A Three-Treatment Comparison Under a Fixed Budget

Why Solve the Allocation Before Recruitment?

A fixed participant budget forces a design choice: how many people should be assigned to the shared control, and how many to each of three treatments? The answer affects the precision of all three treatment-control comparisons. It also determines whether one multi-arm protocol makes better use of the budget than running three separate two-arm trials.

The comparison needs a clear objective. Here, assume the three treatment-control comparisons are equally important, participants cost the same to recruit and follow, and the main design goal is to make their average sampling variance small. This is an allocation exercise, not a claim that any treatment works or that a particular error rate or power has been achieved.

Definition: Let \(n_0\) be the number of participants assigned to one common control and \(n_j\) the number assigned to treatment \(j\), for \(j=1,2,3\). With a fixed total budget \(N\), the allocation must satisfy \(n_0+n_1+n_2+n_3=N\). For a continuous outcome, the estimated treatment-control difference is \(\bar{x}_j-\bar{x}_0\). Under independent observations with a common within-group variance \(\sigma^2\), its variance is \(\sigma^2(1/n_j+1/n_0)\).

That variance formula assumes that outcomes from different participants are independent, each arm has the same outcome variance, and the shared control is suitable for comparison with all three treatments. It also assumes complete outcome data for the calculation. Equal recruitment costs and equally important comparisons are additional design assumptions, not consequences of the variance formula.

A Useful Allocation for Three Comparisons

If the treatment groups have equal sizes \(x\) and the control has size \(y\), the total budget condition is \(y+3x=N\). The sum of the three contrast variances, apart from the common factor \(\sigma^2\), is \(3(1/y+1/x)\). Minimizing that sum gives a useful allocation rule. The derivation treats group sizes as positive continuous quantities first; actual sample sizes must then be whole numbers.

f(x)=3\left(\frac{1}{N-3x}+\frac{1}{x}\right), \qquad 0<x<\frac{N}{3}.

Differentiating and setting the derivative to zero gives the minimum. The second derivative is positive throughout the feasible range, so this stationary point is a minimum rather than a maximum.

f'(x)=\frac{9}{(N-3x)^2}-\frac{3}{x^2}=0 \quad\Longrightarrow\quad N-3x=\sqrt{3}\,x.

Result: When three comparisons are equally important, recruitment costs are equal, and the groups have a common outcome variance, the continuous allocation that minimizes the sum of the three contrast variances has \(n_0=\sqrt{3}\,n_j\) for each treatment \(j\). Equivalently, the control-to-each-treatment allocation ratio is \(\sqrt{3}:1\). This rule minimizes the sum of marginal variances under these assumptions; it is not a universal optimum for every design objective.

The control is larger than any one treatment group because its observations contribute to all three contrasts. The result does not say to make the control three times the size of a treatment: the optimal ratio is about 1.732 to 1. If treatment effects have different priorities, variances differ across arms, or recruitment costs are unequal, the objective and allocation can change.

Worked Example: Allocating a 240-Participant Budget

Suppose a hypothetical trial has a budget of 240 participants, one shared control, and three treatments. Assume equal costs, common outcome variance, and equal importance for the three comparisons. Find a practical allocation close to the variance-minimizing allocation.

First solve for the continuous group sizes. With \(n_0=\sqrt{3}x\) and \(n_0+3x=240\),

x=\frac{240}{3+\sqrt{3}}\approx 50.72, \qquad n_0=\sqrt{3}(50.72)\approx 87.85.

These values cannot be used exactly because participants are indivisible. Rounding to 51 participants in each of the three treatment groups leaves \(240-3(51)=87\) participants for control. The feasible allocation is therefore \(n_0=87\), \(n_1=n_2=n_3=51\), which uses the full budget and is close to the continuous allocation.

The ratio is \(87/51\approx1.706\), close to the target \(\sqrt{3}\approx1.732\). The rounding is a practical approximation, not a guarantee that this allocation is optimal for every possible integer-valued objective.

Compare With Three Separate Trials

A fair comparison holds the total participant budget fixed. If the same 240 participants instead enter three separate two-arm trials of equal size, each trial can enroll 80 people: 40 on its own control and 40 on its treatment. Across the three trials this uses 120 control participants and 120 treatment participants. The shared-control allocation uses 87 control participants overall and 153 treatment participants overall.

For a numerical precision comparison, suppose—hypothetically—that the outcome has a common standard deviation of 10 points in every arm. The comparison assumes independent participant outcomes and the same outcome measurement and eligibility conditions in each design. It compares the standard error of each treatment-control mean difference; it does not compare power or account for other design features.

Worked Example: Standard Errors Under the Two Designs

For each shared-control comparison, use \(SE=\sigma\sqrt{1/n_0+1/n_j}\). With 87 control participants and 51 in each treatment group, the standard error is

SE_{\text{shared}} =10\sqrt{\frac{1}{87}+\frac{1}{51}} =10\sqrt{0.0311021} \approx 1.764 \text{ points}.

In each separate trial, there are 40 participants in control and 40 in treatment. Its treatment-control standard error is

SE_{\text{separate}} =10\sqrt{\frac{1}{40}+\frac{1}{40}} =10\sqrt{0.05000} \approx 2.236 \text{ points}.

Thus, under the stated assumptions, each shared-control comparison has a smaller marginal standard error: about 1.764 rather than 2.236 points. Its variance relative to a separate-trial comparison is

\frac{1/87+1/51}{1/40+1/40} =\frac{0.0311021}{0.05000} \approx 0.6220.

This means the shared-control contrast variance is about 62.2% of the separate-trial contrast variance in this example. It does not mean that the treatment effect is 62.2% as large, or that the probability of a successful result has increased by a particular amount.

