Why Trace Treatment History?
In a crossover study, a participant receives treatments in successive periods. The outcome in a period is intended to reflect the treatment given then, but residual drug or a lingering pharmacodynamic effect from an earlier period could also contribute. This possibility makes the treatment immediately before an administration important: the same current treatment may follow different formulations in different sequences.
A schedule audit therefore asks more than how many times each formulation appears or whether formulations are spread across periods. It records, for each administration, which treatment immediately preceded it. The resulting directed transitions reveal whether a current formulation follows the test treatment, the reference treatment, or no earlier study treatment. A transition is a design feature that may signal a risk to investigate; it is not proof that carryover occurred.
Use \(T\) for test and \(R\) for reference. A transition audit follows the actual order within each sequence, not just the total number of \(T\) and \(R\) administrations. It is also important to distinguish assigned schedule from received treatment: if a participant misses or does not take an administration, the relevant history for that participant may differ from the planned sequence.
How to Read a Schedule
For each sequence, list the administration in each period and annotate it with its immediate predecessor. Then count the directed transitions across the sequence groups. Keep \(T \to R\) and \(R \to T\) separate: their possible residual effects need not be equal. Likewise, \(T \to T\) and \(R \to R\) are distinct transitions, not interchangeable instances of “same treatment.”
For a compact summary, let \(N_{ab}\) be the number of observed scheduled transitions from treatment \(a\) to treatment \(b\), where \(a,b \in \{T,R\}\). Counts can be reported within each sequence and summed across sequence groups. When sequence groups have different numbers of participants, totals should account for those group sizes rather than treating each sequence pattern as if it were equally common.
Worked Example: Auditing a Four-Period Full-Replicate Schedule
Suppose a hypothetical study assigns equal numbers of participants to sequences \(TRTR\) and \(RTRT\). Each participant receives each formulation twice. Annotate the immediate predecessor for every period and compare the transition counts.
For \(TRTR\), the predecessor annotations are: period 1, none; period 2, \(T\); period 3, \(R\); period 4, \(T\). The transitions are \(T \to R\), \(R \to T\), and \(T \to R\), giving two \(T \to R\) and one \(R \to T\) per participant in this sequence.
For \(RTRT\), the annotations are: period 1, none; period 2, \(R\); period 3, \(T\); period 4, \(R\). The transitions are \(R \to T\), \(T \to R\), and \(R \to T\), giving one \(T \to R\) and two \(R \to T\) per participant.
With equal group sizes, the combined counts per matched pair of sequence participants are \(N_{TR}=2+1=3\) and \(N_{RT}=1+2=3\). Neither sequence alone has equal directional transitions, but the paired allocation balances them in aggregate. The schedule still does not establish that residual exposure is absent; it shows how the planned predecessor patterns are distributed.
What Transition Balance Can and Cannot Show
A useful way to see why predecessor balance matters is to write a simple first-order response model. This is a conceptual model for the audit, not a claim that carryover has been measured or a recommendation to use a particular inferential procedure:
Here \(Y_{ij}\) is participant \(i\)'s response in period \(j\); \(b_i\) represents a participant effect; \(\pi_j\) is a period effect; \(d_{ij}\) is the current treatment; \(\tau\) is its direct effect; and \(\kappa\) represents a possible effect of the immediately preceding treatment. In period 1, there is no previous study treatment, so the predecessor contribution is defined as zero. This model assumes that any carryover relevant to the response depends on the immediately previous treatment and enters additively. Those assumptions may be inadequate if effects persist over more than one period or depend on the particular current–previous combination.
The implication for a current-treatment comparison follows by averaging the model within each current-treatment group. If \(\bar{\kappa}_T\) is the average predecessor contribution among observations currently receiving \(T\), and \(\bar{\kappa}_R\) is the corresponding average among those currently receiving \(R\), then the expected difference in those group means includes:
The “other terms” include, for example, period contributions if current treatments occur in different periods and period effects are not accounted for. Thus, when predecessor histories differ between current-treatment groups, an apparent current-treatment contrast could include a carryover component under this model. If predecessor contributions are balanced, that particular component cancels in the contrast; the audit cannot prove the contributions are equal or that the model captures the biology.
Worked Example: Checking a Three-Period Reference-Replicate Layout
Consider the hypothetical sequence groups \(TRR\), \(RTR\), and \(RRT\), with equal numbers of participants assigned to each. This is a reference-replicate arrangement: each sequence contains two \(R\) administrations and one \(T\) administration. Count its directed transitions.
In \(TRR\), period 1 has no predecessor, period 2 follows \(T\), and period 3 follows \(R\). Its transitions are \(T \to R\) and \(R \to R\).
In \(RTR\), the transitions are \(R \to T\) and \(T \to R\). In \(RRT\), they are \(R \to R\) and \(R \to T\).
