Enrollment Balance Across Crossover Sequences

Why Enrollment Balance Matters

A crossover study assigns participants to sequences: each sequence specifies the order of treatments across periods. If enrollment differs substantially among sequences, the study may have fewer participants contributing a particular treatment-period combination than expected. That can complicate interpretation and reduce the information available for comparisons. The relevant counts depend on the sequence schedules, so equal enrollment across sequences does not automatically guarantee equal treatment counts in every period.

The practical aim is usually approximate rather than perfect balance. The planned total may not divide evenly by the number of sequences, and the sequence set may make exact treatment-by-period balance mathematically impossible. A useful audit therefore considers both sequence counts and the treatment exposures implied by those counts.

Definition: Let \(S\) be the set of planned sequences and let \(n_s\) be the number of enrolled participants assigned to sequence \(s\). The sequence enrollment counts are balanced when their values are as equal as possible for the given total enrollment. For treatment \(t\) and period \(p\), the planned treatment-period count is the number of enrolled participants whose assigned sequence specifies treatment \(t\) in period \(p\).

These are counts of assigned exposures. If a participant misses a period, withdraws, or does not receive the assigned treatment, the number of actual exposures can differ. It is useful to keep enrollment, assigned schedule, and treatment actually received distinct in study records and summaries.

Dividing Enrollment Across Sequences

Suppose a study plans to enroll \(N\) participants across \(K\) sequences. Divide \(N\) by \(K\):

N=qK+r,\qquad 0\le r<K.

Here \(q\) is the number assigned to every sequence in an initially equal allocation, and \(r\) is the remainder. An allocation as even as possible gives \(r\) sequences \(q+1\) participants each and the other \(K-r\) sequences \(q\) participants each. The total is \(r(q+1)+(K-r)q=qK+r=N\).

This allocation is not just a convenient rule. It is the most even possible integer allocation. To see why, consider any two sequence counts \(a\) and \(b\) with \(a\ge b+2\). Moving one participant from the larger count to the smaller changes their sum of squares by

(a-1)^2+(b+1)^2-a^2-b^2=-2(a-b)+2\le -2.

Thus, whenever two counts differ by at least two, this move strictly reduces the sum of squared counts. Repeating such moves must terminate: the sum of squares is a nonnegative integer and decreases at every move. At termination no two counts differ by more than one. Conversely, a collection of integer counts summing to \(N\) with all differences at most one must consist of \(r\) counts equal to \(q+1\) and \(K-r\) equal to \(q\). This proves the stated allocation rule.

Worked Example: Allocating 23 Participants to Four Sequences

A hypothetical study has four planned sequences and a target enrollment of 23 participants. Find the most even sequence allocation.

Divide the total by the number of sequences: \(23=5(4)+3\). Thus \(q=5\) and \(r=3\). Three sequences receive \(q+1=6\) participants, and one receives \(q=5\).

The allocation is therefore \(6,6,6,5\), in any order across the four sequences. It sums to \(6+6+6+5=23\), and the largest and smallest counts differ by one. An allocation such as \(8,5,5,5\) also sums to 23, but it is less even: the difference between the largest and smallest counts is three.

From Sequence Counts to Treatment-Period Counts

Sequence balance is only one part of the audit. To calculate the treatment-period counts, make a schedule indicator for each sequence: it is 1 when sequence \(s\) assigns treatment \(t\) in period \(p\), and 0 otherwise. The count for treatment \(t\) in period \(p\) is the sum of the enrolled participants in sequences that assign that treatment in that period.

C_{t,p}=\sum_{s\in S} n_s I(s\text{ assigns }t\text{ in period }p),

where \(I(\cdot)\) equals 1 if its statement is true and 0 otherwise. This formula makes clear that the sequence schedule and the sequence counts jointly determine each treatment-period count. In a complete planned schedule with no missing assigned periods, summing \(C_{t,p}\) over treatments for a given period gives the total number enrolled.

Worked Example: Auditing Treatment-Period Counts in a Three-Period Design

Consider a hypothetical three-period design with sequences RRT, RTR, and TRR, where R denotes the reference treatment and T the test treatment. Suppose their enrollment counts are 4, 4, and 3, respectively. Calculate the planned count for each treatment in each period.

In period 1, RRT and RTR assign R, while TRR assigns T. The counts are \(C_{R,1}=4+4=8\) and \(C_{T,1}=3\).

In period 2, RRT and TRR assign R, while RTR assigns T. Thus \(C_{R,2}=4+3=7\) and \(C_{T,2}=4\).

In period 3, RTR and TRR assign R, while RRT assigns T. Thus \(C_{R,3}=4+3=7\) and \(C_{T,3}=4\).

Period	Reference (R)	Test (T)	Total
1	8	3	11
2	7	4	11
3	7	4	11

The sequence counts differ by only one, yet period 1 has five more planned R than T exposures. The aggregate schedule still has 22 R exposures and 11 T exposures: each participant is assigned two R periods and one T period, so \(11(2)=22\) and \(11(1)=11\). This example shows why a sequence-count summary cannot substitute for a treatment-by-period count.

