Purpose of a Three-Period Reference-Replicate Schedule
A reference-replicate sequence gives each participant the reference formulation in two of three study periods and the test formulation in the remaining period. The two schedules considered here are TRR and RRT. The letters record the treatment administered in periods 1, 2, and 3, in that order. The design therefore repeats the reference within each participant, while changing the period in which the test is given between the two sequence groups.
The schedule is a prescription for treatment order, not a statement that the periods are consecutive calendar days or that treatment is administered continuously between periods. Each period has its own planned treatment administration and assessments, with the protocol specifying the interval and procedures between periods. The treatment sequence alone does not establish that the interval is adequate or that the study is free of carryover; those require design and conduct decisions in their own right.
Writing and Checking the Schedules
The schedules can be displayed as a treatment-by-period table. Each row is one sequence; each column is a study period.
| Sequence | Period 1 | Period 2 | Period 3 | Participant-level count |
|---|---|---|---|---|
| TRR | T | R | R | 1 T, 2 R |
| RRT | R | R | T | 1 T, 2 R |
The key check is row-wise: count the treatments received by one participant across all three periods. A TRR participant has one \(T\) and two \(R\) administrations; an RRT participant also has one \(T\) and two \(R\) administrations. Thus, assignment to either sequence meets the individual exposure-count requirement. The order differs, but the treatment count per participant does not.
A separate check is column-wise: count how many participants receive each treatment in each period. For equal numbers assigned to TRR and RRT, period 1 and period 3 are balanced between \(R\) and \(T\), while period 2 is \(R\) for everyone. This is a property of the pair of sequences when their group sizes are equal; it is not a property of an individual sequence by itself.
Worked Example: Checking One Participant in Each Sequence
In a hypothetical study, one participant is assigned to TRR and another to RRT. List each participant’s treatment by period and verify the required counts.
For the TRR participant, the period record is \(T,R,R\). Counting the three entries gives \(1\) test period and \(2\) reference periods.
For the RRT participant, the record is \(R,R,T\). It also contains \(1\) test period and \(2\) reference periods.
Across these two participants, there are \(2 \times 3 = 6\) treatment-period assignments. The reference count is \(2+2=4\), and the test count is \(1+1=2\). The totals agree with the individual schedules: two participants each receive reference twice and test once.
Why the Pair of Sequences Has a Useful Period Pattern
When the two sequence groups are the same size, their combination places each formulation equally often in periods 1 and 3. Period 2 remains a reference period for every participant. This can be shown directly from the schedule, without making assumptions about any pharmacokinetic outcome.
Suppose \(m\) participants are assigned to each sequence. In period 1, the \(m\) TRR participants receive \(T\), and the \(m\) RRT participants receive \(R\). In period 2, all \(2m\) participants receive \(R\). In period 3, the TRR group receives \(R\) and the RRT group receives \(T\). Therefore, the period-specific treatment counts are:
The total number of reference administrations is \(m+2m+m=4m\), and the total number of test administrations is \(m+0+m=2m\). There are \(2m\) participants, so each participant contributes three treatment-period assignments. The total is \(4m+2m=6m=3(2m)\), as required.
This calculation establishes the allocation pattern, not a claim that treatment and period effects are fully separable under every model or study circumstance. In particular, period 2 has no test administrations. The schedules should be understood as a specific replicate design with this period structure, rather than as a design in which both formulations occur in every period.
Worked Example: Period Counts in a Balanced Group
Suppose, hypothetically, that 12 participants are assigned: 6 to TRR and 6 to RRT. Calculate the treatment counts in each period and the overall numbers of administrations.
Period 1. The 6 TRR participants receive \(T\), and the 6 RRT participants receive \(R\). The counts are \(R=6\) and \(T=6\).
Period 2. Both sequence groups receive \(R\), so the counts are \(R=6+6=12\) and \(T=0\).
Period 3. The 6 TRR participants receive \(R\), and the 6 RRT participants receive \(T\). The counts are \(R=6\) and \(T=6\).
There are \(12 \times 2=24\) reference administrations because every participant receives reference twice, and \(12 \times 1=12\) test administrations. The period counts give the same totals: \(6+12+6=24\) for \(R\), and \(6+0+6=12\) for \(T\). In total, \(24+12=36\) treatment-period assignments, equal to \(12 \times 3=36\).
Constructing an Allocation and Checking It
A protocol or randomization list should distinguish the assigned sequence from the actual treatment record for each period. Once the sequence is assigned, the schedule determines which formulation is planned in each period. A clear table reduces the risk of reversing a sequence label, duplicating a period, or recording an incorrect formulation. The assignment process should also be carried out according to the study’s specified randomization procedures; a schedule table alone does not make assignment random.
