Why Specify the Four Periods Explicitly?
Replicate crossover designs are used when it matters that a participant receives the same formulation more than once. In a four-period full replicate, each participant receives both the test formulation and the reference formulation twice. Writing the schedule period by period makes that repetition visible and helps detect allocation or recording errors. It also clarifies what is meant by assigning a participant to a sequence: the sequence is the ordered list of formulations that participant is scheduled to receive.
This tutorial focuses on two alternating four-period sequences: TRTR and RTRT. Here, T denotes the test formulation and R the reference formulation. The letters specify which formulation is administered in each period; they do not by themselves specify dose, washout duration, measurement times, or the number of participants assigned to a sequence.
The TRTR and RTRT Sequences
The two schedules are written from period 1 through period 4. Each row below represents one participant assigned to that sequence, not a different treatment group within the participant.
| Sequence | Period 1 | Period 2 | Period 3 | Period 4 | Participant’s exposures |
|---|---|---|---|---|---|
| TRTR | T | R | T | R | T twice; R twice |
| RTRT | R | T | R | T | T twice; R twice |
A participant assigned TRTR receives T in periods 1 and 3, and R in periods 2 and 4. A participant assigned RTRT receives R in periods 1 and 3, and T in periods 2 and 4. Thus, both schedules satisfy the full-replication count, even though the formulation given first differs.
In both schedules the formulation alternates at every adjacent period boundary. The repeated administrations of a formulation occur two periods apart: T is repeated from period 1 to period 3 or from period 2 to period 4, depending on sequence; R is repeated in the other pair of positions. The schedules therefore describe not only how many times each formulation is administered, but also its order and period locations.
Worked Example: Writing Both Sequences by Period
A hypothetical study protocol calls for four treatment periods and asks for two alternating sequences, one beginning with T and one beginning with R. Write the assignments and check the exposure counts for each participant.
Sequence beginning with T. Alternation gives T in period 1, R in period 2, T in period 3, and R in period 4. The sequence is TRTR. Its counts are:
The first sum counts T in periods 1 and 3; the second counts R in periods 2 and 4.
Sequence beginning with R. Alternation gives R, T, R, T across periods 1 through 4. The sequence is RTRT. Its counts are:
Here T occurs in periods 2 and 4, and R in periods 1 and 3. The assignments meet the stated four-period full-replicate requirement for a participant in either sequence.
Why Alternation Produces These Two Patterns
The construction can be justified directly. Let the formulation in period \(j\) be \(x_j\), where \(x_j\) is either T or R. For an alternating schedule, each adjacent pair must differ: periods 1 and 2 differ, periods 2 and 3 differ, and periods 3 and 4 differ. Once the first formulation is chosen, each subsequent formulation is forced.
If \(x_1=\mathrm{T}\), then \(x_2=\mathrm{R}\); because \(x_3\) must differ from R, \(x_3=\mathrm{T}\); and because \(x_4\) must differ from T, \(x_4=\mathrm{R}\). This gives TRTR. If instead \(x_1=\mathrm{R}\), the same reasoning gives RTRT. These exhaust the possibilities because there are only two choices for the first period, and alternation fixes the next three.
Each resulting sequence has two positions occupied by T and two by R, so the alternating condition also delivers full replication in this four-period case. The argument establishes the two alternating patterns; it does not establish that these are the only possible four-period schedules with two administrations of each formulation. For example, a nonalternating pattern could still have two T assignments and two R assignments while failing the alternating condition.
Worked Example: Checking a Proposed Sequence
A hypothetical schedule lists the assignments for one participant as T, R, T, R. Verify the period sequence, count the administrations, and check whether adjacent periods alternate.
Write the entries in period order: period 1 = T, period 2 = R, period 3 = T, and period 4 = R. Reading these four entries gives TRTR.
T occurs in periods 1 and 3, so its count is \(1+1=2\). R occurs in periods 2 and 4, so its count is \(1+1=2\). The schedule meets the full-replicate count.
For the alternation check, compare each adjacent pair: period 1 differs from period 2 (T then R), period 2 differs from period 3 (R then T), and period 3 differs from period 4 (T then R). All three comparisons satisfy alternation. Therefore this is the TRTR schedule.
From a Sequence Label to a Participant Record
A sequence label is a compact design instruction. To implement it, the study record needs to preserve the assignment for every period. A label alone is not evidence that the participant actually received the scheduled formulation; administration records and deviations must be checked separately. For a schedule check, read the four period entries in order, verify that each entry is T or R, count each formulation, and then confirm the intended order.
