Sequence Coverage in Partial-Replicate Designs

Why Audit a Partial-Replicate Sequence Set?

A partial-replicate design schedules participants to receive formulations over multiple periods, with at least one formulation repeated and another not necessarily repeated. The order of treatments matters: a participant receiving test then reference has a different ordered transition from one receiving reference then test. Before randomizing participants, a design can therefore be checked for two distinct properties: whether its sequence set covers the required adjacent-period transitions, and whether its allocation plan balances formulation assignments across periods.

These are design properties, not evidence about pharmacological effects. A scheduled transition such as R-to-T identifies the order of treatments in adjacent periods; it does not establish that carryover occurs or is absent. Likewise, period balance concerns assigned treatments, not whether outcomes are comparable or the design is free of bias.

Definition: Let R denote the reference formulation and T the test formulation. A sequence is the ordered list of formulations assigned across the study periods; for example, TRR assigns T in period 1 and R in periods 2 and 3. An adjacent-period transition is an ordered pair of treatments in consecutive periods. Thus TRR contains the transitions T-to-R and R-to-R. A sequence set has transition coverage for a specified required set if every required ordered transition occurs at least once somewhere in the set.

The required transitions must be specified for the design being audited. For the three-period reference-replicate set TRR, RTR, RRT, the commonly required adjacent transitions are R-to-R, R-to-T, and T-to-R. T-to-T is not part of that requirement. For a three-period test-replicate set TTR, TRT, RTT, the corresponding set is T-to-T, T-to-R, and R-to-T. Coverage is therefore not a demand to include every theoretically possible pair; it is a check against the protocol's stated design requirements.

Auditing Ordered Transition Coverage

To check coverage, write out each sequence's adjacent pairs and take their union. The direction of each pair must be retained. For example, R-to-T and T-to-R are separate transitions, even though both involve the same two formulations. In a three-period sequence there are two adjacent transitions; in a longer sequence there are one fewer transitions than periods.

Sequence	Period 1 to 2	Period 2 to 3
TRR	T-to-R	R-to-R
RTR	R-to-T	T-to-R
RRT	R-to-R	R-to-T

The union in this reference-replicate set is {R-to-R, R-to-T, T-to-R}, so all three specified transitions are represented. This audit is about whether a transition appears at all. It does not require that every transition occur equally often, nor does it by itself show that the number of participants assigned to each sequence is balanced.

Worked Example: Checking Coverage in a Reference-Replicate Set

A hypothetical three-period design proposes the sequences TRR, RTR, and RRT. The protocol requires R-to-R, R-to-T, and T-to-R transitions. Check whether the set meets the requirement.

Step 1: List the transitions. TRR contributes T-to-R and R-to-R. RTR contributes R-to-T and T-to-R. RRT contributes R-to-R and R-to-T.

Step 2: Combine the distinct transitions. Across the three sequences, the set is {T-to-R, R-to-R, R-to-T}.

Step 3: Compare with the requirement. Each required transition occurs at least once, so the sequence set has the specified transition coverage. The set does not contain T-to-T, but that is not a coverage failure because T-to-T was not required for this reference-replicate design.

Period Balance Depends on Sequence Allocations

Transition coverage is a property of the sequences included. Period balance also depends on how many participants are assigned to each sequence. Let \(n_s\) be the number assigned to sequence \(s\), and let \(N_{T,p}\) be the total number assigned T in period \(p\). Define \(N_{R,p}\) similarly. For a two-formulation design, period balance means that the formulation assignment counts are the same from period to period:

N_{T,1}=N_{T,2}=\cdots=N_{T,P} \quad\text{and}\quad N_{R,1}=N_{R,2}=\cdots=N_{R,P}.

If every assigned participant has one scheduled treatment in every period, then \(N_{T,p}+N_{R,p}\) is the total number assigned in period \(p\). When the same participants are scheduled in every period, equal T counts across periods imply equal R counts as well. It is still useful to report both counts: the formulation totals make the intended balance explicit and can expose inconsistencies in an allocation table.

