What the Three-Period Partial-Replicate Layout Specifies
A schedule audit checks what each participant is assigned to receive in each study period. For a three-period partial-replicate layout, the two sequences are TRR and RRT: T denotes the test formulation and R the reference formulation. Participants assigned to TRR receive test in period 1 and reference in periods 2 and 3. Participants assigned to RRT receive reference in periods 1 and 2 and test in period 3.
The word partial describes which formulation is repeated within a participant: reference is given twice, while test is given once. The schedule is not a design in which both formulations are repeated for every participant. The central audit therefore has two parts: count formulations in each period, and separately verify the number of each formulation received by each participant. Period totals alone cannot establish the participant-level repetition.
The audit assumes that T and R are consistently labelled, that each participant has one assigned formulation in each of the three periods, and that the schedule records the intended assignments accurately. It is useful to keep assigned treatment distinct from treatment actually administered: an assignment table audits the planned design, while a record of administrations audits implementation. The examples below identify which is being counted.
Constructing the Schedule and Its Expected Counts
Write each sequence across the three period columns before listing participants. Then give every participant one row and record that participant's assigned sequence and period-by-period formulation. This makes the repeated reference visible within a row, rather than relying on totals at the bottom of the table.
| Sequence | Period 1 | Period 2 | Period 3 | Per-participant total |
|---|---|---|---|---|
| TRR | T | R | R | 1 T, 2 R |
| RRT | R | R | T | 1 T, 2 R |
Suppose \(a\) participants are assigned to TRR and \(b\) are assigned to RRT. The period counts follow directly from the sequence definitions. In period 1, TRR participants receive T and RRT participants receive R. Every participant receives R in period 2. In period 3, TRR participants receive R and RRT participants receive T.
Across all periods, the number of T administrations is \(a+b\), one for each participant. The number of R administrations is \(2a+2b=2(a+b)\), two for each participant. Thus the total number of administrations is \(3(a+b)\), as it must be for a complete three-period schedule. These are useful consistency checks, but they do not replace checking each row.
If the two sequences have equal enrollment, \(a=b\), periods 1 and 3 each have equal numbers of T and R. Period 2 is different by design: all participants receive R. It is therefore incorrect to require each formulation to appear equally often in every period. Sequence counts, period counts, and participant-level repetition answer different audit questions.
Worked Example: Building a Balanced Eight-Participant Schedule
Consider a hypothetical cohort of eight participants, with four assigned to TRR and four to RRT. Construct the schedule, then check the counts by period and by participant.
| Participant | Assigned sequence | Period 1 | Period 2 | Period 3 |
|---|---|---|---|---|
| P01 | TRR | T | R | R |
| P02 | TRR | T | R | R |
| P03 | TRR | T | R | R |
| P04 | TRR | T | R | R |
| P05 | RRT | R | R | T |
| P06 | RRT | R | R | T |
| P07 | RRT | R | R | T |
| P08 | RRT | R | R | T |
Period 1. P01–P04 receive T, so the T count is 4. P05–P08 receive R, so the R count is 4.
Period 2. All eight participants receive R. The period counts are 0 T and 8 R.
Period 3. P01–P04 receive R and P05–P08 receive T. The counts are 4 R and 4 T.
Participant check. Each of the four TRR rows contains one T and two R administrations. Each of the four RRT rows also contains one T and two R administrations. Overall there are \(4+4=8\) T administrations and \(8+8=16\) R administrations, or \(24=8\times3\) total administrations. The participant-level and period-level checks agree.
Why the Counts Follow From the Sequences
The count formulas can be justified by adding the contributions of each sequence. Each TRR participant contributes one T to period 1, one R to period 2, and one R to period 3. Each RRT participant contributes one R to period 1, one R to period 2, and one T to period 3. Multiplying these contributions by the number assigned to each sequence gives the table of expected counts.
This derivation is also a practical debugging aid. If a reported period count does not match the formula for the actual sequence sizes, either the sequence totals or one or more period entries need review. Matching the formulas is necessary, but—as the next example shows—not sufficient to establish that every participant followed a valid sequence.
Worked Example: Auditing Unequal Sequence Counts
Suppose a hypothetical schedule has five participants assigned to TRR and four assigned to RRT. Compute expected formulation counts for each period, then compare them with a schedule summary reporting period 1 as 5 T and 4 R, period 2 as 0 T and 9 R, and period 3 as 4 T and 5 R.
Here \(a=5\) and \(b=4\). In period 1, TRR contributes five T administrations and RRT contributes four R administrations: 5 T and 4 R. In period 2, all \(a+b=9\) participants receive R: 0 T and 9 R. In period 3, RRT contributes four T administrations and TRR contributes five R administrations: 4 T and 5 R. The reported summary agrees with all three expected period counts.
Across periods, the expected T total is \(5+4=9\), one for each of the nine participants. The expected R total is \(4+9+5=18\), two for each participant. Together the counts give \(9+18=27=9\times3\) administrations. If the participant rows also show five TRR patterns and four RRT patterns, the schedule passes these structural checks. The unequal enrollment does not invalidate the layout; it simply makes the period 1 and period 3 margins unequal.
