Why Summarize Enrollment by Time?
A trial can have nearly equal total numbers assigned to two arms while assigning most early participants to one arm and most late participants to the other. If prognosis or care changes over calendar time, those totals conceal a potentially important feature of the trial. An arm-by-time enrollment table makes the pattern visible by showing who was assigned to each arm in each enrollment period.
This table is a practical companion to allocation-probability timelines: the timeline displays the probabilities used by the randomization procedure, while the table displays the assignments that actually occurred. Both matter. A probability can shift without the realized counts changing much in a small period, and random variation can make realized counts differ from the probabilities. A table also allows outcome availability to be summarized alongside enrollment, provided the time point at which availability is assessed is stated.
Define the Table Before Reading It
Choose intervals that are meaningful for the trial, such as calendar months, enrollment cohorts, or protocol-defined epochs. State the boundaries and use the same enrollment-date rule throughout. A participant belongs to the interval containing the date of random assignment, not the date when the outcome is measured or reported. If enrollment is sparse, many narrow intervals produce a noisy table with small cell counts; overly broad intervals can conceal changes. The choice is descriptive and should be made transparently rather than selected to make a particular pattern appear stronger.
Let \(n_{kt}\) denote the number assigned to arm \(k\) in time interval \(t\), and let \(N_t\) denote the total number assigned in that interval. The realized allocation proportion for arm \(k\) in interval \(t\) is \(n_{kt}/N_t\), when \(N_t>0\). These are proportions of assignments actually observed, not the randomization probabilities at each assignment. With two arms, the within-interval proportions add to one.
For outcome availability, use a separate count. Let \(r_{kt}\) be the number of participants assigned to arm \(k\) in interval \(t\) whose specified outcome is available at the stated cutoff. The availability proportion is \(r_{kt}/n_{kt}\), provided \(n_{kt}>0\). Do not use the number with observed outcomes as the denominator for the allocation proportion: doing so would describe the observed-outcome subset, not the assignments in that interval.
Availability is time-dependent. For a delayed outcome, a participant assigned recently may still be pending rather than lost to follow-up. The table should therefore include a data cutoff and, where useful, distinguish resolved outcomes, pending outcomes, and dropout. That distinction prevents a recent enrollment period from appearing to have poor outcome availability simply because participants have had less time to reach the outcome assessment.
Worked Example: A Shift Hidden by Equal Arm Totals
Worked Example: A Shift Hidden by Equal Arm Totals
Suppose a hypothetical two-arm trial enrolls 120 participants in three consecutive periods. The observed assignments are:
| Enrollment period | Arm A | Arm B | Total | Arm A proportion |
|---|---|---|---|---|
| Early | 30 | 10 | 40 | 30/40 = 0.750 |
| Middle | 20 | 20 | 40 | 20/40 = 0.500 |
| Late | 10 | 30 | 40 | 10/40 = 0.250 |
| Total | 60 | 60 | 120 | 60/120 = 0.500 |
Within each period, divide the Arm A count by that period's total. For example, in the early period, \(30/40=0.750\), so three quarters of its participants were assigned to Arm A. The Arm B proportion is \(10/40=0.250\). The corresponding Arm A proportions are 0.500 in the middle period and 0.250 in the late period.
The pooled totals are exactly equal: \(30+20+10=60\) for Arm A and \(10+20+30=60\) for Arm B. Yet allocation is strongly separated by period. In the early period, the Arm A minus Arm B proportion is \(30/40-10/40=0.500\), or 50 percentage points. In the late period it is \(10/40-30/40=-0.500\), or minus 50 percentage points. Equal overall counts do not imply similar calendar-time distributions.
This table flags a pattern worth investigating alongside calendar-time outcomes and the recorded allocation probabilities. It does not establish that assignment probabilities were incorrectly implemented, nor does it show that time caused an outcome difference. Those questions require information and analyses beyond the counts themselves.
A Simple Check on Pooled Summaries
The overall proportion assigned to an arm is a weighted average of its interval-specific proportions. The weights are the interval shares of total enrollment, not equal weights unless all intervals contain the same number of participants. This identity explains how an overall proportion can look unremarkable while the period-specific allocation differs substantially.
To verify it, distribute each weight across the period-specific proportion:
In the example, each period has 40 of 120 participants, so the three weights are \(40/120=1/3\). The overall Arm A proportion is \((1/3)(0.750)+(1/3)(0.500)+(1/3)(0.250)=0.500\). The pooled value averages over the shifts; it does not make them disappear.
Worked Example: Outcome Availability at a Data Cutoff
Worked Example: Outcome Availability at a Data Cutoff
At a specified hypothetical data cutoff, an early enrollment interval contains 24 participants assigned to Arm A and 16 assigned to Arm B. Outcomes are resolved for 20 Arm A participants and 12 Arm B participants. In each arm, three outcomes are still pending and one participant has dropped out.
The realized allocation proportions use all assigned participants: Arm A is \(24/(24+16)=24/40=0.600\), and Arm B is \(16/40=0.400\).
Availability proportions use assigned participants in the corresponding arm as their denominators. For Arm A, \(20/24=0.8333\) (approximately); for Arm B, \(12/16=0.750\). The unrounded difference is \(20/24-12/16=0.0833\) (approximately), or about 8.33 percentage points. This describes availability at the stated cutoff, not a difference in treatment outcomes.