What the Comparison Does—and Does Not—Show

With the budget fixed, the shared-control design assigns more participants to each treatment than the separate-trial design does: 51 rather than 40. The separate trials assign more participants to control overall, but those participants are divided among three controls. A participant in one separate trial's control contributes to that trial's comparison only; a participant in the shared control contributes to all three comparisons.

The contrasts in a shared-control design are not independent. Under the common-variance model, the estimated differences \(\bar{x}_1-\bar{x}_0\) and \(\bar{x}_2-\bar{x}_0\) share the control mean. Their covariance is \(\sigma^2/n_0\), which is positive when \(\sigma^2>0\). The standard-error calculations above are marginal calculations for each contrast. They do not justify treating all three contrasts as independent when making joint conclusions.

That distinction matters: a smaller standard error for each comparison is not by itself a complete analysis of a multi-arm trial. The choice of hypotheses and control of error across several comparisons are separate design and analysis questions. The allocation calculation here does not set those rules.

Worked Example: An Equal-Allocation Benchmark

One alternative is to divide the same 240 participants equally among all four groups, giving 60 participants to control and 60 to each treatment. Under the same hypothetical standard deviation of 10 points, the standard error for each treatment-control difference is

SE_{\text{equal}} =10\sqrt{\frac{1}{60}+\frac{1}{60}} =10\sqrt{0.0333333} \approx 1.826 \text{ points}.

The equal-allocation standard error is larger than the approximately 1.764 points under the 87/51 allocation, though smaller than the 2.236 points in each separate trial. For the sum of the three contrast variances, the equal allocation gives

3(10^2)\left(\frac{1}{60}+\frac{1}{60}\right) =10.00 \text{ points}^2.

For the 87/51 allocation, the corresponding sum is

3(10^2)\left(\frac{1}{87}+\frac{1}{51}\right) \approx 9.331 \text{ points}^2.

So, for this particular objective and model, the allocation that gives the common control more participants has a smaller sum of marginal contrast variances than equal allocation. The numerical difference is modest, and the result does not imply that equal allocation is always inferior: other priorities, costs, or variances can change the comparison.

Assumptions, Diagnostics, and Limits

Before using the calculation to justify a real design, check whether its assumptions are plausible. In particular, verify that the shared control answers the same scientific question for every treatment. Differences in eligibility, calendar time, background care, or outcome assessment may make a nominally common control less comparable to one treatment than to another. A shared group is useful only if the comparisons it supports are scientifically interpretable.

The common-variance calculation also deserves scrutiny. Historical data or blinded internal pilot information may suggest that outcome variability differs across groups. In that case, the variance of a difference generally depends on the variance in each group, not on a single common \(\sigma^2\). Missing outcomes or unequal recruitment costs can also make enrolled group sizes differ from the effective information available for analysis. Recalculate the allocation under an appropriate model rather than treating the simple ratio as a rule.

A practical diagnostic is to check the planned group sizes against both the budget and the objective: do the counts sum to \(N\), are all groups adequately represented, and does the allocation still reflect the relative importance of the comparisons? Also ask whether recruitment can reliably deliver those counts. If the control is slow to recruit or treatment arms have different costs, a mathematically attractive allocation may not be feasible.

The separate-trial comparison is most informative when the trials would genuinely use the same budget, population, outcome, follow-up, and control condition. If separate studies occur at different times or sites, their control groups may differ for reasons unrelated to treatment. Conversely, a single multi-arm protocol brings operational demands, including consistent conduct across all arms. Neither design is automatically superior in every setting.

Common Pitfalls

Dividing the budget equally without naming the objective. Equal group sizes are easy to implement, but the shared-control variance formula shows why a larger control can be useful when it supports several equally important comparisons.
Comparing designs with different total budgets. A larger trial may have more precise comparisons simply because it enrolls more people. Here the comparison fixes the total at 240.
Counting the shared control three times. The control is used in three contrasts, but its participants are enrolled once. The total remains \(87+3(51)=240\), not 327.
Reading standard error as a treatment effect or success probability. Standard error describes sampling variability under the model. It does not tell us the size of an effect or the probability that a treatment is effective.
Ignoring the shared-control dependence. Each contrast has the stated marginal standard error, but the contrasts share a control mean and are correlated.

Key takeaway: Under equal costs, a common outcome variance, and three equally important treatment-control comparisons, a useful fixed-budget allocation has a control about \(\sqrt{3}\) times the size of each treatment group. Check the assumptions and round to feasible counts; the resulting precision comparison is not, on its own, a full analysis of the multi-arm design.

Check Your Understanding

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

With 120 participants and three equally important treatments, what continuous control and per-treatment group sizes follow from the allocation rule?
For a shared-control design with 60 participants in control and 40 in a treatment, and a common standard deviation of 8 units, calculate the standard error of the treatment-control difference.
Why is it not valid to count a shared control group as three separately enrolled control groups when checking a fixed budget?
State one reason the \(\sqrt{3}:1\) allocation may not be appropriate in a particular trial.

Solutions. (1) The per-treatment size is \(x=120/(3+\sqrt{3})\approx25.36\), and the control size is \(\sqrt{3}x\approx43.92\); these continuous values must be rounded to feasible integer counts. (2) \(SE=8\sqrt{1/60+1/40}=8\sqrt{0.0416667}\approx1.633\) units. (3) Participants are assigned to the shared control once, so the total is \(n_0+n_1+n_2+n_3\); the same observations contribute to multiple comparisons but are not multiple enrollments. (4) For example, the comparisons may have different priorities, costs may vary by arm, or outcome variances may differ, changing the allocation objective or its solution.