Summing one participant from each sequence gives \(N_{TR}=2\), \(N_{RT}=2\), \(N_{RR}=2\), and \(N_{TT}=0\). The layout balances the two directions between different formulations across these equal-sized groups, while also containing reference-to-reference transitions. It has no test-to-test transition. That asymmetry is a property to note when considering possible residual exposure; it is not by itself a reason to declare the layout invalid. The scientific relevance depends on the formulations and the possible residual effects.
Unequal Enrollment and a Second Replicate Pattern
A transition pattern that is balanced across sequence labels can become unbalanced in the enrolled sample if sequence-group sizes differ. For example, suppose the three groups in the preceding example had sizes \(n_{TRR}=12\), \(n_{RTR}=8\), and \(n_{RRT}=10\). Weight each sequence’s transition counts by its participants:
The transition totals are now not equal, even though each sequence pattern contributes the same number of participants in the equal-allocation audit. This calculation describes planned exposure histories under the stated enrollment; it does not account for deviations from assigned treatment or missing administrations.
Worked Example: Auditing a Three-Period Test-Replicate Layout
Now consider hypothetical sequence groups \(TTR\), \(TRT\), and \(RTT\), with equal numbers of participants. Each participant receives \(T\) twice and \(R\) once. Determine which preceding treatments occur before each later administration.
For \(TTR\), the transitions are \(T \to T\) and \(T \to R\). For \(TRT\), they are \(T \to R\) and \(R \to T\). For \(RTT\), they are \(R \to T\) and \(T \to T\).
The pooled counts for one participant in each group are \(N_{TT}=2\), \(N_{TR}=2\), \(N_{RT}=2\), and \(N_{RR}=0\). The two between-formulation directions occur equally often, while this layout contains test-to-test but not reference-to-reference transitions. As in the reference-replicate example, the zero count is a design fact, not evidence about the magnitude of any residual effect.
Assumptions, Diagnostics, and Failure Modes
A transition audit is descriptive and depends on an accurate account of treatment actually received and the ordering of periods. Interpreting its counts as relevant to residual exposure also requires a scientific rationale: for instance, that the previous formulation could plausibly leave drug, a metabolite, or a pharmacodynamic effect capable of influencing a later response. Schedule labels alone contain no concentration or effect measurements.
Several checks make the audit more informative:
- Check the first period. It has no preceding study formulation; do not count it as an \(R \to T\) or \(T \to R\) transition.
- Retain direction. Do not combine \(T \to R\) with \(R \to T\) unless there is a justified reason to treat their risks as equivalent.
- Use actual group sizes. If sequence enrollment differs, weight sequence-specific transition counts by participants rather than comparing sequence patterns alone.
- Check treatment received. If doses were missed or sequence deviations occurred, the planned predecessor may not be the actual predecessor.
- Consider the scope of the audit. Immediate-predecessor counts do not describe effects persisting from two or more periods earlier, nor do they establish how much exposure remains.
One tempting response to a concerning pattern is to rely on a statistical test for carryover and treat a non-significant result as proof of no carryover. That conclusion is not justified: failure to detect an effect does not demonstrate its absence, and the test may have limited ability to distinguish carryover from other design features. The schedule audit instead identifies where the design makes prior treatment histories differ and where scientific or pharmacokinetic evidence may be needed.
Comparing Design Questions
Period balance, treatment-exposure balance, and transition balance are complementary rather than interchangeable. Period balance asks whether treatments are represented across periods. Treatment-exposure balance asks how often participants receive each formulation. Transition balance asks which formulation immediately precedes each administration. A schedule may do well on one and still show an asymmetry on another.
A first-order transition table is also only one representation of history. It is useful when the immediate predecessor is the relevant concern, but it cannot represent longer memory or an interaction between current and previous formulation without additional structure. Conversely, a table with balanced transitions does not eliminate a biological carryover effect shared across transition types. The limitation is important: the audit diagnoses features of a design, not a treatment’s pharmacokinetic behavior.
Check Your Understanding
For each question, try the audit first, then compare your reasoning with the solutions.
- In sequence \(RTT\), what are the two transitions, and which period has no predecessor?
- Across equal-sized groups \(TRT\) and \(RTR\), how many \(T \to R\) and \(R \to T\) transitions occur per pair of participants?
- Why can equal numbers of \(T\) and \(R\) administrations fail to guarantee equal numbers of \(T \to R\) and \(R \to T\) transitions?
- A schedule has equal directed transition counts. Does this establish that carryover is absent? Explain.
- What does a first-order transition audit omit if an effect persists from two periods earlier?
Solutions. (1) The transitions are \(R \to T\) and \(T \to T\); period 1 has no predecessor. (2) \(TRT\) contributes one \(T \to R\) and one \(R \to T\); \(RTR\) contributes one of each as well, for two in each direction per pair. (3) Exposure counts ignore order: the same totals can occur in sequences with different predecessor patterns. (4) No. Equal counts describe the schedule, not whether residual drug or effects occur or whether their contributions are equal. (5) It records only the immediate predecessor, so it does not distinguish histories that differ in treatments given two or more periods earlier.