The status counts reconcile: for Arm A, \(20+3+1=24\); for Arm B, \(12+3+1=16\). Reporting pending and dropout separately matters because those categories have different meanings. In a recent interval, an elevated pending count may reflect shorter follow-up rather than a greater rate of loss.
Reading Imbalance and Overlap
For two arms, a useful descriptive contrast within interval \(t\) is the difference between their realized allocation proportions, \(\widehat{q}_{At}-\widehat{q}_{Bt}\). Its magnitude indicates how far the observed split is from equal allocation in that interval. It is not a test of randomization and has no universal cutoff for being “substantial”; interpretation depends on period size, the planned allocation, and the purpose of the review. If the intended allocation was not 1:1, compare observed proportions with the relevant planned or recorded probabilities rather than treating 0.500 as the target.
Overlap concerns whether both arms are represented in the same periods. If one arm has very few assignments in a period, comparisons involving that period may be unstable or uninformative, especially when outcomes are also sparse. A zero count means there is no within-period comparison between arms in that interval. Small nonzero counts are not the same as zero, but they still call for caution. There is no single cell-count threshold that guarantees adequate overlap in every design; a review may specify an operational flag in advance, while reporting the actual counts so readers can judge their importance.
Worked Example: Finding Sparse Periods Despite Balanced Totals
A hypothetical trial has four enrollment weeks, with the following assignments:
| Week | Arm A | Arm B | Total | Arm A proportion |
|---|---|---|---|---|
| 1 | 18 | 2 | 20 | 0.900 |
| 2 | 15 | 5 | 20 | 0.750 |
| 3 | 5 | 15 | 20 | 0.250 |
| 4 | 2 | 18 | 20 | 0.100 |
| Total | 40 | 40 | 80 | 0.500 |
The overall totals are balanced: each arm has 40 assignments. In week 1, the Arm A proportion is \(18/20=0.900\), and the Arm B proportion is \(2/20=0.100\), a difference of \(0.800\), or 80 percentage points. Week 4 has the reverse pattern: \(2/20-18/20=-0.800\).
Suppose this trial's monitoring plan uses fewer than five assignments in either arm during a week as a descriptive flag for sparse representation. Then weeks 1 and 4 are flagged because one arm has two assignments in each. This is an operational rule for review, not a general statistical guarantee or a formal conclusion about bias. The table shows why equal totals alone would not reveal the sparse periods.
Construction, Comparisons, and Failure Modes
A clear table keeps denominators visible and distinguishes three different questions: how many people were assigned, what share of the period's assignments went to each arm, and what share of each arm's assigned participants had an outcome available. A compact report can include counts and proportions side by side, with an explicit cutoff for outcome status.
| Summary | Numerator | Denominator | Question answered |
|---|---|---|---|
| Arm-by-period count | Assignments to arm \(k\) in period \(t\) | None | How many were assigned? |
| Allocation proportion | Assignments to arm \(k\) in period \(t\) | All assignments in period \(t\) | What was the realized split? |
| Outcome availability proportion | Assigned participants with outcome available | Assignments to that arm in that period | What share had an outcome at this cutoff? |
A pooled count is a useful check on totals, but it discards the time index. An allocation-probability timeline answers a different question by showing the probabilities offered by the adaptive procedure over time; it does not replace realized assignment counts. Conversely, counts alone cannot verify that probabilities were implemented as recorded. Reviewing both can distinguish an intended shift from an unusual realized split, although neither summary by itself establishes an outcome effect.
Common construction errors include assigning participants to periods by outcome date, changing period boundaries without saying so, omitting periods with zero enrollment, and mixing pending outcomes with dropout. Another error is reporting only percentages: a 50% allocation proportion could represent 1 of 2 participants or 50 of 100, which have very different evidential weight. Always show the cell counts and period totals alongside proportions.
The table is descriptive. It does not adjust for calendar-time differences, prove that imbalance is more than random variation, or estimate a treatment effect. A sharp pattern should prompt verification of dates, arm labels, randomization records, and outcome status, and may motivate further analysis. Small cells, coarse intervals, and changing outcome follow-up can all limit what the table can show.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In a period, 18 participants are assigned to Arm A and 12 to Arm B. What are the within-period allocation proportions?
- In a period, 8 of 10 Arm A participants and 9 of 15 Arm B participants have outcomes available. Calculate each arm's availability proportion.
- Two periods each enroll 20 participants. Arm A receives 16 in the first and 4 in the second; Arm B receives 4 and 16. What are the pooled totals, and what pattern do those totals conceal?
- Why should a recent period's pending outcomes not automatically be counted as dropout?
Solutions. (1) The period total is \(18+12=30\); the proportions are \(18/30=0.600\) for Arm A and \(12/30=0.400\) for Arm B. (2) Use assigned participants in each arm as the denominator: Arm A availability is \(8/10=0.800\), and Arm B availability is \(9/15=0.600\). (3) Arm A totals \(16+4=20\), and Arm B totals \(4+16=20\). The pooled totals are balanced, but the first period has 80% assigned to A and the second has 20% assigned to A. (4) Pending outcomes have not yet been resolved at the cutoff; participants may still complete follow-up. Dropout is a distinct status and should be reported separately.