Why Allocate Participants Across Sampling Schedules?
At an interim analysis, some participants may already have contributed data, while others are yet to be assigned a sampling schedule. Different schedules can inform different parameters to different degrees: an early sample may help characterize distribution, while a later sample may better distinguish elimination. The allocation question is therefore not simply how many participants remain, but which feasible assignment is expected to improve precision for the parameters that matter.
The design comparison is conditional on the interim information and on a model for the observations. It does not guarantee that the eventual estimates will be precise: realized data can be more or less informative than expected. The calculation is a principled way to compare planned allocations before the remaining participants are observed.
Information and a Precision Criterion
Suppose \(K\) candidate sampling schedules are available. A schedule specifies the planned sampling times for one participant. Let \(J_k\) be the expected Fisher information matrix contributed by one participant following schedule \(k\), evaluated under the model and parameter values or parameter distribution chosen for the design calculation. Let \(n_k\) be the integer number of remaining participants assigned to that schedule. Let \(I_0\) denote the information already available at the interim analysis. Under independent participants and a correctly specified likelihood, expected information adds across participants:
Here \(\mathbf n=(n_1,\ldots,n_K)\). The matrices must refer to the same parameterization and information convention. If observations within a participant are correlated, the schedule-specific matrix \(J_k\) must account for that joint observation model; it is not generally valid to add the information from each time point as if those observations were independent.
One common choice is weighted A-optimality. For a positive semidefinite weight matrix \(W\), define
When \(I(\mathbf n)\) is nonsingular and the usual local large-sample approximation is suitable, \(I(\mathbf n)^{-1}\) approximates the covariance matrix of the parameter estimator. The criterion is then a weighted sum of approximate parameter variances. With \(W\) equal to the identity matrix, it sums the variances on the chosen parameter scale; a different \(W\) can emphasize selected parameters or units. Smaller \(\Phi\) means better precision according to this specific criterion, not according to every possible design objective.
The allocation must also satisfy operational and scientific constraints. A basic constraint is \(\sum_k n_k=N\), where \(N\) is the number of remaining participants. Other constraints may specify minimum or maximum numbers on a schedule, requirements within patient strata, or schedules that are unavailable for particular participants. The decision set is discrete: participants cannot ordinarily be divided fractionally across schedules.
Why the Criterion Rewards Information
For a short justification, consider a continuous relaxation in which \(n_k\) can vary smoothly, and write \(I=I(\mathbf n)\). For a small increase \(dn_k\), the information changes by \(dI=J_k\,dn_k\). Differentiating \(I I^{-1}=I_d\), where \(I_d\) is the identity matrix, gives \(d(I^{-1})=-I^{-1}(dI)I^{-1}\). Therefore,
The negative derivative indicates that adding information in a useful direction tends to reduce the criterion. The quantity \(\operatorname{tr}(WI^{-1}J_kI^{-1})\) is a local, continuous approximation to the precision gain from one more participant on schedule \(k\). It can help explain a proposed allocation, but it is not a complete integer-allocation algorithm: the exact gain depends on the current allocation, and constraints can rule out the locally attractive choice. In practice, compare the criterion over feasible integer allocations or use an integer optimization method that evaluates the stated objective.
Worked Example: Balancing Information Between Two Parameters
Suppose four remaining participants can each follow schedule A or B. Under a hypothetical model, the interim information and per-participant schedule information, for two parameters on their stated scales, are
Assume independent participants, these matrices are the appropriate expected information at the interim analysis, and the objective is the unweighted A-criterion \(W=I_d\). If \(a\) participants receive A, then \(4-a\) receive B, for integer \(a=0,1,2,3,4\). The total information is
Because this matrix is diagonal, \(\Phi(a)=1/(3+1.5a)+1/(9-1.5a)\). For example, at \(a=2\), the two diagonal information values are both \(6\), so \(\Phi(2)=1/6+1/6=0.3333\). Evaluating all feasible choices gives:
| Participants on A, \(a\) | Information diagonal | \(\Phi(a)\) |
|---|---|---|
| 0 | (3, 9) | 0.4444 |
| 1 | (4.5, 7.5) | 0.3556 |
| 2 | (6, 6) | 0.3333 |
| 3 | (7.5, 4.5) | 0.3556 |
| 4 | (9, 3) | 0.4444 |
The allocation of two participants to each schedule minimizes this criterion. The calculation illustrates why assigning everyone to the schedule with the largest information for one parameter need not be optimal: the objective accounts for precision in both parameters.
Compare Feasible Allocations, Not Just Schedules
A schedule’s information contribution is not itself the decision. Allocation is a comparison of complete feasible participant assignments, including the information already accumulated. If participants have different eligibility, covariates, or allowable schedules, the information contribution can depend on the participant. In that case, represent the assignment at the participant level or use schedule-specific matrices appropriate to each subgroup; do not apply a single \(J_k\) indiscriminately.
Worked Example: Checking Candidate Allocations
Consider four hypothetical candidate allocation patterns for four remaining participants. Each pattern assigns all four participants to one candidate schedule; the operational plan makes these four patterns the feasible choices. The interim information is zero on the two-parameter scale used here. Suppose the per-participant information matrices for the four schedules are diagonal, with entries \((1,2)\), \((1.4,1.275)\), \((1.6,1.05)\), and \((1.8,0.825)\). Assume independent participants and use the unweighted A-criterion.
For a pattern assigning all four participants to a schedule with per-participant diagonal information \((j_1,j_2)\), total information is \(\operatorname{diag}(4j_1,4j_2)\) and the criterion is \(1/(4j_1)+1/(4j_2)\). Thus the four candidate values are:
- Pattern 0: \(1/4+1/8=0.3750\).
