Worked Four-Period Full-Replicate Layout

What the Four-Period Full-Replicate Layout Shows

A four-period full-replicate layout gives each participant two planned administrations of the test formulation and two of the reference formulation. The sequence labels show the treatment assigned in each successive period: T denotes test and R denotes reference. In the TRTR/RTRT layout, participants are assigned to one of two sequences, each of which contains both formulations twice.

The key checks are distinct. First, inspect each sequence to confirm that it repeats both formulations within a participant. Second, count the number assigned to each formulation in each period to assess treatment-by-period balance. Equal total numbers of T and R administrations do not, by themselves, guarantee that the treatments are balanced within every period.

This is a schedule-level audit: it describes planned assignments, not whether participants completed every period or received every assigned dose. Its basic assumptions are that T and R consistently identify the same formulations throughout the table, each participant follows one complete four-period sequence as assigned, and the displayed sequence counts describe the participants being audited. If actual administrations are incomplete, the planned layout and the observed dosing records should not be conflated.

Build the Sequence Table

Start with periods as columns and the sequence assignments as rows. The sequence pattern makes the within-participant replication visible before any participants are counted.

This table establishes the replication property for either assigned sequence: every row contains two T entries and two R entries. It does not yet establish period balance in a particular study. That depends on how many participants are assigned to each row.

Count Treatments Within Each Period

Let \(n_A\) be the number of participants assigned to TRTR and \(n_B\) the number assigned to RTRT. Each participant contributes one planned treatment assignment to each period. Consequently, the count in each treatment-by-period cell follows directly from the sequence row.

For example, in period 1 every TRTR participant receives T and every RTRT participant receives R. In period 2 those assignments reverse; the same pattern repeats in periods 3 and 4. This yields a simple condition for exact treatment-by-period balance: the two sequence counts must be equal.

A related check concerns the total number of planned administrations across all periods. Each participant contributes two T and two R administrations, so the total T and R counts are equal for any sequence allocation, even when the counts differ between sequences. This equality across the whole study is not the same as equality of T and R within each period.

Why Equal Sequence Counts Give Period Balance

The balance condition can be justified by considering the two possible entries in each period. In periods 1 and 3, TRTR participants receive T and RTRT participants receive R, so the difference between the T and R counts is \(n_A-n_B\). In periods 2 and 4, the assignments switch, so the difference is \(n_B-n_A\). All four differences are zero precisely when \(n_A=n_B\).

If \(n_A=n_B\), both expressions equal zero, proving balance in every period. Conversely, if period 1 is balanced, then \(n_A-n_B=0\), which requires \(n_A=n_B\). Thus the condition is both sufficient and necessary for exact balance under this two-sequence layout.

Diagnostics, Alternatives, and Limits

A practical audit should display the assigned sequence counts and the treatment counts by period, rather than relying on a statement that the study is “balanced.” A useful diagnostic table has one row per period and separate T and R counts, as in the examples. If a period total does not equal the number assigned to the study, check for a transcription error, a missing sequence assignment, or a table that mixes planned assignments with observed administrations.

Also verify that each listed sequence has two entries of each formulation. A sequence containing three T assignments and one R assignment is not the stated full-replicate layout, even if the overall study happens to have equal numbers of T and R administrations. The count should be checked at both levels: within each participant's assigned sequence and across participants within each period.

The four-period full replicate differs from shorter alternatives in what it repeats. A two-period crossover assigns each participant one T and one R, so it does not provide repeated administrations of each formulation within the participant. A three-period partial-replicate layout repeats one formulation but not both for every participant. Those alternatives can be appropriate for different design goals, but neither has the same per-participant replication pattern as TRTR/RTRT. The earlier material on period balance in replicate designs provides broader context for why period-specific counts matter.

The schedule alone does not demonstrate that participants completed all periods, that missing observations are absent, or that any other design concern has been resolved. In particular, equal treatment-by-period counts are a property of assigned counts, not proof that observed data are complete or that the design is free of all possible bias. The audit answers a narrower question: whether this sequence layout repeats both formulations as intended and balances their planned counts by period.