Use a fixed order—period 1, period 2, period 3—before writing any treatment letters.
Enter TRR as \(T,R,R\) and RRT as \(R,R,T\), keeping the letters aligned with their periods.
Confirm that the row contains exactly two \(R\) entries and one \(T\) entry.
Count participants per sequence, then calculate period-specific treatment counts. Equal sequence group sizes give the balanced pattern in periods 1 and 3.
Worked Example: Finding and Correcting a Schedule Error
A hypothetical allocation table lists three participants as TRR and three as RRT, but one participant’s recorded period treatments are \(R,T,R\). Determine whether that record follows either intended schedule, and correct the problem if the participant was assigned to TRR.
The recorded row \(R,T,R\) contains two reference administrations and one test administration, so its treatment count is correct. However, its order is neither TRR nor RRT: period 1 is \(R\), period 2 is \(T\), and period 3 is \(R\). It therefore does not follow the assigned TRR sequence.
If the participant was assigned to TRR, the planned record is \(T,R,R\). The correction is not to relabel \(R,T,R\) as TRR; the sequence name is defined by the period order. The study team would need to reconcile the treatment record against the source documentation and protocol procedures. The schedule check identifies a discrepancy but cannot determine from the sequence table alone what treatment was actually administered.
For the intended allocation of three participants per sequence, the planned counts are \(R=3,T=3\) in period 1, \(R=6,T=0\) in period 2, and \(R=3,T=3\) in period 3. Any discrepancy in an individual record should prompt a recount of the actual group totals rather than an assumption that balance remains intact.
Assumptions, Diagnostics, and Alternatives
The schedules are simple to write, but their use in a crossover study depends on implementation and design conditions. The formulation labels must refer consistently to the intended reference and test products, the period order must be unambiguous, and participants must receive the assigned treatment as specified. The study also needs suitable procedures between periods and comparable measurement conditions. The schedule itself does not guarantee adequate washout, prevent carryover, or ensure that a response is unaffected by period.
Useful checks include comparing the randomization list with the treatment-administration records, verifying the treatment count for every participant, and tabulating sequence membership by period. If the two sequence groups are meant to be equal in size, the allocation counts should be checked before interpreting the group-level period pattern. Withdrawals or treatment deviations can make the numbers of observed records differ from the numbers originally assigned; planned balance should not be confused with the available data.
The TRR/RRT pair is shorter than the four-period full-replicate arrangements covered in earlier material: each participant has three treatment periods and repeats only \(R\). That shorter schedule also means it does not repeat \(T\) within a participant. The design choice depends on the study question and the information needed from repeated administrations; the schedule alone does not establish which design is preferable for a particular study. The central distinction here is the treatment order and count: both TRR and RRT give each participant two references and one test.
Common Mistakes
- Reading the letters as groups rather than periods. TRR does not mean that a participant receives test and reference at the same time. It means \(T\) in period 1, followed by \(R\) in periods 2 and 3.
- Checking only the total number of each treatment. A row such as \(R,T,R\) has the right count but is not the assigned TRR or RRT order. Verify both treatment counts and period placement.
- Assuming the sequence pair is balanced when group sizes differ. Equal numbers in TRR and RRT yield equal \(R\) and \(T\) counts in periods 1 and 3. Unequal group sizes do not.
- Assuming a schedule resolves carryover or period effects. The letters describe planned treatment order; they do not prove that washout is sufficient or that period-related influences are absent.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Write the treatment received in each period for a participant assigned to RRT. How many administrations of each formulation does that participant receive?
- In a hypothetical allocation of 4 participants to TRR and 4 to RRT, how many participants receive \(R\) and \(T\) in each period?
- For 10 participants, 6 assigned to TRR and 4 to RRT, calculate the treatment counts in periods 1, 2, and 3. Are periods 1 and 3 balanced?
- A participant’s record is \(T,R,T\). Does it follow TRR or RRT, and does it meet the required reference-replicate count?
- State one reason why a correct TRR/RRT schedule does not, by itself, establish that the crossover study is free of carryover.
Solutions. (1) RRT is \(R,R,T\): two reference administrations and one test administration. (2) Period 1 has \(R=4,T=4\); period 2 has \(R=8,T=0\); period 3 has \(R=4,T=4\). (3) Period 1 has \(R=4,T=6\), period 2 has \(R=10,T=0\), and period 3 has \(R=6,T=4\). Periods 1 and 3 are not balanced because the sequence groups have different sizes. (4) \(T,R,T\) is neither TRR nor RRT. It has two \(T\) administrations and one \(R\), so it fails the required count of two references and one test. (5) The sequence lists treatment order but does not establish whether the interval between periods is adequate to prevent residual drug effects; washout and carryover require separate consideration.