Treatment-Period Balance in Crossover Schedules

Why check treatment-period balance?

A crossover schedule assigns participants to an order of treatments across study periods. A basic design question is whether each treatment is scheduled equally often in each period. If one treatment appears much more often in an early period than another, the schedule has a treatment-period imbalance. Counting assignments before a study begins can reveal this feature without analyzing any pharmacokinetic outcomes.

The check is about the structure of the proposed schedule, not whether treatment differences are statistically significant. It does not establish that the treatments have the same effect, nor does it show that the schedule was randomized appropriately. It answers a narrower question: how many participants are assigned to receive each treatment in each period?

For a complete crossover schedule, each participant contributes one treatment assignment to each period. The period total is therefore the number assigned to the schedule, provided every participant has an assignment in every period. The table can be built from the planned treatment sequences: count how many sequences place each treatment in each period, then multiply by the number assigned to each sequence.

Targets and departures from balance

Suppose \(J\) treatments are intended to appear equally often in each period and \(N_k\) participants are scheduled for period \(k\). The exact target for each treatment in that period is \(N_k/J\). When this is not an integer, exact equality cannot be achieved; the closest integer counts will differ by at least one. For a balanced complete schedule with the same \(N\) participants in every period, \(N_k=N\).

A useful first diagnostic is simply to compare the cells in each period column. If a compact numerical summary is helpful, the largest absolute departure, \(\max_{j,k}|d_{jk}|\), identifies the most discrepant cell from the equal-count target. Another descriptive measure is the within-period range, the largest treatment count minus the smallest treatment count. These summaries describe the size of the departure; neither is a hypothesis test or a universal acceptability threshold. The design objective and practical context determine whether a small discrepancy matters.

Check the table margins as well as its cells. A balanced total number of assignments to each treatment across the entire study does not guarantee balance within periods: a treatment may be overrepresented in one period and underrepresented in another. Conversely, a table can be balanced within periods while the participants are distributed unevenly among the named sequences in a more complex schedule. The cell counts answer the treatment-by-period question, while sequence totals describe a different feature of the schedule.

Worked examples

Why the two-sequence check is especially simple

In a two-period, two-sequence AB/BA schedule, treatment-period balance is equivalent to equal sequence counts. Let \(n_{AB}\) and \(n_{BA}\) be the assigned numbers. Period 1 has \(n_{AB}\) assignments to A and \(n_{BA}\) to B; period 2 has \(n_{BA}\) to A and \(n_{AB}\) to B. Therefore, both periods have equal treatment counts exactly when \(n_{AB}=n_{BA}\).

This gives a short justification for the shortcut: setting the two treatment counts equal in period 1 requires \(n_{AB}=n_{BA}\); the same equality then makes period 2 balanced as well. The shortcut depends on this particular two-period, two-sequence structure. For schedules with more treatments, periods, or sequences, construct the treatment-by-period table rather than assuming that equal sequence totals are either necessary or sufficient for the desired cell balance.

How to audit a proposed schedule

What balance does and does not show

Treatment-period balance is a design property expressed in counts. It can make the schedule more even in the representation of treatments across periods, but the count table alone cannot show that assignment was random, that treatment groups are comparable in every other respect, or that the study will meet a bioequivalence criterion. Those questions require information and analyses beyond this arithmetic audit. The table also cannot diagnose outcome-based issues: it contains assignment counts, not concentration measurements or participant responses.

When reviewing a schedule, keep three checks distinct. First, examine the treatment-by-period cells for the balance specified by the design. Second, inspect sequence totals to understand how participants are distributed among treatment orders. Third, verify that the planned counts describe the assignments actually intended; if participants later fail to complete periods or have missing observations, realized counts may differ from the planned table. A schedule can be balanced as proposed and become unbalanced in the observed data. Report which table is being assessed.

A useful comparison is with an overall treatment total. Overall totals collapse the period dimension and can conceal offsetting excesses and deficits, as in the AB/BA example with 26 and 14 participants. Sequence totals retain the order information but are not a substitute for cell counts in a general design. The treatment-by-period table is the direct tool for the particular question of whether each treatment appears equally often in each period.

Common mistakes and limitations

• Checking only row totals. Equal totals for each treatment across the whole study can conceal an excess in one period and a deficit in another. Compare treatments within each period.
• Counting sequence names instead of participants. If one sequence has 12 assigned participants and another has 8, the schedule has counts 12 and 8—not one and one.
• Assuming exact equality is always possible. With 20 participants and three treatments in a period, equal integer counts are impossible. State the target and the unavoidable rounding pattern.
• Treating balance as evidence of randomization or equivalence. Equal assignment counts are descriptive properties of a schedule; they do not prove random allocation or a treatment effect.
• Using a two-sequence shortcut in a more complex schedule. Derive the cells from the actual sequences whenever the structure is not the two-period AB/BA design.

A further limitation is that “balanced” depends on the intended target. Some schedules do not aim for equal representation of every treatment in every period, for example when constraints require unequal allocation. In that case, compare observed planned counts with the stated target rather than labeling every departure from equality a design error. A clear audit names the target, the counts, and the size of any departure.