Deriving the Shared-Control Sample-Size Saving

Why Derive the Saving?

A multi-arm trial can compare several experimental treatments with one control group. The control participants contribute to each treatment-versus-control comparison, but each person is recruited only once. In separate two-arm trials, by contrast, each comparison has its own control group. Counting the participants in these two layouts makes the recruitment saving precise.

The calculation is deliberately conditional: it compares designs with the same number of participants in every group and the same set of treatment-versus-control comparisons. It does not, by itself, establish that two designs have the same power under every analysis plan. The per-comparison sample-size requirement must be comparable for the count difference to isolate the effect of sharing a control.

Definition: Let \(K\) be the number of experimental treatment arms, with one planned comparison of each treatment against a common control. Let \(n\) be the number of participants in each group in either design. In separate trials, each of the \(K\) comparisons has its own treatment group and control group of size \(n\). In the shared-control design, there are \(K\) treatment groups of size \(n\) and one control group of size \(n\).

The count comparison assumes that each participant is counted once, the separate trials recruit distinct participants, and the multi-arm trial has one control group used in all \(K\) planned comparisons. It also assumes that the same group size \(n\) is appropriate for every comparison in both layouts. For example, the comparisons might share the same outcome definition, target population, and per-comparison sample-size requirement. These assumptions let us compare the layouts without changing the treatment-group sizes or the number of planned comparisons.

Deriving the Participant Counts

In one two-arm trial, there are \(n\) participants in the treatment group and \(n\) in the control group, for \(2n\) total. If this design is repeated as \(K\) separate trials, the total is \(K\) times \(2n\). In one shared-control trial, there are \(K\) treatment groups plus one control group, each of size \(n\). Subtracting the second total from the first gives the absolute saving.

N_{\text{separate}} = K(n+n) = 2Kn

N_{\text{shared}} = Kn+n = (K+1)n

S = N_{\text{separate}}-N_{\text{shared}} =2Kn-(K+1)n =(K-1)n

Thus, under the stated assumptions, the shared-control design uses \((K-1)n\) fewer participants. The same result can be seen by counting control participants directly: the separate trials have \(K\) control groups, or \(Kn\) control participants, while the shared-control trial has \(n\). The treatment groups total \(Kn\) in both layouts. Therefore the entire count saving is the reduction of \((K-1)n\) control participants.

A percentage saving needs a reference total. Here the natural baseline is the total for the separate trials. Dividing the absolute saving by that baseline gives:

\text{Relative saving} =\frac{N_{\text{separate}}-N_{\text{shared}}}{N_{\text{separate}}} =\frac{(K-1)n}{2Kn} =\frac{K-1}{2K}

To express this as a percentage, multiply the fraction by 100%. The group size \(n\) cancels: under equal allocation, the percentage saving depends on the number of treatment comparisons, not on the size of each group. When \(K=1\), there is only one comparison, so the two layouts are the same and the saving is zero.

Result: With \(K\) treatment comparisons and \(n\) participants per group, separate two-arm trials require \(2Kn\) participants, whereas one trial with a shared control requires \((K+1)n\). The shared-control design saves \((K-1)n\) participants, or \(\frac{K-1}{2K}\) of the separate-trial total, provided the group-size and comparison assumptions above hold.

Worked Comparisons

Worked Example: Three Treatment Comparisons

Suppose a hypothetical programme is planning three treatment-versus-control comparisons. Assume each group needs 72 participants, and the same group size is appropriate in either layout. How many participants does each layout require, and what is the saving?

Here \(K=3\) and \(n=72\). With separate trials, each comparison uses \(2(72)=144\) participants, so:

N_{\text{separate}}=2Kn=2(3)(72)=432

With one shared-control trial, there are three treatment groups and one control group:

N_{\text{shared}}=(K+1)n=(3+1)(72)=288

The difference is \(432-288=144\) participants. Using the separate-trial total as the denominator, the percentage saving is:

\frac{144}{432}\times 100\%=33.33\%

The count also follows from the control groups: the separate trials have \(3(72)=216\) control participants, while the shared-control design has 72. The reduction is \(216-72=144\), matching the total-count calculation. Both layouts still have \(3(72)=216\) treatment participants.

Worked Example: Four Comparisons and a Percentage Check

Consider a hypothetical set of four treatment comparisons with 60 participants required in each group. Calculate both totals and check the percentage saving directly.