Allocation Choices When Exact Balance Is Impossible

The appropriate allocation target depends on what balance is intended to accomplish. When the main goal is similar representation in each sequence, the quotient-and-remainder allocation is a direct target. When treatment-by-period counts are also important, calculate those counts for candidate sequence allocations before enrollment. The best available allocation may not make every period equally balanced, particularly when the design has an unequal number of T and R exposures or when the target total is small.

Worked Example: Comparing Equally Even Allocations

Suppose a hypothetical study plans to enroll 10 participants across the same sequences, RRT, RTR, and TRR. Since \(10=3(3)+1\), the most even sequence counts are permutations of \(4,3,3\). Compare two choices for the sequence receiving four participants.

Choice A: Assign counts 4, 3, and 3 to RRT, RTR, and TRR, respectively. In period 1, R receives \(4+3=7\) participants and T receives 3. In period 2, R receives \(4+3=7\) and T receives 3. In period 3, R receives \(3+3=6\) and T receives 4.

Choice B: Assign counts 3, 4, and 3 to RRT, RTR, and TRR. In period 1, R receives \(3+4=7\) and T receives 3. In period 2, R receives \(3+3=6\) and T receives 4. In period 3, R receives \(4+3=7\) and T receives 3.

Both choices have equally balanced sequence counts, but they place the relatively larger treatment-period difference in different periods. Neither can make all three periods exactly 5 R and 5 T: period 1 has T only in TRR, so that exact count would require \(n_{\mathrm{TRR}}=5\). Period 2 has T only in RTR, requiring \(n_{\mathrm{RTR}}=5\), and period 3 has T only in RRT, requiring \(n_{\mathrm{RRT}}=5\). Those three requirements would total 15 participants, not 10. The schedule and target enrollment therefore rule out exact balance in every period.

Common practical approaches include setting sequence-specific enrollment targets, choosing a feasible target total that divides more evenly across sequences, and monitoring realized counts during enrollment. If important participant characteristics are used to organize allocation, balance should be considered within those groups as well as overall; many small groups can make close balance difficult. Allocation procedures must preserve the study’s planned random assignment and concealment safeguards. The particular construction and management of randomization lists is a separate design task.

When treatment-period balance is a specific concern, candidate allocations can be compared using simple summaries such as the largest absolute difference between treatment counts across periods, \(\max_p |C_{T,p}-C_{R,p}|\). This is a descriptive planning criterion, not a universal statistical optimality rule. It should be considered alongside sequence balance and the design’s scientific objectives; minimizing one count difference does not by itself establish that an allocation is best for every analysis.

Diagnostics, Pitfalls, and Limitations

An enrollment audit should list both the number assigned to each sequence and the treatment-by-period counts implied by those assignments. Compare the realized sequence counts with the planned targets, then recompute the treatment-period counts from the actual schedule. A mismatch between the two summaries can reveal a counting error, an incorrect sequence label, or an unanticipated allocation pattern. If actual treatment receipt is being audited, it needs a separate summary because missed periods and withdrawals can change those counts.

Do not assume equal sequence counts mean equal treatment-period counts. The schedule determines which sequences contribute to each treatment-period cell.
Do not confuse assigned exposures with completed exposures. A participant assigned to a treatment in a period may not actually receive it.
Do not interpret a small difference as proof of good implementation. Check the intended targets and allocation records, not just a final summary.
Do not force an impossible balance. Integer totals and sequence structure can make exact balance unattainable.

A further limitation is that counts alone do not describe all aspects of a design or its analysis. Similar sequence counts do not guarantee similar participant characteristics, complete observations, or equal precision for every scientific contrast. Enrollment balance is a planning and monitoring tool, not a substitute for evaluating the design’s assumptions or analyzing the observed data.

Key takeaway: First calculate the most even feasible sequence counts from the total enrollment. Then use the sequence schedules to calculate treatment-by-period counts. Report both, and distinguish planned assignments from treatments actually received.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A study plans 26 participants across five sequences. What is the most even allocation?
In a two-sequence, two-period design, 9 participants are assigned TR and 8 are assigned RT. Give the T and R counts in each period.
For sequences RRT, RTR, and TRR, counts are 2, 3, and 4. Calculate the R and T counts in each of the three periods.
Why can equal enrollment across sequences fail to produce equal T and R counts in every period?

Solutions. (1) Since \(26=5(5)+1\), one sequence receives 6 participants and each of the other four receives 5. (2) In period 1, TR assigns T to 9 and RT assigns R to 8; in period 2, the counts reverse, giving R=9 and T=8. (3) Period 1: R=\(2+3=5\), T=4. Period 2: R=\(2+4=6\), T=3. Period 3: R=\(3+4=7\), T=2. Each period total is 9, as required. (4) The sequence schedules determine which treatment appears in each period. If the schedule or total enrollment constrains the number of participants assigned to each treatment in a particular period, equal sequence counts may still yield unequal treatment-period counts.