- Pattern 1: \(1/5.6+1/5.1=0.1785714286+0.1960784314=0.3746498600\), or 0.3746 to four decimal places.
- Pattern 2: \(1/6.4+1/4.2=0.15625+0.2380952=0.3943452\), or 0.3943 to four decimal places.
- Pattern 3: \(1/7.2+1/3.3=0.1388889+0.3030303=0.4419192\), or 0.4419 to four decimal places.
Pattern 1 has the lowest criterion, narrowly improving on Pattern 0. This conclusion is limited to these feasible choices and the specified information matrices and criterion; it does not show that Pattern 1 is best under a different parameter weighting or another design objective.
Respecting Participant-Level Constraints
Constraints can change the best feasible allocation. For instance, a schedule may require a minimum number of participants because it is needed to estimate a parameter, while a maximum may reflect burden or a limited number of available sampling slots. Constraints should be written before ranking designs and included in the optimization, rather than applied informally after selecting the unconstrained minimum.
Worked Example: Applying Minimum and Maximum Allocation Rules
Suppose five hypothetical participants remain and can follow schedule A or B. At least two must receive A, at least one must receive B, and no more than four may receive A. Thus \(n_A+n_B=5\), \(2\leq n_A\leq4\), and \(n_B\geq1\). Assume independent participants, interim information \(I_0=\operatorname{diag}(2,1)\), per-participant information \(J_A=\operatorname{diag}(1.5,0.25)\), and \(J_B=\operatorname{diag}(0.25,1)\). Use the unweighted A-criterion.
The constraints leave \(n_A=2,3,4\), with \(n_B=5-n_A\). For \(n_A=2\), the total information diagonal is \((2+2(1.5)+3(0.25),\,1+2(0.25)+3(1))=(5.75,4.5)\). Therefore, \(\Phi=1/5.75+1/4.5=0.173913+0.222222=0.3961\).
For \(n_A=3\), the diagonal is \((7,3.75)\), giving \(\Phi=1/7+1/3.75=0.142857+0.266667=0.4095\). For \(n_A=4\), the diagonal is \((8.25,3)\), giving \(\Phi=1/8.25+1/3=0.121212+0.333333=0.4545\). The feasible minimum is \(n_A=2,n_B=3\). More A participants would improve the first parameter’s information but reduce precision for the second enough to worsen the selected overall criterion.
Assumptions, Diagnostics, and Alternatives
The calculation requires a model that can supply schedule-specific expected information and an interim information contribution on a common parameter scale. The likelihood or observation model must represent the sampling process adequately, including residual error and within-participant dependence. The usual interpretation of inverse information as approximate estimator covariance also relies on regularity conditions such as local identifiability, an interior parameter value, and sufficient information for a useful large-sample approximation. If the total information matrix is singular or nearly singular, the inverse is undefined or unstable; the stated precision criterion cannot be trusted without resolving that issue.
Diagnostics should examine more than the final scalar criterion. Check the eigenvalues or condition number of total information, inspect parameter-specific variances and correlations, and confirm that all candidates use the same parameterization, weights, and information convention. Verify each candidate satisfies the integer and participant-level constraints. Where interim parameter uncertainty is material, a single plug-in parameter value can produce a fragile ranking. One alternative is to evaluate the criterion across plausible parameter scenarios and average or otherwise summarize the resulting values using a prespecified rule. The scenario weights and summary rule are part of the design objective and should not be selected after seeing which allocation wins.
A-optimality is not the only defensible criterion. A determinant-based criterion emphasizes overall covariance volume, while a criterion based on the largest eigenvalue targets the worst-informed direction. A weighted A-criterion can focus the objective on parameters important to the scientific question. These approaches can rank allocations differently. Equal allocation is simple and may be operationally robust, but it does not generally minimize a chosen precision criterion. The appropriate comparison depends on which parameter uncertainties matter and on the costs and feasibility of each schedule.
A common mistake is to assign every remaining participant to the schedule with the greatest information for the parameter currently estimated least precisely. That reasoning ignores how information for other parameters changes, how information combines with the interim data, and whether the schedule satisfies constraints. The correction is to compute the stated criterion for each feasible allocation using the full information matrix, then choose the feasible allocation with the smallest value. Another pitfall is to call the resulting criterion a guarantee of future precision: it is a model-based expectation or approximation, not a promise about realized data.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Why must the information from multiple sampling times within one participant account for their joint observation model?
- In the first worked example, what is the criterion for allocating three participants to A and one to B, and which allocation has the smaller criterion: this allocation or two on each schedule?
- Why does a schedule with the largest information for one parameter not necessarily minimize the weighted A-criterion?
- What does a nearly singular total information matrix indicate for the proposed precision calculation?
- If interim parameter uncertainty is substantial, name one way to reduce reliance on a single plug-in value.
Solutions. (1) Measurements within a participant can be correlated, so treating them as independent can misstate the schedule’s information; \(J_k\) should reflect their joint model. (2) At three on A, the information diagonal is \((7.5,4.5)\), so the criterion is \(1/7.5+1/4.5=0.133333+0.222222=0.3556\). Two on each gives 0.3333, which is smaller. (3) The criterion combines uncertainty across the weighted parameters, so a gain for one parameter may be outweighed by a loss for another. (4) It indicates weakly informed parameter direction(s); inversion can be unstable, making the reported criterion unreliable. (5) Evaluate candidate allocations across plausible parameter scenarios and summarize the criterion using a prespecified rule.