Participant-Level Checks and Failure Modes
A reliable audit keeps the assigned sequence beside the three period entries and compares them row by row. For each participant, check both the sequence order and the formulation totals. The totals should be one T and two R, but the order must also match TRR or RRT. A row with R, T, R has one T and two R, yet it is neither allowed sequence.
Period totals can conceal offsetting participant-level mistakes. For example, one participant might receive T twice while another receives R three times. Their entries can offset in the aggregate, leaving period totals that look exactly as expected. Therefore, a successful margins check cannot certify the within-participant reference repetition.
Worked Example: Detecting Errors Hidden by Correct Period Totals
Four hypothetical participants are intended to be assigned two each to TRR and RRT. The observed assignment table is below. Check its period totals and then audit each row against the valid sequences.
| Participant | Intended sequence | Observed period 1 | Observed period 2 | Observed period 3 |
|---|---|---|---|---|
| P01 | TRR | T | R | R |
| P02 | RRT | R | R | T |
| P03 | TRR | T | R | T |
| P04 | RRT | R | R | R |
Period totals. In period 1, P01 and P03 receive T and P02 and P04 receive R: 2 T and 2 R. Period 2 contains 4 R and 0 T. In period 3, P02 and P03 receive T and P01 and P04 receive R: 2 T and 2 R. These are exactly the expected margins for \(a=b=2\).
Participant rows. P01 follows TRR and P02 follows RRT. P03's row is TRT, so P03 receives T twice and R once, rather than the required one T and two R. P04's row is RRR, so P04 receives no T and R three times. Both rows disagree with their intended sequence. The errors offset: the total remains four T and eight R, but the schedule does not give every participant a repeated reference administration. The two participant rows must be corrected or reconciled with the administration record; correct aggregate counts are not enough.
How to Audit a Schedule Systematically
A useful audit proceeds from the sequence definitions to the individual entries and then back to the totals. This order avoids treating plausible margins as proof that the participant assignments are valid.
Write TRR as T, R, R and RRT as R, R, T, in period order. Confirm that the sequence labels used in the schedule have this meaning.
Verify that all three periods are present and that the row matches the participant's assigned sequence. Count one T and two R within the row.
Tabulate T and R separately in each period and compare the results with the counts implied by the TRR and RRT enrollment totals.
For \(N=a+b\) complete participants, confirm \(N\) T administrations, \(2N\) R administrations, and \(3N\) total period entries. Investigate any mismatch rather than adjusting totals without checking source rows.
Comparison, Interpretation, and Limits
The two-period crossover uses two treatment periods, whereas TRR/RRT adds a third period so that reference is repeated within each participant. The important structural distinction is not simply the extra row or period: it is the planned within-participant repetition of R, while T appears once. A four-period full-replicate layout has a different repetition structure because it repeats both formulations; it should not be mistaken for the three-period partial replicate.
The TRR/RRT layout also does not provide equal representation of both formulations in every period. Period 2 is deliberately all reference, and the formulation mix in periods 1 and 3 depends on how many participants are assigned to each sequence. Thus, a reviewer should not call the schedule unbalanced merely because period 2 contains no test administrations. The appropriate check is whether the observed counts follow from the specified sequence allocation.
Schedule validity is only a statement about assignments and administrations. It does not by itself establish that the study was randomized properly, that periods are free of carryover, that washout is adequate, or that all planned observations are available for analysis. Those questions require separate design and data checks. Nor does a valid TRR/RRT table prove that any pharmacokinetic or equivalence conclusion is correct. Its role is narrower and essential: to confirm the intended treatment sequence and repeated-reference structure.
Common Pitfall
Check Your Understanding
Answer each question by using the sequence definitions and participant-level checks above. Brief solutions follow the questions.
- For six TRR and six RRT participants, how many T and R administrations are expected in each period?
- A participant's period entries are R, R, T. Which sequence is this, and how many times is reference administered?
- Can period 1 counts of 7 T and 5 R be consistent with this layout? If so, how many participants are assigned to each sequence?
- A schedule has the correct overall totals of 10 T and 20 R for 10 participants. Does this prove each participant received reference twice? Explain.
- What is wrong with a participant row of T, R, T, even though it contains both formulations and has three entries?
Solutions. (1) With \(a=b=6\), period 1 has 6 T and 6 R; period 2 has 0 T and 12 R; period 3 has 6 T and 6 R. (2) The sequence is RRT, with two R administrations and one T. (3) Yes. Period 1 T count equals \(a=7\), and its R count equals \(b=5\), so the allocation is seven TRR and five RRT participants. (4) No. The totals are necessary, but participant-level errors can offset; inspect each row for one T and two R in the assigned order. (5) T, R, T gives two T and one R, so it does not follow TRR or RRT and does not repeat reference within that participant.