Why Distribute Sampling Times Across Participants?
A rich pharmacokinetic profile collects many samples from each participant, but that can impose substantial burden. A sparse design collects fewer samples per person. If every participant is sampled at the same few times, however, the cohort may leave important parts of the time course poorly represented. One way to broaden coverage is to assign different planned time points to different participants while keeping the number of planned samples per participant equal.
The key is to distinguish two ideas. Participant-level balance concerns how many samples each participant is scheduled to provide. Time-point coverage concerns how many participants are scheduled at each planned time. A schedule can be balanced in the first sense but uneven in the second. A useful design checks both.
Equal coverage is a possible additional goal, not part of the definition of participant-level balance. A plan may be balanced across participants yet intentionally schedule more observations at certain times. Whether that is appropriate depends on the study’s objectives; this tutorial focuses on constructing and checking the allocation, not choosing a statistically optimal design.
Representing and Checking a Schedule
Suppose there are \(N\) participants and \(K\) candidate sampling times. Define \(A_{ij}=1\) if participant \(i\) is scheduled for a sample at time \(j\), and \(A_{ij}=0\) otherwise. Here each row represents one participant, not a group. The row sum is that participant’s planned number of samples, and the column sum is the number of participants scheduled at that time.
Participant-level balance means \(r_i=m\) for every \(i\). The value \(c_j\) describes planned cohort coverage at time \(j\). If schedules are displayed in a group-level table to make them easier to read, each group’s size must be used when calculating the column totals. A group of four assigned to a time contributes four participants, not one.
A simple accounting identity helps audit any proposed plan. The total number of scheduled samples can be counted either by adding the samples per participant or by adding the participants scheduled at each time:
Thus, if all \(K\) candidate times are to have exactly the same number of participants, that number must be \(Nm/K\). Since a participant count must be an integer, \(Nm\) must be divisible by \(K\). This arithmetic condition is necessary, although it does not by itself guarantee that the desired schedule can be formed from the permitted sampling combinations.
Worked Example: Equal Coverage With Group-Assigned Schedules
A hypothetical study plans three samples for each of 12 participants. There are six candidate times, \(T_1\) through \(T_6\). The study team proposes four schedules, assigning three participants to each schedule. Does this create balanced participant burden and equal coverage?
| Assigned group | Participants in group | Times assigned to each participant |
|---|---|---|
| Group A | 3 | \(T_1,T_2,T_3\) |
| Group B | 3 | \(T_1,T_4,T_5\) |
| Group C | 3 | \(T_2,T_4,T_6\) |
| Group D | 3 | \(T_3,T_5,T_6\) |
Each participant in each group has three planned samples, so \(m=3\) and the participant-level allocation is balanced. To calculate time-point totals, multiply each group’s assignment by its group size. For example, \(T_1\) appears in Groups A and B, each of size three, so \(c_1=3+3=6\).
The same calculation gives \(c_2=3+3=6\), \(c_3=3+3=6\), \(c_4=3+3=6\), \(c_5=3+3=6\), and \(c_6=3+3=6\). There are \(12\times3=36\) planned samples in total, and the column totals also sum to \(6\times6=36\). Therefore, every participant has three planned samples and every candidate time has six participants scheduled.
The group table is a compact description, not an incidence matrix with one row per participant. If it were used to count coverage by simply counting group rows containing a time, it would incorrectly report two rather than six participants at \(T_1\).
When Equal Coverage Is—and Is Not—Possible
The accounting identity gives a quick feasibility check before schedules are assigned. If \(N=10\), each participant is planned for \(m=2\) samples, and there are \(K=5\) times, then \(Nm=20\), so equal coverage would require four participants at each time. The divisibility condition holds. By contrast, with \(N=10\), \(m=2\), and \(K=6\), the average is \(20/6=3.333\) participants per time. Exact equal coverage is impossible, though coverage can be made as even as possible.
Even when the arithmetic permits equal counts, the available schedules may constrain what can be assigned. For instance, a protocol might prohibit certain combinations of times for operational reasons. The allocation must still meet the per-participant sample limit and distribute participants among allowed combinations. Counts should therefore be checked from the proposed assignments, rather than inferred from the average alone.
Worked Example: Auditing an Uneven but Participant-Balanced Plan
A hypothetical cohort has 10 participants and five candidate times. Each participant is assigned two samples. Four participants receive \(T_1,T_2\); two receive \(T_2,T_3\); two receive \(T_3,T_4\); and two receive \(T_4,T_5\). Is the schedule balanced, and how even is its time-point coverage?
Every participant is assigned two times, so the schedule is balanced across participants. To count the time-point totals, multiply the number assigned to each schedule by whether that time appears in the schedule:
The totals sum to \(4+6+4+4+2=20\), matching the \(10\times2=20\) planned samples. But the time-point counts are not equal: \(T_2\) has six participants and \(T_5\) has two. This confirms that equal samples per participant do not guarantee equal cohort coverage. The plan covers all five times, but its coverage is concentrated at \(T_2\) relative to \(T_5\).
The average coverage is \(20/5=4\) participants per time, but an average does not make the observed schedule balanced across times. The calculated totals are what reveal the difference.
