Why Audit Transitions?
A crossover schedule specifies more than which treatments each participant receives: it also determines the treatment that precedes each dose. If a response in a period may depend on the treatment given in the previous period, then the schedule’s treatment transitions are relevant design information. An audit makes those transitions visible before interpreting results.
The audit is descriptive. It can show that some transitions occur more often than others, or that a transition occurs only in a particular period. Such patterns can flag a design feature to examine alongside washout planning and other carryover risks. Counts alone do not establish that carryover occurred, nor do they measure its size.
How to Construct the Audit
For each sequence, write its treatment in every period. Then move from period 2 onward, recording the preceding treatment and the current treatment separately. Do not count period 1 as a transition: it has no preceding study-period treatment. Keep one count for each ordered pair in each period, and add those period-specific counts only when a pooled summary is useful.
Formally, let \(s\) index the assigned sequences, \(n_s\) be the number of participants assigned to sequence \(s\), and \(a_{sp}\) be the treatment assigned in period \(p\) of sequence \(s\). The number of \(i\to j\) transitions into period \(p\) is
where \(I(\cdot)\) equals 1 when its condition is true and 0 otherwise, and \(K\) is the number of periods. The pooled count is \(N_{ij,\mathrm{pool}}=\sum_{p=2}^{K}N_{ij,p}\). It counts transition occasions, not necessarily distinct participants: a participant can contribute a transition in several periods.
A useful check follows directly from the definition. If all \(N\) assigned participants have a complete schedule, the four possible transitions between two treatments satisfy
These equalities are arithmetic checks, not assumptions about biological responses. If the audit is based on observed treatment administrations rather than the assigned schedule, missing or discontinued periods can make the observed opportunity count smaller; report that basis explicitly.
Worked Example: Auditing a Four-Period Full-Replicate Schedule
Suppose a hypothetical study assigns 10 participants to T–T–R–R and 10 to R–R–T–T. Count each transition into periods 2, 3, and 4, then pool across those periods.
For T–T–R–R, the transitions are \(T\to T\) into period 2, \(T\to R\) into period 3, and \(R\to R\) into period 4. Each occurs for 10 participants. For R–R–T–T, they are \(R\to R\), \(R\to T\), and \(T\to T\), respectively, also for 10 participants.
| Transition | Into period 2 | Into period 3 | Into period 4 | Pooled |
|---|---|---|---|---|
| T→T | 10 | 0 | 10 | 20 |
| T→R | 0 | 10 | 0 | 10 |
| R→T | 0 | 10 | 0 | 10 |
| R→R | 10 | 0 | 10 | 20 |
| Period total | 20 | 20 | 20 | 60 |
The pooled totals are \(T\to T=20\), \(T\to R=10\), \(R\to T=10\), and \(R\to R=20\). The period-4 column totals 20 and includes both \(T\to T\) and \(R\to R\). As a check, each of the three transition occasions has 20 participants, and the total is \(3(20)=60\). The directed transitions are not evenly represented: the two same-treatment transitions occur twice as often as either switch when pooled. Also, each switch occurs only into period 3. This is a feature to note in the schedule audit, not evidence that a response was affected by the preceding treatment.
Reading Period-Specific Patterns
Pooling can conceal where an imbalance occurs. A pooled total might make two transitions appear similarly represented even though one occurs early and the other late. Period-specific columns therefore come first; pooling is a secondary summary. Sequence-level inspection is useful too, because a total may be driven by one highly enrolled sequence.
Worked Example: Finding a Transition Concentrated in One Period
Consider a hypothetical three-period schedule with 8 participants assigned to each of three sequences: T–R–R, R–T–R, and R–R–T. Count the transitions into periods 2 and 3.
Into period 2, the sequences contribute \(T\to R\), \(R\to T\), and \(R\to R\), respectively, so each count is 8. Into period 3, T–R–R contributes \(R\to R\); R–T–R contributes \(T\to R\); and R–R–T contributes \(R\to T\). Thus each of those latter transitions also has count 8. When grouped by transition type across both periods, \(T\to R\) totals 16? No: it occurs once in period 2 (8 participants) and once in period 3 (8 participants), while \(R\to T\) likewise occurs once in each period. The complete table is:
| Transition | Into period 2 | Into period 3 | Pooled |
|---|---|---|---|
| T→T | 0 | 0 | 0 |
| T→R | 8 | 8 | 16 |
| R→T | 8 | 8 | 16 |
| R→R | 8 | 8 | 16 |
| Period total | 24 | 24 | 48 |
There are \(3(8)=24\) participants and two transition occasions per participant, giving \(2(24)=48\) transition counts. The table verifies this. It also shows that \(T\to T\) is absent, while the other three transition types are equally frequent overall and in each period. An absent transition can matter if a later analysis or design question requires that contrast, but this count table by itself does not establish that the absent transition causes bias.
