterion must be calculated after conversion instead of assuming that rounding always reduces or preserves it.
Rank Checks and Conversion Failure
A conversion can satisfy the sample budget and still fail as a design. In particular, rounded samples may all fall at times with sensitivities in the same parameter direction. Then the information matrix is singular even if the fractional matrix was nonsingular. Rank is therefore a basic diagnostic alongside the determinant: inspect whether the chosen times jointly inform every parameter direction in the local model.
Worked Example: A Rounded Schedule Becomes Singular
Suppose a hypothetical two-parameter design has standardized sensitivity vectors \(A=(1,0)^\mathsf{T}\), \(B=(2,0)^\mathsf{T}\), and \(C=(0,1)^\mathsf{T}\), and plans \(n=2\) samples. A proposed approximate allocation is \((w_A,w_B,w_C)=(0.40,0.30,0.30)\). The target counts are \((0.80,0.60,0.60)\). Under a tie-breaking rule that assigns the two remaining samples to the largest remainders, rounding could produce \((1,1,0)\).
That schedule has \(I=\begin{pmatrix}1&0\\0&0\end{pmatrix}+\begin{pmatrix}4&0\\0&0\end{pmatrix}=\begin{pmatrix}5&0\\0&0\end{pmatrix}\), so \(\det(I)=0\). The two samples provide no information in the second parameter direction. By contrast, the fractional matrix for two planned observations is \(I_{\mathrm{approx}}=2\left[0.40\begin{pmatrix}1&0\\0&0\end{pmatrix}+0.30\begin{pmatrix}4&0\\0&0\end{pmatrix}+0.30\begin{pmatrix}0&0\\0&1\end{pmatrix}\right]=\begin{pmatrix}3.20&0\\0&0.60\end{pmatrix}\), with determinant \(1.92\).
A feasible repair is \((0,1,1)\), allocating one sample to \(B\) and one to \(C\). It gives \(I=\begin{pmatrix}4&0\\0&1\end{pmatrix}\), with determinant 4 and full rank. This repair does not follow the largest-remainder allocation, but it prevents the singularity and improves the D-criterion among the schedules compared. If other clinical or operational requirements constrain the choice, those must also be applied before selecting the final schedule.
Checks, Alternatives, and Limitations
Before accepting a converted schedule, verify the assumptions and the calculation in context. The nominal PK parameters used for sensitivities should be stated, since individual local information depends on them. The assumed observation error model must be appropriate for constructing \(M(t_j)\), and the calculation must use the same model and parameterization when comparing schedules. Check that counts are nonnegative integers summing to \(n\), that the information matrix has adequate rank, and that the chosen schedule meets any required sampling rules. A numerically large determinant does not compensate for a violated constraint or a poorly justified nominal model.
The main approaches have different roles:
| Approach | Strength | Limitation |
|---|---|---|
| Optimize approximate weights | Convenient continuous allocation; useful for identifying promising times and proportions. | May require fractional counts and may not be attainable at the planned sample size. |
| Round weights to integer counts | Fast and easy to implement as a first conversion. | Rounding can alter the criterion, break a tie unfavorably, or produce a singular schedule. |
| Compare feasible exact schedules | Evaluates allocations that can actually be collected and can check rank directly. | Can be computationally demanding when the candidate set or sample budget is large. |
A frequent mistake is to interpret \(w_j=0.30\) as a prescription to take exactly 30% of a single individual's sample at time \(t_j\), or to treat \(nw_j\) as an integer merely because the total number of samples is fixed. The correction is to regard weights as fractional design proportions, convert them to integer counts whose sum is exactly \(n\), and then recalculate the information for that schedule. Another mistake is to assume the rounded schedule is optimal because its weights are close to the approximate weights. Proximity in counts does not prove optimality, and a rank check can reveal a serious failure that a rounding rule alone misses.
The comparison is local: information is calculated at specified nominal parameters, under a specified model and error structure. A schedule that is favorable by the local D-criterion is not guaranteed to be best for every plausible individual or parameter value. Nor does conversion alone resolve the choice of candidate times or operational restrictions; it addresses the allocation between the specified candidate times. The useful conclusion is narrower and practical: an approximate design can guide an exact schedule, but feasibility and information must be checked at the integer level.
Check Your Understanding
Use the definitions and calculations above to answer each question. Brief solutions are included so you can check both the arithmetic and the interpretation.
- A plan has \(n=7\) samples and approximate weights \((0.20,0.50,0.30)\). What are the fractional target counts, and why are they not yet an exact schedule?
- For two parameters, an exact information matrix is \(\begin{pmatrix}4&1\\1&3\end{pmatrix}\). Calculate its determinant and state whether it is singular.
- An approximate design has total determinant 12. An exact schedule has total determinant 10. For a two-parameter model, calculate the determinant ratio and D-efficiency of the exact schedule relative to that approximate design.
- Why can a rounded schedule have determinant zero even when its approximate design has a positive determinant?
Solutions. (1) The target counts are \(7(0.20,0.50,0.30)=(1.4,3.5,2.1)\). They sum to seven but are fractional, so they cannot be collected as counts; an integer allocation summing to seven is still required. (2) The determinant is \(4(3)-1(1)=11\), which is positive, so the matrix is nonsingular. (3) The determinant ratio is \(10/12=0.8333\). D-efficiency is \((10/12)^{1/2}=0.9129\), or 91.29%. It compares the determinant-based information for these two matrices; it does not state a probability. (4) Rounding may place all samples at times whose sensitivity vectors lie in a lower-dimensional subspace. The resulting exact matrix then lacks rank, even though fractional contributions at additional times made the approximate matrix full rank.