Worked Example: Constructing an Equal-Coverage Allocation
A second hypothetical plan has 10 participants, five candidate times, and two planned samples per participant. The team assigns two participants to each of the following pairs: \(T_1,T_2\); \(T_2,T_3\); \(T_3,T_4\); \(T_4,T_5\); and \(T_5,T_1\). Verify the allocation.
Each participant receives two samples, so the total planned sample count is \(10\times2=20\). Each time appears in two of the listed pairs. Since each pair is assigned to two participants, each time has \(2+2=4\) participants scheduled. Thus the five column totals are \((4,4,4,4,4)\), which sum to 20. Both participant burden and time-point coverage are balanced.
This construction illustrates a simple design principle: use complementary schedules so each candidate time occurs the same number of times across the cohort. It does not establish that this particular allocation is optimal for estimating any PK parameter.
Assumptions, Diagnostics, and Limitations
A schedule table describes planned samples. To interpret its balance as a property of the data actually collected, several practical conditions matter. The planned times must be defined consistently, such as relative to the same dose or study reference point. Each participant should contribute no more than one sample at a given planned time unless repeated samples at that time are explicitly part of the design. Most importantly, scheduled observations can be lost through missed visits, unusable specimens, or timing deviations.
It is also important to consider how participants are assigned to schedules. If participant characteristics related to PK are unevenly distributed across schedules, differences in observed concentrations at different times may partly reflect who was assigned there, rather than time-course differences alone. Random assignment to schedules, or an appropriate allocation strategy, can reduce systematic differences, but it cannot guarantee identical groups in a finite study. The schedule counts should be reviewed alongside relevant participant characteristics and actual sample collection.
Balanced sparse sampling supports broad cohort-level coverage while limiting each participant’s planned burden. It does not give every participant a complete concentration-time profile. Interpreting observations from different participants as information about a shared population time course depends on the population PK model and its assumptions. A participant with only a few observations may provide limited information about that individual’s entire profile, even when the overall cohort covers many times.
Useful checks include confirming that each participant has the intended row sum, that weighted group assignments give the intended column totals, and that all planned times are represented. After collection, repeat the checks using actual sample records: planned balance can be altered by missed or mistimed samples. Large gaps or highly unequal actual counts are signals to investigate, not proof on their own that a model will fail.
How Balanced Sparse Sampling Compares With Alternatives
| Approach | What is distributed | Main benefit | Important limitation |
|---|---|---|---|
| Rich sampling | Many times within each participant | Detailed within-participant profiles | Greater participant burden and more samples per person |
| Balanced sparse sampling | Different time subsets across participants, with equal planned counts per person | Can broaden cohort time coverage with a fixed per-person burden | Each participant has an incomplete profile; time coverage still needs checking |
| Unbalanced sparse allocation | Different numbers of samples per participant, or uneven time-point totals | May accommodate practical or study-specific priorities | Burden or coverage may be concentrated; totals must be evaluated explicitly |
Rich sampling and balanced sparse sampling answer different design needs rather than representing a universal ranking. Compared with assigning everyone the same few times, distributing complementary schedules can cover more of the candidate time grid. Compared with an unbalanced allocation, equal per-person counts make planned sampling burden more uniform. Neither feature alone establishes that the schedule will precisely estimate a particular parameter; that requires considering the study’s objectives and the model, a separate design question.
Common Mistakes
- Equating equal burden with equal coverage. The wrong reasoning is that if everyone has two samples, every time must have the same number of participants. The correction is to calculate each column total; equal row sums do not imply equal column sums.
- Counting schedule groups as if each were one participant. A group of four assigned to \(T_1\) contributes four scheduled observations at \(T_1\). Multiply by group size, or use one incidence-matrix row per participant.
- Confusing planned and observed balance. A balanced protocol may produce uneven data if visits are missed. Audit both the planned assignment and the samples actually collected.
- Assuming cohort coverage gives each person a full profile. Complementary time points across participants broaden population-level coverage, but no participant should be treated as though they supplied the cohort’s entire time course.
Check Your Understanding
Answer each question, then compare with the solutions below.
- There are 12 participants, each assigned three samples, at six candidate times. What is the average number of participants per time if coverage is equal? Is exact equal coverage arithmetically possible?
- Two participants are assigned to \(T_1,T_2\), and four participants are assigned to \(T_2,T_3\). How many participants are scheduled at each of \(T_1,T_2,T_3\)?
- Can a plan with 10 participants, two samples per participant, and six candidate times have exactly equal time-point coverage? Explain.
- A plan assigns every participant two samples, but one time is scheduled for twice as many participants as another. Is participant-level balance present? Is time-point coverage equal?
- Why should the study team recheck balance after sample collection?
Solutions. (1) There are \(12\times3=36\) planned samples, so equal coverage gives \(36/6=6\) participants per time. Exact equal coverage is arithmetically possible because 36 is divisible by 6, though the assigned combinations must still permit it. (2) \(T_1\) has 2 participants, \(T_2\) has \(2+4=6\), and \(T_3\) has 4. (3) No. The total is \(10\times2=20\), and \(20/6\) is not an integer, so equal participant counts at all six times are impossible. (4) Participant-level balance is present because everyone has two planned samples, but time-point coverage is not equal. (5) Missed visits, unusable samples, or timing deviations can make actual counts differ from planned counts, so the collected data may no longer have the intended balance.