Definition: For a sequence set with sequence allocations \(n_s\), the number assigned formulation T in period \(p\) is \(N_{T,p}=\sum_s n_s I(s_p=T)\), where \(s_p\) is the treatment in period \(p\) of sequence \(s\), and \(I(s_p=T)\) equals 1 when that condition holds and 0 otherwise. Replace T by R to obtain \(N_{R,p}\). Period balance is assessed from these allocation-weighted counts, not simply from the number of sequence labels in the list.

For the set TRR, RTR, RRT, write the sequence allocations as \(n_{TRR}\), \(n_{RTR}\), and \(n_{RRT}\). T appears in period 1 only in TRR, in period 2 only in RTR, and in period 3 only in RRT. Therefore:

N_{T,1}=n_{TRR},\qquad N_{T,2}=n_{RTR},\qquad N_{T,3}=n_{RRT}.

It follows directly that T assignments are equal across the three periods if and only if \(n_{TRR}=n_{RTR}=n_{RRT}\). Since the total number assigned in each period is then the same, R assignments are equal too. This is a useful design-specific result, not a general rule that all partial-replicate designs must use equal sequence allocations.

Worked Example: Deriving the Allocation Needed for Period Balance

A hypothetical study uses TRR, RTR, and RRT. Let the number assigned to each sequence be \(a\), \(b\), and \(c\), respectively. Derive the condition for period balance.

Count T assignments. Period 1 has T only in TRR, period 2 has T only in RTR, and period 3 has T only in RRT. Thus \(N_{T,1}=a\), \(N_{T,2}=b\), and \(N_{T,3}=c\).

Apply the balance condition. Equal T counts require \(a=b=c\). Each period contains \(a+b+c\) assigned participants in total, so \(N_{R,p}=a+b+c-N_{T,p}\). If \(a=b=c\), the R count is the same in each period as well. Conversely, if the T counts differ, the R counts differ by the corresponding opposite amounts. Therefore period balance holds exactly when \(a=b=c\).

Check with numbers. If \(a=b=c=20\), then each period has 20 T assignments and 40 R assignments, for 60 participants scheduled per period. The assignments are balanced by period.

Coverage and Balance Are Separate Checks

A sequence set can cover every required transition while failing period balance. Conversely, counts could be balanced in each period while some required transition is absent. The audit must test both properties independently.

Worked Example: Full Transition Coverage Without Period Balance

A hypothetical study assigns 20 participants to TRR and 20 to RRT, with no participants assigned to RTR. The required transitions are R-to-R, R-to-T, and T-to-R. Check coverage and period balance.

Coverage. TRR contains T-to-R and R-to-R; RRT contains R-to-R and R-to-T. Their union includes all three required transitions, so coverage is complete.

Period assignments. In period 1, TRR contributes 20 T assignments and RRT contributes 20 R assignments. In period 2, both sequences assign R, giving 0 T and 40 R. In period 3, TRR assigns R and RRT assigns T, giving 20 T and 20 R.

(N_{T,1},N_{T,2},N_{T,3})=(20,0,20),\qquad (N_{R,1},N_{R,2},N_{R,3})=(20,40,20).

The sequence set meets the transition-coverage requirement but is not balanced by period. In particular, period 2 has no T assignments. Adding the omitted sequence is not automatically enough; its allocation must be planned so that the resulting period counts meet the intended balance.

A Practical Audit and Its Limits

A design audit should use the planned sequence list and the planned allocation counts. If the audit concerns the realized study rather than the randomization plan, use the actual numbers assigned to each sequence and state whether the counts refer to randomized participants or to participants with records in all periods. Withdrawals can change the observed counts even when the planned assignment was balanced.

