Converting Stage Sizes to Total Enrollment

From Stage Allocations to Participant Counts

A MAMS design can compare several active treatments with a control group, and it can enroll participants in successive stages. Turning the planned numbers for each arm and stage into a total enrollment count is an accounting task, but it requires a clear unit of counting. The total is the number of unique participants randomized, not the sum of every treatment-versus-control comparison’s sample size.

This distinction matters because a shared-control participant is one person, even if that participant’s data contribute to more than one treatment comparison. Adding the sample sizes of all comparisons as though each had its own control group would count those people repeatedly.

Definition: Let $j=1,\ldots,J$ index stages and let $A_j$ be the set of active treatment arms open for randomization in stage $j$. Write $n_{0j}$ for the number randomized to the shared control in that stage and $n_{ij}$ for the number randomized to active arm $i\in A_j$. The stage enrollment is $$ N_j=n_{0j}+\sum_{i\in A_j}n_{ij}. $$ The cumulative enrollment through stage $r$ is $$ N_{\le r}=\sum_{j=1}^{r}N_j. $$ These quantities count each randomized participant once. They do not count screened people who were not randomized.

The formulas assume that the listed arm counts are numbers of distinct participants randomized in that stage, that each participant is randomized only once, and that the stage groups do not overlap as participant groups. Counts can differ across arms and stages; the sum does not require equal allocation. If a design uses separate control groups rather than a shared control, each distinct control group must be counted as its own arm. Always identify what the table’s counts represent before adding them.

Stage Enrollment and Comparison Counts

Stage enrollment answers, “How many unique participants were randomized in this stage?” A comparison-specific count answers a different question: “How many observations are available for this treatment-versus-control contrast?” In a stage with active arm $i$ and a shared control, the two arm counts are $n_{ij}$ and $n_{0j}$. They sum to the number of observations in that stage’s comparison, but they do not define a second, separate enrollment total.

For example, when all active arms are compared with a shared control, the same control group can contribute to several comparisons. That reuse can induce dependence among comparison statistics; it does not create additional participants. The number enrolled is obtained by counting the control group once and each active-arm group once.

Worked Example: One Stage With Two Active Arms

Suppose one stage randomizes 40 participants to control, 40 to Treatment A, and 40 to Treatment B. Calculate the stage enrollment and the observation count in each treatment comparison.

The unique participant total is $$ N_1=40+40+40=120. $$ Thus, 120 participants are randomized in the stage.

The Treatment A versus control comparison has $40+40=80$ arm observations, and the Treatment B versus control comparison also has $40+40=80$. Summing those comparison counts gives $80+80=160$, but this is not the enrollment: the 40 control participants were included in both comparison counts. The correct enrollment remains 120.

The same accounting works when counts vary by stage. For a cumulative total, add the stage enrollment totals, not the comparison totals. This is valid when the stages represent disjoint groups of newly randomized participants. In contrast, analyses may use data from participants enrolled at earlier stages again at later analyses; repeated use of the same data in an analysis does not mean those participants were enrolled again.

Worked Example: Three Stages With Unequal Stage Sizes

Consider a design with three active treatments and one shared control. All four arms are open in each stage. The planned numbers randomized are 40 per arm in Stage 1, 30 per arm in Stage 2, and 25 per arm in Stage 3. Find the enrollment in each stage and the cumulative total.

In Stage 1, the shared control is counted once and each of the three active arms is counted once: $$ N_1=40+40+40+40=160. $$ In Stage 2, $$ N_2=30+30+30+30=120, $$ and in Stage 3, $$ N_3=25+25+25+25=100. $$

The cumulative enrollment through Stage 2 is $160+120=280$. Through Stage 3 it is $$ N_{\le 3}=160+120+100=380. $$ For any one treatment-versus-control comparison, there are $40+30+25=95$ active-arm participants and 95 shared-control participants across these stages, or 190 comparison-specific arm observations. Across the three comparisons, adding 190 three times would give 570 comparison-specific observations, not 570 unique participants. The unique total is 380.

Why the Counting Formula Works

The stage formula follows from partitioning the randomized participants into mutually exclusive arm groups. Every participant randomized in stage $j$ belongs to exactly one of the groups: the shared control or one of the active arms open in that stage. Therefore, the number of participants in the union of those disjoint groups equals the sum of their sizes.

\left|\{\text{participants randomized in stage }j\}\right| = \left|\{\text{control participants}\}\right| + \sum_{i\in A_j}\left|\{\text{participants in active arm }i\}\right| = n_{0j}+\sum_{i\in A_j}n_{ij}.

