Encoding Candidate Sampling Designs

Why the Design Must Be Explicit

A simulation can generate plausible drug concentrations and still answer the wrong design question if its records do not reflect the proposed sampling plan. For example, a schedule with six samples per participant is not equivalent to one with three, even if both schedules contain the same early and late time points across the study as a whole. The simulation must encode not only which times are considered, but also which participant is scheduled at each time.

The task here is to represent candidate sampling designs: the planned sampling times, the number of participants assigned to each schedule or group, and any rule that determines that assignment. A sampling design specifies intended collection. Whether a scheduled sample is ultimately observed is a separate issue: the schedule should remain identifiable even when a simulated record has a missing observation.

The definition assumes that the simulation’s sampling unit is identified (usually a participant), that each time is defined relative to a stated anchor, and that the allocation rule can be applied to the simulated participants. If timing depends on dosing, for instance, “2 hours” must mean 2 hours after a specified dose, not simply 2 hours after an unspecified study start. A design can instead use clock time or time after an event, provided that event and its timing are defined consistently.

A useful representation separates a schedule table from a participant allocation table. The schedule table lists the nominal times in each schedule. The allocation table lists how many participants, or which simulated participants, receive each schedule. This avoids repeating timing information for every person while preserving the link between an individual and their planned samples.

Counting Planned Samples

For a fixed design, suppose there are \(G\) allocation groups. Group \(g\) has \(n_g\) participants and schedule \(A_g\), containing \(m_g\) distinct nominal sampling times per participant. If every participant in that group receives that schedule, the total number of planned sample records is

This count is a structural check on the encoded design. It is not the number of usable observations: realized records may have fewer observed values because of missingness or other processes represented separately. The formula assumes one planned sample per participant at each listed time. If the protocol schedules multiple samples at the same nominal time on different occasions, represent occasions separately so they are counted correctly.

Worked Examples

Fixed Counts and Allocation Rules

A design may specify exact counts, such as 12 participants per sparse schedule, or a rule that assigns schedules probabilistically. For a probabilistic rule, define the eligible participant information, the possible schedules, and the assignment probabilities before generating the assignments. If there are \(J\) schedules, their probabilities \(q_1,\ldots,q_J\) must satisfy \(q_j\geq 0\) and \(\sum_{j=1}^{J}q_j=1\). A simulated cohort may then have realized schedule counts that differ from their expected counts.

For example, if each participant is independently assigned to schedule A with probability 0.6 and schedule B with probability 0.4, then among 50 participants the expected counts are 30 and 20. These are expectations, not guaranteed quotas. If the design instead requires exactly 30 on A and 20 on B, encode a fixed-count allocation, not independent assignments with those probabilities. The two procedures can yield different simulated designs.

Allocation may also be stratified: the assignment rule is applied within levels of a prespecified variable. State whether quotas are fixed within strata or whether probabilities apply within strata, and identify which participant information is available when assignment occurs. If schedule choice depends on a measured outcome that is generated later, it is not a baseline allocation rule; it would require an explicitly defined sequential design.

Justification: Why the Count Check Works

The total planned sample count follows by partitioning the participants according to their assigned schedule. In group \(g\), each of \(n_g\) participants contributes exactly \(m_g\) scheduled records, so the group contributes \(n_gm_g\). The groups do not overlap and together include all assigned participants. Adding their contributions gives the total:

This derivation also shows the assumptions behind the check: every participant belongs to one group, each group has a defined schedule, and every listed time contributes one planned record per participant. If schedules vary within a group, split that group into schedule-specific groups or calculate the number of times individually as \(\sum_i |A_i(D)|\). Do not apply the group formula with one \(m_g\) when participants in that group actually have different numbers of scheduled times.

Practical Encoding and Diagnostics

A robust design representation should make it possible to reconstruct every participant’s planned samples without guessing. A long-format schedule table can use one row per schedule-time pair, with fields for schedule identifier, occasion, nominal time, time unit, and anchor. A separate allocation table can contain participant identifier, stratum or group, and schedule identifier. Alternatively, a participant-level record can store an explicit list of times; this can be convenient for small designs but harder to audit for many patterns.

Before using an encoded candidate design, check the following:

• Allocation totals: exact quotas sum to the intended cohort size, or probabilistic allocation probabilities sum to one.
• Time meaning: every nominal time has a unit and a clear anchor, such as hours after dose or minutes after infusion start.
• Schedule integrity: times are ordered and duplicates are intentional and distinguished by occasion when necessary.
• Participant mapping: every participant receives exactly the schedule or schedules allowed by the rule.
• Count reconciliation: the sum of schedule-specific counts agrees with the total planned sample records.
• Design separation: scheduled samples remain distinguishable from observations that are later absent or unusable.

A common diagnostic is to summarize the encoded data by participant and schedule. Participants with zero scheduled samples, unexpected sample counts, or times outside their assigned schedule indicate an encoding error unless the protocol explicitly permits that structure. Also inspect counts by time and stratum: a correct overall total can conceal an allocation mistake, such as assigning too many participants to one sparse pattern and too few to another.

Common Mistakes and Alternatives

Common mistake: “There are six sampling times in the study, so every participant has six planned samples.” This confuses the union of times across a sparse design with the schedule assigned to each participant. In the four-pattern example, six times appear across the study, but each participant has only three. The correction is to encode participant-to-schedule assignment and count the times within each assigned schedule.

Another frequent error is mixing time anchors or units—for example, storing one schedule in hours after dose and another in minutes after study start, then treating both as the same time variable. Convert to a common unit only after the event anchors are unambiguous. A further error is to represent a fixed quota using random assignment probabilities and assume the exact quota will result. Use the allocation mechanism that matches the intended design.

A single complete-profile schedule is simple and transparent when all participants have the same planned times. Multiple schedule patterns are more compact for sparse designs and make assignment explicit. A participant-by-time indicator matrix, with entries marking whether a sample is scheduled, is useful for visual checks and for designs with many overlapping schedules. The choice of representation does not change the design, but the encoding must preserve the same participant-level assignments and counts.

These representations have limits. A static schedule table is not sufficient when sampling times are deliberately adapted in response to information observed during the study; such a design needs a specified decision rule and its inputs. Nor does an encoded schedule by itself establish that samples are feasible, that timing is clinically appropriate, or that the resulting design estimates a particular quantity well. Those are separate design and evaluation questions. For this step, the requirement is narrower: simulated planned records must faithfully represent the candidate schedule and allocation.