Unequal Sequence Enrollment and Interpretation
A schedule can have one pattern of transitions by design and another in its realized enrollment if sequence sizes differ. The audit should therefore use the actual \(n_s\), rather than assuming equal sequence allocation. This is especially important when reporting realized counts or checking a randomization list against enrollment.
Worked Example: Unequal Numbers Assigned to Two Sequences
In a hypothetical four-period study, 12 participants are assigned to T–T–R–R and 8 to R–R–T–T. Derive the transition table using the sequence-specific contributions.
T–T–R–R contributes 12 each to \(T\to T\) in period 2, \(T\to R\) in period 3, and \(R\to R\) in period 4. R–R–T–T contributes 8 each to \(R\to R\) in period 2, \(R\to T\) in period 3, and \(T\to T\) in period 4.
| Transition | Into period 2 | Into period 3 | Into period 4 | Pooled |
|---|---|---|---|---|
| T→T | 12 | 0 | 8 | 20 |
| T→R | 0 | 12 | 0 | 12 |
| R→T | 0 | 8 | 0 | 8 |
| R→R | 8 | 0 | 12 | 20 |
| Period total | 20 | 20 | 20 | 60 |
The column totals equal the 20 assigned participants in each period after period 1, and the pooled total is \(3(20)=60\). The switch counts differ: \(T\to R=12\), whereas \(R\to T=8\). Sequence enrollment explains the difference in these realized counts; it does not by itself establish that the treatments have different carryover effects. The table also makes clear that both switch directions occur only into period 3.
What the Counts Can and Cannot Show
Transition balance is distinct from several other design summaries. Period balance asks how many participants receive each treatment in each period. Sequence balance describes how participants are allocated among complete treatment orders. A transition audit instead counts adjacent ordered pairs. These summaries answer different questions: balanced treatment totals by period do not guarantee balanced transitions, and balanced transitions do not guarantee balanced sequence enrollment.
The standard first-order carryover concern is that the treatment in the preceding period could influence a later response. The transition table identifies which preceding-current combinations the schedule provides and where they occur. It cannot show whether such an influence exists. A statistical model that represents period, current treatment, and potentially prior treatment is a different analysis and requires its own assumptions and adequate design information. A transition count is not a test of a carryover effect.
A practical audit follows this order:
Use the assigned schedule or, if auditing administered treatments, state how incomplete records are handled.
For each sequence, identify the preceding and current treatment for every period after the first.
Check each period total against its number of transition opportunities and the pooled total against the number of participants times \(K-1\), when schedules are complete.
Flag imbalances, absent transitions, or transitions confined to a period; do not treat them as evidence that carryover occurred.
Common Pitfall and Diagnostics
Wrong reasoning: “There are more \(T\to R\) than \(R\to T\) transitions, so treatment T causes more carryover.” This confuses a schedule count with a response effect. The counts describe how often adjacent assignments occur; they do not compare outcomes or identify a causal effect.
Correction: State the imbalance as a property of the schedule, give the period and sequence pattern that produces it, and treat it as a design feature relevant to considering carryover. Any claim about an effect requires outcome data and an appropriate model, not the transition table alone.
When checking a table, look for arithmetic failures first: period totals that do not match sequence enrollment, pooled cells that do not equal their period-specific cells added together, or a grand total inconsistent with the number of transitions. Then inspect structural features: a zero cell, a direction represented only in one period, or a large contribution from one sequence. These are prompts for review, not automatic evidence of a defective design. A transition audit is limited to adjacent treatments; it does not summarize every possible influence of treatment history, adherence, washout, or response measurement.
Check Your Understanding
Answer each question using the transition-counting rules above. Solutions follow the questions.
- For the sequence R–T–T, identify the transitions into periods 2 and 3.
- A hypothetical study has 6 participants in each of T–R–T and R–T–R. How many transitions in total occur across periods 2 and 3, assuming complete schedules?
- For the schedules T–T–R and R–R–T with 5 participants per sequence, calculate each transition type pooled across periods 2 and 3.
- What does a zero \(T\to T\) count establish, and what does it not establish?
Solutions. (1) Into period 2 the pair is \(R\to T\); into period 3 it is \(T\to T\). (2) There are \(2(6+6)=24\) transitions, because 12 participants each contribute two transition occasions. (3) T–T–R contributes \(T\to T\) and \(T\to R\), each 5; R–R–T contributes \(R\to R\) and \(R\to T\), each 5. Pooled counts are therefore 5 for each of the four transition types, totaling 20. (4) It establishes that the audited schedule or records contain no \(T\to T\) transition in the periods counted. It does not establish that carryover occurred, that it is absent, or that any outcome effect has a particular direction or size.