Worked Example: Auditing a Hypothetical Participant Record
Suppose a hypothetical participant’s administration record contains the following entries. The protocol assignment for this participant is RTRT.
| Period | Protocol assignment | Recorded formulation |
|---|---|---|
| 1 | R | R |
| 2 | T | T |
| 3 | R | R |
| 4 | T | R |
Check the protocol sequence. Reading the protocol column gives R, T, R, T, which is RTRT. The assignment contains R in periods 1 and 3 and T in periods 2 and 4; hence the scheduled counts are two R and two T.
Check the recorded sequence. Reading the recorded column gives R, T, R, R. Its counts are R in periods 1, 3, and 4, for \(1+1+1=3\), and T in period 2, for a count of \(1\). The final-period record does not match the protocol assignment.
The protocol schedule is a valid RTRT full replicate, but the participant’s recorded administrations do not follow that schedule and do not show two exposures to each formulation. The discrepancy should be investigated and represented accurately in study records; it should not be silently changed to match the planned sequence.
Assumptions, Implementation, and Diagnostics
The sequence is a planned allocation, not a guarantee of how a study is conducted. Interpreting the schedule as a four-period crossover requires that each participant has four ordered periods, that one formulation is assigned in each period, and that the recorded treatment identities correspond to the actual administrations. The intended replication further requires that each participant completes the scheduled administrations, or that incomplete records are identified rather than treated as complete schedules.
Conduct also matters. Washout intervals and other protocol conditions must be appropriate for the study; repeating treatment labels cannot itself prevent carryover or ensure that periods are comparable. The earlier material on pharmacokinetic and pharmacodynamic carryover and washout planning addresses those issues. Likewise, the sequence notation does not establish that participants were allocated as intended or that treatment administration was blinded.
A practical schedule audit can be made in four checks:
Check that the record has four distinct ordered periods, labelled 1 through 4, with no missing or duplicated period identifiers.
Write the treatment in each period in order as a four-letter string. Do not infer order from an unordered list of exposures.
For a full replicate, confirm two T assignments and two R assignments for the participant’s planned schedule.
Compare the protocol sequence with the administration record. Flag deviations, missing administrations, or inconsistent labels rather than assuming that the planned sequence was completed.
Common warning signs include a record with fewer than four periods, an entry other than T or R, a sequence with unequal counts, and a mismatch between the assigned and recorded period treatment. A missing period should not be filled in by guessing from the sequence label. The label can indicate what was planned, but it cannot establish what was administered.
What This Design Does—and Does Not—Specify
A conventional two-period crossover has one administration of each formulation per participant, so it is not a full replicate: neither formulation is repeated within that participant. In TRTR and RTRT, each is repeated, which provides within-participant replication. The earlier material on why replicate crossover designs are used explains the motivation for that feature; the schedule here specifies where the repeated administrations occur.
The schedule is also distinct from allocation across participants. A sequence says what one participant receives over four periods. It does not say how many participants should receive TRTR versus RTRT, whether the sequence groups are balanced, or how participants are randomized to them. Those are separate design decisions. Nor does the sequence alone establish that period effects, carryover, or other sources of variation have been removed.
A useful mistake to avoid is reasoning, “There are two T labels and two R labels in the study, so every participant has a full replicate.” That conclusion does not follow: the counts must be checked within each participant’s four-period schedule. A cohort could have several participants with different individual counts, even if its overall totals happen to look equal. The correction is to verify the ordered assignments participant by participant, and then separately consider how the sequence groups were allocated.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- For TRTR, which formulation is scheduled in periods 1 and 3, and how many times is each formulation scheduled?
- Write the four-period alternating sequence that begins with R. Verify its T and R counts.
- A proposed sequence is T, T, R, R. Does it have two administrations of each formulation? Is it one of the two alternating sequences discussed here?
- A participant’s protocol assignment is RTRT, but the recorded sequence is R, T, R, R. What can be concluded from the record?
- Does assigning some participants to TRTR and some to RTRT, by itself, establish that the sequence groups are balanced? Explain.
Solutions. (1) T is scheduled in periods 1 and 3, and R in periods 2 and 4; each occurs twice. (2) The sequence is RTRT: T occurs in periods 2 and 4, while R occurs in periods 1 and 3, so each occurs twice. (3) TTRR has two T and two R assignments, but it is not alternating and is neither TRTR nor RTRT. (4) The protocol assignment is a valid RTRT schedule, but the recorded sequence differs in period 4 and contains three R entries and one T entry. It is a discrepancy to investigate, not a completed RTRT record. (5) No. The sequence labels specify individual schedules; they do not specify the numbers allocated to each sequence or establish balance across groups.