Why the Layout Matters
A two-period, two-treatment crossover study has a compact schedule, but its notation must be read carefully. Each participant is scheduled to receive both treatments, one in each period. The order is represented by a two-letter sequence: TR means test treatment in period 1 and reference treatment in period 2; RT means reference in period 1 and test in period 2. Writing out the assignments makes the design explicit and prevents sequence labels from being mistaken for participant-level records.
The layout describes the planned treatment schedule. It does not, by itself, say how participants are assigned to sequences, whether a participant completes both periods, or whether a planned treatment was actually administered. Those are separate questions. The earlier material on within-subject comparisons explains why observations from the same participant are paired; here the focus is the structure that makes those two treatment observations possible.
Writing the Two Sequences
A sequence label records treatment order, not a treatment contrast or an outcome. Read its first letter as the planned treatment in period 1 and its second letter as the planned treatment in period 2. A schedule table makes this convention visible:
| Sequence | Period 1 | Period 2 |
|---|---|---|
| TR | T | R |
| RT | R | T |
There are exactly two valid sequences under the stated layout. There are two treatments to arrange in two periods, and each treatment must appear exactly once for a participant. If the first-period treatment is \(T\), the remaining treatment \(R\) must be in period 2, giving TR. If the first-period treatment is \(R\), the remaining treatment \(T\) must be in period 2, giving RT. These are the two possible orders; TT and RR do not meet the requirement that each participant receive both treatments once.
For participant-level notation, let \(A_{i,j}\) denote the planned treatment assigned to participant \(i\) in period \(j\), where \(j\) is 1 or 2. For a participant labelled M12 on the TR sequence, the assignments are \(A_{\mathrm{M12},1}=T\) and \(A_{\mathrm{M12},2}=R\). The participant identifier and period index have distinct roles: the first subscript identifies the participant, and the second identifies the period. The sequence label is a shorthand for these two assignments, not a substitute for specifying them when individual records are being described.
Worked Example: Expanding Sequence Labels Into Assignments
In a hypothetical study, participant P04 is scheduled on TR and participant P05 is scheduled on RT. Write each person’s planned assignment in both periods.
For P04, TR means \(T\) first and \(R\) second. Thus \(A_{\mathrm{P04},1}=T\) and \(A_{\mathrm{P04},2}=R\).
For P05, RT means \(R\) first and \(T\) second. Thus \(A_{\mathrm{P05},1}=R\) and \(A_{\mathrm{P05},2}=T\).
The period index is essential. For example, \(A_{\mathrm{P04},1}=T\) states P04’s planned treatment in period 1; it does not describe period 2 or another participant.
From a Sequence List to a Participant Schedule
A useful layout has one row per participant and one column for each period. The sequence column summarizes the row, while the period columns show exactly which treatment is planned. The table below is a hypothetical schedule for six participants; it illustrates how to map labels to assignments, not how to decide which participant receives which label.
| Participant | Sequence | Period 1 planned treatment | Period 2 planned treatment |
|---|---|---|---|
| P01 | TR | T | R |
| P02 | RT | R | T |
| P03 | RT | R | T |
| P04 | TR | T | R |
| P05 | TR | T | R |
| P06 | RT | R | T |
For each row, the sequence label should agree with the two period entries. For example, P03’s RT row says \(A_{\mathrm{P03},1}=R\) and \(A_{\mathrm{P03},2}=T\). The table’s purpose is to make participant-by-period assignments auditable. It does not imply that the six participants were selected in any particular way, nor does the layout alone establish that all planned treatments were administered.
Worked Example: Reconstructing a Schedule From Period Records
Suppose a hypothetical schedule lists these planned assignments: participant K21 receives \(R\) in period 1 and \(T\) in period 2; participant K22 receives \(T\) in period 1 and \(R\) in period 2. Determine each sequence and write the assignments using participant and period indices.
K21’s first letter is \(R\) and second letter is \(T\), so K21 is on RT. In indexed form, \(A_{\mathrm{K21},1}=R\) and \(A_{\mathrm{K21},2}=T\).
K22’s first letter is \(T\) and second letter is \(R\), so K22 is on TR. In indexed form, \(A_{\mathrm{K22},1}=T\) and \(A_{\mathrm{K22},2}=R\).
The sequence is recovered by reading across a single participant’s row from period 1 to period 2. Reading down a period column instead would compare different participants and would not define any one participant’s sequence.
Checking a Layout
The defining checks apply participant by participant. A complete planned row should have one treatment in each of the two periods, and the two treatments should be different: the row must be either T followed by R or R followed by T. These conditions are a direct consequence of the layout definition. They detect common transcription errors, but they do not confirm that the observed study conduct matched the plan.
