Why Count Exposures by Sequence?
A replicate crossover design assigns participants to ordered sequences of test and reference treatments. To check whether a design matches its objective, it is not enough to count how many test and reference administrations appear in the whole study. The count for each assigned sequence matters: every participant assigned to that sequence follows its treatment order, so the sequence determines how many times that participant is scheduled to receive each formulation.
This is a design audit, not an analysis of pharmacokinetic outcomes. Exposure counts can show whether the schedule calls for repeated administrations of the intended formulation. They cannot show whether a participant completed those administrations, whether the administrations were carried out as planned, or what variability the resulting measurements will have.
For example, the sequence \(TRRT\) schedules a participant to receive test, reference, reference, and test across four periods. It has \(n_{sT}=2\), \(n_{sR}=2\), and four scheduled administrations. The same counts occur in \(RTTR\), but the order differs. Both the counts and the ordered sequence are useful: counts verify the replication objective, while the sequence records when each administration is scheduled.
A Basic Counting Check
For a sequence with \(k\) scheduled periods, each period contributes exactly one scheduled formulation. Define an indicator \(I_{sjT}\) to equal 1 when period \(j\) of sequence \(s\) assigns \(T\), and 0 otherwise. Define \(I_{sjR}\) in the same way for \(R\). The exposure counts are sums across periods:
Since each period assigns exactly one of these two formulations, \(I_{sjT}+I_{sjR}=1\) for every period. Summing that equality across all \(k\) periods gives:
This identity is a useful first check, but it is not sufficient by itself. A four-period sequence with three \(T\) administrations and one \(R\) administration still sums to four; it may nevertheless fail a design objective requiring two administrations of each formulation. A proper audit checks the total number of periods and the formulation-specific counts.
Worked Examples of Replication Counts
Worked Example: Checking a Four-Period Full-Replicate Schedule
Suppose a hypothetical four-period full-replicate design uses sequences \(TRTR\) and \(RTRT\). Check the number of planned administrations of each formulation for participants assigned to either sequence.
For \(TRTR\), the positions are \(T,R,T,R\). Thus \(n_{sT}=1+1=2\) and \(n_{sR}=1+1=2\). The total is \(2+2=4\), matching the four scheduled periods.
For \(RTRT\), the positions are \(R,T,R,T\). Thus \(n_{sT}=1+1=2\) and \(n_{sR}=1+1=2\), again totaling four.
Both sequences schedule two test and two reference administrations per participant. That matches the full-replication count objective for these two formulations. It does not, by itself, establish that the sequence groups are balanced in size or that participants will complete all four periods.
Worked Example: Checking a Three-Period Reference-Replicate Schedule
Consider the hypothetical three-period sequences \(TRR\), \(RTR\), and \(RRT\). The intended count objective is one test administration and two reference administrations for each assigned participant.
For \(TRR\), there is one \(T\) and two \(R\) labels, so \(n_{sT}=1\) and \(n_{sR}=2\). For \(RTR\), the count is also \(n_{sT}=1\), \(n_{sR}=2\). For \(RRT\), the count is again \(n_{sT}=1\), \(n_{sR}=2\). Each sequence has three periods, and in every case \(1+2=3\).
The schedule therefore meets the stated reference-replication objective sequence by sequence: a participant assigned to any of these sequences is scheduled for two reference and one test administration. The introductory material on sequence coverage addresses which treatment orders are represented; the count check here answers the distinct question of how often each formulation occurs within each order.
Worked Example: Finding a Sequence That Breaks a Test-Replicate Objective
A hypothetical three-period design is described as a test-replicate design, with two test administrations and one reference administration required for every participant. The proposed sequence list is \(TTR\), \(TRT\), and \(RRT\). Audit each sequence.
For \(TTR\), \(n_{sT}=2\) and \(n_{sR}=1\); it meets the objective. For \(TRT\), \(n_{sT}=2\) and \(n_{sR}=1\); it also meets the objective. For \(RRT\), \(n_{sT}=1\) and \(n_{sR}=2\); it does not.
All three sequences contain three periods, so checking only the period totals would miss the error. The third sequence gives a participant one test and two reference administrations, the opposite replication pattern from the stated objective. If the intended set includes the three orders with two \(T\) labels and one \(R\), replacing \(RRT\) with \(RTT\) gives counts \(n_{sT}=2\), \(n_{sR}=1\) for that sequence. The full sequence list should then be checked again against any separate order-coverage requirements.
From Sequence Counts to Study-Wide Totals
Per-sequence counts are the primary check because the replication objective applies to each assigned participant. Study-wide totals can provide a secondary arithmetic check when the number of participants assigned to each sequence is known. If \(m_s\) participants are assigned to sequence \(s\), the total number of scheduled test administrations across the design is \(\sum_s m_s n_{sT}\), and the reference total is \(\sum_s m_s n_{sR}\). These totals weight each sequence’s count by its assigned group size.
For instance, suppose a hypothetical reference-replicate design assigns 10 participants to each of \(TRR\), \(RTR\), and \(RRT\). Each participant is scheduled for one test and two reference administrations. Across the 30 assigned participants, the scheduled totals are \(30\times1=30\) test administrations and \(30\times2=60\) reference administrations, for \(30+60=90\) administrations across \(30\times3=90\) planned periods. These pooled totals agree with the sequence-level counts, but they do not replace them. If a sequence were misclassified or entered incorrectly, pooled totals might not make clear which participants’ schedules violate the objective.