A compact audit can be organized as follows:

Specify the requirement.
List the required ordered adjacent-period transitions for the design; do not assume that every possible pair is required.

Expand the sequences.
For each sequence, record every adjacent pair in its actual order and check whether each required pair appears.

Apply allocation counts.
For each period, sum the participants assigned to R and T across all sequences.

Report both findings.
State separately whether transition coverage is complete and whether formulation assignments are balanced across periods.

The reference-replicate and test-replicate three-period arrangements illustrate why the specified requirement matters. For TRR, RTR, RRT, the required set typically includes R-to-R, R-to-T, and T-to-R. For TTR, TRT, RTT, it typically includes T-to-T, T-to-R, and R-to-T. With equal allocation to the three sequences in either arrangement, each period has the same formulation counts: the reference-replicate arrangement has one-third T and two-thirds R assignments in every period; the test-replicate arrangement has two-thirds T and one-third R assignments in every period. These are different design targets, not interchangeable sequence lists.

An alternative allocation can be considered if the protocol has a reason for unequal sequence sizes, but its consequences should be calculated rather than assumed away. Unequal allocation may preserve transition coverage and still produce period imbalance. Also, transition coverage only confirms that a scheduled ordered pair is represented; it does not ensure equal frequencies of transitions, adequate precision, or absence of carryover. Those questions require their own design and analysis considerations.

Common Mistakes and Diagnostics

Ignoring order. Counting R-to-T as if it were the same as T-to-R can make a missing transition look covered. Write each pair in sequence order.
Checking only the sequence names. A list may look familiar but omit a needed sequence or include it with zero allocation. Expand the actual sequences and use their assigned counts.
Assuming coverage guarantees balance. The TRR/RRT example covers all specified transitions but assigns no T in period 2. Calculate counts period by period.
Assuming equal numbers of sequence labels means equal allocation. A list with one occurrence of each sequence is balanced only if the participant allocation is also equal, for the three-sequence reference-replicate set considered here.
Calling all possible transitions “required.” The requirement depends on the intended replicate design. Check against the design specification rather than adding an unneeded pair.

A useful diagnostic is to display a transition table and a period-count table side by side. A blank required-transition cell indicates a coverage failure; unequal counts in a period-count row indicate allocation imbalance. If the tables disagree with the protocol, check for a transcription error, an omitted sequence, or a mismatch between planned and realized allocations.

Key takeaway: Audit partial-replicate sequence coverage and period balance separately. Coverage is based on the union of ordered adjacent treatment pairs; balance is based on allocation-weighted formulation counts in each period.

Check Your Understanding

Answer each question using the stated requirement and allocation counts.

A hypothetical design uses RTR and RRT. Does it cover R-to-T, T-to-R, and R-to-R?
A reference-replicate design uses TRR, RTR, and RRT, with 12 participants assigned to each. How many T and R assignments are scheduled in each period?
For the same three sequences, allocations are 15 to TRR, 15 to RTR, and 30 to RRT. Give the T assignment count in each period and decide whether the design is period-balanced.
A test-replicate design uses TTR, TRT, and RTT. Which set of distinct adjacent transitions does it cover?

Solutions. (1) RTR contains R-to-T and T-to-R; RRT contains R-to-R and R-to-T. The union is {R-to-T, T-to-R, R-to-R}, so all three required transitions are covered. (2) Period 1 has 12 T assignments from TRR and 24 R assignments from RTR and RRT. Period 2 has 12 T from RTR and 24 R from TRR and RRT. Period 3 has 12 T from RRT and 24 R from TRR and RTR. The counts are balanced across periods. (3) The T counts are 15 in period 1, 15 in period 2, and 30 in period 3. They are unequal, so the design is not period-balanced. (4) TTR contributes T-to-T and T-to-R; TRT contributes T-to-R and R-to-T; RTT contributes R-to-T and T-to-T. The distinct transitions are {T-to-T, T-to-R, R-to-T}.