For four separate trials, each with one treatment and one control group:

N_{\text{separate}}=2(4)(60)=480

For one multi-arm trial, the four treatment groups and one shared control group give:

N_{\text{shared}}=(4+1)(60)=300

The absolute saving is \(480-300=180\) participants. The percentage saving is \(180/480=0.375\), or 37.5%. The formula gives the same result:

\frac{K-1}{2K}=\frac{4-1}{2(4)}=\frac{3}{8}=0.375

The 180 participants saved are exactly the three control groups no longer recruited: \(3(60)=180\). The percentage is not \(180/300\); that would compare the saving with the shared-control total rather than with the separate-trial baseline.

Worked Example: Calculating the Saving From the Design

A hypothetical multi-arm trial has five treatment groups and one control group, with 45 participants per group. Find the total in the shared-control design, the total for five separate trials with the same group sizes, and the absolute and relative savings.

There are \(K=5\) comparisons and \(n=45\). The shared-control design has six groups:

N_{\text{shared}}=(5+1)(45)=270

The separate-trial design has five pairs of groups:

N_{\text{separate}}=2(5)(45)=450

Therefore, the absolute saving is \(450-270=180\) participants. Relative to the separate-trial total, this is \(180/450=0.400\), or 40.0%. Checking the control count gives the same absolute saving: five separate control groups require \(5(45)=225\) people, compared with 45 in the shared group, a difference of \(225-45=180\).

What the Count Difference Does—and Does Not—Show

The two layouts have the same total number of treatment participants, \(Kn\). Their difference is that one shared control group replaces \(K\) separate control groups. This is the core accounting reason for the saving. In a shared-control design, the same control observations are included in each treatment-versus-control comparison; they are not counted as new participants for every comparison.

The calculation describes participant totals under a matched group-size assumption. It does not show that the shared-control design automatically has the same inferential properties in every setting. The sample size \(n\) must remain appropriate for the planned comparisons under the analysis and design assumptions. If, for example, a design choice changes the required group size, the correct comparison is to substitute the appropriate group sizes for each design rather than apply the simple formula unchanged. The formula also does not include extra participants needed for losses to follow-up unless those losses have already been incorporated into \(n\).

The separate-trial layout gives each treatment comparison its own control group. The shared-control layout uses fewer total participants when \(K>1\) under the assumptions above, while keeping \(n\) participants in each treatment group and in the control group. The choice between layouts may depend on considerations beyond this count, such as whether the comparisons can appropriately use a common control. The count formulas answer the narrower question: how many participants are needed when the group sizes and comparisons are held fixed?

Common Mistakes and Checks

Counting the shared control repeatedly. It is used in \(K\) comparisons, but it is one group of \(n\) unique participants. Count it once in the multi-arm total.
Subtracting treatment participants. Both layouts include \(K\) treatment groups of size \(n\). The saving comes from control participants, not from removing a treatment group.
Using the wrong percentage denominator. The relative saving above is measured against the separate-trial total, \(2Kn\), not against the shared-control total.
Comparing different group-size assumptions without saying so. If the required group sizes differ across layouts, \((K-1)n\) is not automatically the saving. State and compare the actual group counts.

A quick diagnostic is to check both the total-count subtraction and the control-group subtraction. If they do not agree, a group may have been omitted or counted more than once.

Count the comparisons.
Set \(K\) equal to the number of treatment arms being compared with the common control.

Fix the group size.
Confirm that \(n\) is the same per group in both layouts and is appropriate for every comparison being counted.

Calculate and check.
Compare \(2Kn\) with \((K+1)n\), then verify that the difference equals the \(K-1\) control groups of size \(n\) removed.

Check Your Understanding

Try each problem, then compare your reasoning with the solutions below. All scenarios are hypothetical and use the equal-group-size assumptions stated in this tutorial.

For \(K=2\) comparisons and \(n=50\) participants per group, find the separate-trial total, the shared-control total, and the saving.
With \(K=3\), what fraction of the separate-trial total is saved? Express it as a percentage.
A planner calculates a saving of \(120\) participants for \(K=4\) comparisons. Under the stated formula, what is \(n\)?
Why is it incorrect to count the shared control group \(K\) times when calculating unique participants in the multi-arm trial?

Solutions. (1) \(N_{\text{separate}}=2(2)(50)=200\), and \(N_{\text{shared}}=(2+1)(50)=150\). The saving is \(200-150=50\) participants. (2) The fraction is \((3-1)/(2(3))=2/6=1/3\), or 33.33% of the separate-trial total. (3) Since \(S=(K-1)n\), \(120=(4-1)n=3n\), so \(n=40\) participants per group. (4) The control participants are the same people used in multiple comparisons; they contribute once to the number of unique participants. Counting them \(K\) times would count comparisons, not people.