When sequence group sizes differ, overall counts also reflect that allocation. For a valid reference-replicate schedule with 8 participants assigned to each of the three sequences above, the totals would be \(24\) test and \(48\) reference administrations. A larger count of reference administrations does not mean that every sequence has been checked; the conclusion follows here only because the per-sequence counts were already verified.
Planned Schedules and Actual Administrations
The sequence table describes what participants are assigned to receive. Study records describe what was actually administered. Those are related but different counts. A participant may withdraw, miss a period, or have a dose withheld; in such a case, the assigned sequence remains the same, but the number of completed administrations is lower than planned.
For example, a participant assigned to the four-period sequence \(TRTR\) has planned counts \(n_{sT}=2\) and \(n_{sR}=2\). If that participant withdraws before period 4 after receiving \(T,R,T\), the record of received administrations is two \(T\) and one \(R\). It would be misleading to rewrite the design sequence as \(TRT\): the assigned schedule was \(TRTR\), while the participant’s completed exposure record was incomplete. Keeping these concepts separate lets the design audit ask whether the intended schedule is correct and the conduct record show whether that schedule was completed.
For an audit, it is useful to retain both fields: the assigned sequence and the period-by-period administration record. A discrepancy between the two calls for review of the record and its reason; it does not change the count specified by the original design objective.
Diagnostics, Alternatives, and Limits
A reliable audit can begin with a compact table that gives each sequence, its number of periods, its \(T\) count, and its \(R\) count. Common warning signs include a sequence with a different length from the others, a formulation count that fails the stated replication objective, an unexpected symbol or blank period, or a pooled total that does not equal the number of scheduled periods across participants. A mismatch should be traced to the sequence entries and group sizes rather than corrected by adjusting a total without explanation.
A spreadsheet can count labels, but the method matters less than preserving the sequence-by-sequence check. Manual counting is transparent for a short schedule; a validated script or spreadsheet is convenient for a larger sequence list. In either case, independently check that the sequence string has the intended number of periods and that its treatment counts sum to that number. Automated summaries can reproduce an incorrectly entered sequence perfectly, so they do not remove the need to verify the source schedule.
| Approach | What it checks well | What it can miss |
|---|---|---|
| Count \(T\) and \(R\) within each sequence | Whether each participant assigned to a sequence receives the intended planned number of each formulation | Whether the sequence groups are allocated as intended or administrations are completed |
| Count administrations across the whole design | Whether totals reconcile with group sizes and planned periods | Which sequence has a wrong formulation count |
| Inspect only the ordered sequence strings | Which formulation is scheduled in each period | A systematic count comparison with the objective, especially in a long list |
The count audit has a deliberately narrow scope. It does not establish that the design is statistically suitable for a particular question, that participants are randomized correctly, or that period effects and other design considerations are adequately addressed. Nor do repeated treatment counts alone establish any particular amount of information about variability. Those questions require other design and analysis considerations; they should not be inferred from a count table.
Common Mistakes
- Checking only the grand total. A design can have the expected total number of \(T\) and \(R\) administrations overall while one sequence gives its participants the wrong replication pattern. Check every sequence first.
- Treating a sequence as an unordered set. \(TRR\) and \(RRT\) have the same counts but different period orders. Preserve the order when recording a sequence, even when the immediate check concerns counts.
- Confusing planned with received administrations. A withdrawal changes the participant’s completed exposure record, not the originally assigned sequence.
- Assuming equal sequence counts imply equal group sizes. The counts describe administrations per participant assigned to a sequence. They say nothing by themselves about how many participants are assigned to each sequence.
- Assuming a correct count proves the entire design is correct. Counts confirm only the stated replication objective. They do not settle other design questions.
Check Your Understanding
Answer each question using the distinction between a sequence’s planned counts and the study-wide totals.
- For the sequence \(RTTR\), give the number of scheduled periods, \(n_{sT}\), and \(n_{sR}\).
- A three-period reference-replicate schedule contains \(TRR\), \(RRT\), and \(RTR\). Does every sequence meet the one-\(T\), two-\(R\) count objective? Show the counts.
- A proposed test-replicate list is \(TTR\), \(TRT\), and \(RRT\). Which sequence fails an objective of two \(T\) and one \(R\) per participant?
- In a four-period assigned sequence \(TRTR\), a participant receives only the first three scheduled administrations before withdrawal. State the planned counts and the received counts separately.
- Why is a correct study-wide total not a sufficient demonstration that every sequence meets its replication objective?
Solutions. (1) \(RTTR\) has four periods, two \(T\) administrations, and two \(R\) administrations: \(n_{sT}=2\), \(n_{sR}=2\), and \(2+2=4\). (2) Yes. \(TRR\) has \((n_{sT},n_{sR})=(1,2)\); \(RRT\) has \((1,2)\); and \(RTR\) has \((1,2)\). Each has three periods and meets the stated objective. (3) \(RRT\) fails: it has one \(T\) and two \(R\) administrations, not two \(T\) and one \(R\). (4) The assigned \(TRTR\) schedule plans two \(T\) and two \(R\) administrations. The received record \(TRT\) contains two \(T\) and one \(R\) administration. (5) Pooled totals combine counts across sequences and participants; they can conceal which sequence has an incorrect count. The objective must be checked for each sequence before using totals as a reconciliation.