It is helpful to keep three records conceptually separate:
- Sequence label: the shorthand order, TR or RT.
- Planned assignment: the treatment specified for a particular participant and period, represented by \(A_{i,j}\).
- Observed administration: what was actually given or received in that period, which may differ from the plan or may be missing from the record.
A schedule audit should first check that each sequence agrees with its period entries. It should then check completeness: a missing period entry cannot be silently inferred to mean that the planned treatment was received. If the sequence label conflicts with the period columns, the record is internally inconsistent and should be resolved against the source schedule before analysis.
Worked Example: Finding and Correcting a Sequence-Label Error
A hypothetical schedule contains the following row for participant L08: sequence TR, period 1 treatment \(T\), and period 2 treatment \(T\). Identify the problem and state what can and cannot be concluded from the row.
The period entries are \(T,T\), which do not match TR: a TR row requires \(T\) in period 1 and \(R\) in period 2. They also fail the layout requirement that the participant be scheduled to receive each treatment once. The row is therefore invalid as written.
If the sequence label and period 1 entry are confirmed as correct, the period 2 entry should be \(R\), and the corrected planned assignments are \(A_{\mathrm{L08},1}=T\) and \(A_{\mathrm{L08},2}=R\). If instead both period entries are verified against the source, the sequence label and possibly the underlying schedule require correction. The row alone does not tell us which field is wrong; that must be checked rather than guessed.
What the Layout Does and Does Not Establish
The TR/RT structure is useful because it makes the within-participant treatment order explicit: each complete planned row contains one \(T\) and one \(R\). That structure is the basis for identifying which observations belong to the same participant and which period each observation represents. It is also a practical communication device: a short sequence label can summarize two assignments once its convention has been stated.
The layout is not a complete study protocol. It does not establish how many participants should be assigned to each sequence, how sequence assignments are generated, whether treatment is balanced across periods in the enrolled sample, or whether a washout is adequate. Those questions require design or conduct information beyond the two-row sequence definition. Nor does the existence of both sequence labels prove that a particular study used them or that its realized records are complete.
The main alternative in broad design terms is a parallel-group arrangement, in which a participant receives one treatment rather than both. The crossover layout instead schedules both treatments for the same participant in different periods. These are different assignment structures; choosing between them depends on the study context and assumptions, not merely on the convenience of writing TR and RT. The sequence table should therefore be understood as a description of a crossover schedule, not as evidence that the design is automatically appropriate.
Common Mistakes and Limitations
- Reversing the order. TR means \(T\) in period 1 and \(R\) in period 2. It does not mean the order in which treatments are listed in a report or data file unless that order is explicitly period 1 then period 2.
- Using indices that identify the wrong record. \(A_{i,j}\) identifies participant \(i\) and period \(j\). For a particular participant, write both indices explicitly, such as \(A_{\mathrm{M12},1}=T\) and \(A_{\mathrm{M12},2}=R\), rather than using a notation that could point to other participants’ period-2 assignments.
- Treating a sequence label as an observed outcome. TR records a planned order; it does not show that both treatments were actually administered or that any pharmacokinetic measurement was obtained.
- Assuming the layout answers every design question. The two sequences specify treatment order only. They do not, on their own, establish the assignment process, period effects, or absence of carryover.
Check Your Understanding
Answer each question by applying the sequence definition and checking the participant-period indices.
- Participant N14 is assigned \(T\) in period 1 and \(R\) in period 2. What is the sequence, and how are the assignments written using \(A_{i,j}\)?
- A row lists participant N15 as RT, with \(R\) in period 1 and \(T\) in period 2. Is the row internally consistent?
- A row lists participant N16 as TR, with \(T\) in both period columns. What layout requirement does it violate?
- Does a TR sequence label show that the participant actually received both planned treatments? Explain.
- Why are there only two valid sequences when each participant must receive both treatments once in two periods?
Solutions. (1) The sequence is TR; \(A_{\mathrm{N14},1}=T\) and \(A_{\mathrm{N14},2}=R\). (2) Yes. RT requires \(R\) in period 1 and \(T\) in period 2, exactly as listed. (3) The row does not schedule each treatment once; it lists \(T,T\), not either TR or RT. (4) No. The label describes the planned schedule, while actual administration must be checked from study records. (5) The first period can contain either \(T\) or \(R\); once that choice is made, the other treatment must occupy period 2. These choices produce TR and RT, respectively, and no other order satisfies the requirement.