If stages also contain disjoint groups of newly randomized participants, then the same reasoning applies across stages: the number randomized through stage $r$ is the sum of the stage totals. A control participant may be included in more than one comparison’s data, but still belongs to just one group in this partition. This is why comparison-specific counts cannot be substituted for participant totals without checking whether they overlap.

When Arms Are Open in Different Stages

The set $A_j$ makes the calculation explicit when not every active arm is randomized in every stage. Include only arms that actually enroll participants in that stage. A treatment comparison’s control count should also reflect the stages in which that treatment and the relevant control participants form the intended comparison. Do not automatically assign every control participant in the trial to every contrast; the design and analysis determine which data are relevant.

Worked Example: Enrollment After an Arm Stops Recruiting

Suppose Stage 1 randomizes 50 participants each to control, Treatment A, and Treatment B. In Stages 2 and 3, only control and Treatment A are open, with 40 participants per arm in Stage 2 and 30 per arm in Stage 3. Calculate each stage’s enrollment and the total through Stage 3.

Stage 1 has three randomized groups: $$ N_1=50+50+50=150. $$ Stage 2 has control and Treatment A: $$ N_2=40+40=80. $$ Stage 3 also has control and Treatment A: $$ N_3=30+30=60. $$

The total randomized through Stage 3 is $150+80+60=290$. The total control count is $50+40+30=120$, and the total Treatment A count is also 120. Treatment B has 50 randomized participants. For the Treatment B contrast in this example, only the 50 participants randomized to control in Stage 1 are paired by stage with Treatment B’s 50 participants. The later control participants remain part of overall trial enrollment even though Treatment B is not open in those stages.

Practical Checks and Common Errors

Before reporting an enrollment total, confirm that every number is on the same scale: participants randomized, rather than, for example, participants per comparison or target numbers per analysis. Then check which groups are distinct and which are shared. A stage-by-stage table is useful because it makes omissions and accidental duplication visible.

Double-counting shared control. Adding each contrast’s active and control counts gives a total of comparison observations, not necessarily a count of unique participants. Count the shared control group once in enrollment.
Adding cumulative totals to stage totals. A cumulative count already includes earlier stages. Add the individual stage totals to build a cumulative count; do not add a cumulative figure to another figure that includes those same stages.
Including an arm that is not recruiting. An arm absent from a stage contributes zero participants to that stage. Do not carry its earlier stage count forward as new enrollment.
Confusing randomized and analyzed counts. The randomized count is not automatically the number with a complete outcome or included in a particular analysis. Missing outcomes and analysis populations require separate reporting.
Assuming a comparison total is an enrollment total. A comparison’s observations may include control participants also used in other contrasts. State whether a reported number means unique participants or comparison-specific observations.

These calculations give planned or realized randomized enrollment for the stages and arms specified. They do not by themselves determine how many participants will actually be enrolled if future stages depend on interim decisions, nor do they calculate an expected enrollment over possible paths. Those questions require additional information about continuation and stopping.

Key takeaway: For each stage, add the counts of the distinct groups randomized in that stage. Add those stage totals for cumulative enrollment. When a control group is shared, count it once for enrollment even though its observations may contribute to multiple treatment comparisons.

Check Your Understanding

Try each question, then compare your work with the solutions below.

A stage randomizes 36 participants to control and 36 to each of two active treatments. What is the unique stage enrollment, and how many arm observations are in each active-versus-control comparison?
In Stage 1, 25 participants are randomized to each of control, A, and B. In Stage 2, 20 are randomized to each of control, A, and B. What is the cumulative enrollment, and what is the cumulative number of control participants?
Three active treatments each have 45 participants in a stage, and they share one control group of 45 participants. A report sums the three comparison totals and calls the result enrollment. What is the correct enrollment, and what quantity did the report calculate?
In Stage 1, control and A each enroll 32 participants. In Stage 2, only control and A are open, with 24 participants per arm. What is the cumulative enrollment, and how many participants were randomized to A across both stages?

Solutions. (1) Enrollment is $36+36+36=108$. Each comparison has $36+36=72$ arm observations. (2) Stage 1 enrollment is $25+25+25=75$, and Stage 2 enrollment is $20+20+20=60$, so cumulative enrollment is $75+60=135$. The control count is $25+20=45$. (3) Correct enrollment is $45+45+45+45=180$, counting the shared control once. The report calculates three comparison totals of 90 each, or 270 comparison-specific arm observations; the shared control is represented three times in that sum. (4) Cumulative enrollment is $(32+32)+(24+24)=112$. Treatment A has $32+24=56$ randomized participants.