Precision at a Fixed Total Sample Size

Why Compare Precision at a Fixed Total?

A shared control can reduce the number of participants recruited across several treatment comparisons. But a recruitment total alone does not describe the statistical consequence: with the same total number of participants, how precisely is each treatment–control difference estimated? This tutorial compares that precision for separate two-arm trials and a design with one control group shared by several experimental arms.

The comparison concerns the variance of an individual estimated difference in means. It is not, by itself, a comparison of power or a complete assessment of a multi-arm design. Those depend on additional choices, including the treatment effects of interest and how multiple comparisons are handled.

Definition: Suppose there are \(K\) experimental arms and one control arm, and the total number recruited across all arms is fixed at \(N\). A separate-trials design uses a distinct control group for each treatment–control comparison. A shared-control design uses one control group for all \(K\) comparisons. Precision for a treatment–control mean difference is summarised by its variance; a smaller variance means a smaller standard error.

Model and Assumptions

Assume a continuous outcome measured once per participant. In every arm, outcomes are independent across participants, have finite variance \(\sigma^2\), and the same variance. The arms are formed by appropriate random allocation, and the outcome is measured in the same way across them. These assumptions let us compare variances without mixing the allocation effect with differences in outcome variability or study procedures. Normality is not needed for the variance calculations below.

For a treatment arm of size \(n\) and a control arm of size \(n_0\), the estimated difference is \(\bar{x}_{T}-\bar{x}_{0}\). Independence of the two groups gives

\operatorname{Var}(\bar{x}_{T}-\bar{x}_{0}) =\operatorname{Var}(\bar{x}_{T})+\operatorname{Var}(\bar{x}_{0}) =\sigma^2\left(\frac{1}{n}+\frac{1}{n_0}\right).

The standard error is the square root of this variance. For a shared-control design, use \(n_0\) for the single control-group size and \(n\) for the size of each equally sized treatment group. For separate trials, each comparison has its own treatment and control groups. We compare designs in which all \(K\) treatment–control comparisons are equally important and receive equal allocation under the symmetry of the assumptions.

Variance Under Separate Trials

Let \(b_j\) be the total number of participants in separate trial \(j\). The constraint is \(\sum_{j=1}^{K} b_j=N\). Within each trial, let the treatment and control sizes be \(n_{T,j}\) and \(n_{0,j}\), so \(n_{T,j}+n_{0,j}=b_j\). The variance of the difference in that trial is

\sigma^2\left(\frac{1}{n_{T,j}}+\frac{1}{n_{0,j}}\right).

For a given trial total, equal allocation minimises this variance: since \((1/n_T+1/n_0)(n_T+n_0)\geq 4\), its variance is at least \(4\sigma^2/b_j\), with equality when \(n_T=n_0=b_j/2\). If all comparisons are equally important, giving them equal trial totals minimises the largest of these per-comparison variances. Thus each trial receives \(N/K\), with \(N/(2K)\) participants per arm. The resulting variance for every comparison is

V_{\mathrm{sep}}=\sigma^2\left(\frac{1}{N/(2K)}+\frac{1}{N/(2K)}\right) =\frac{4K\sigma^2}{N}.

This allocation is a symmetric planning benchmark. With integer sample sizes, exact equal division may not be possible; the actual arm counts should be used in the variance formula.

Variance Under a Shared Control

For the shared-control design, allocate \(n_0\) participants to the common control and \(n\) to each of the \(K\) treatment arms. The fixed-total constraint is \(n_0+Kn=N\). Define the control-to-treatment allocation ratio as \(r=n_0/n\). Then \(n=N/(K+r)\), and the variance of any one treatment–control difference is

V_{\mathrm{shared}}(r) =\sigma^2\left(\frac{1}{n_0}+\frac{1}{n}\right) =\frac{\sigma^2}{N}(K+r)\left(1+\frac{1}{r}\right) =\frac{\sigma^2}{N}\left(K+r+\frac{K}{r}+1\right).

For a fixed total, this expression is smallest at \(r=\sqrt{K}\): the terms depending on \(r\) satisfy \(r+K/r\geq 2\sqrt{K}\), with equality at that ratio. Therefore, the variance-minimising allocation under these assumptions has \(n_0/n=\sqrt{K}\), and its variance is

V_{\mathrm{shared,min}}=\frac{\sigma^2}{N}(1+\sqrt{K})^2.

The common control is larger than each treatment group when \(K>1\). This is not a claim that the control should always be larger in practice: it follows from the particular objective of minimising the variance of each equally important difference under a fixed total, equal participant costs, and a common outcome variance.

Key result: Under the stated assumptions and variance-minimising allocations, \(V_{\mathrm{sep}}=4K\sigma^2/N\) and \(V_{\mathrm{shared,min}}=(1+\sqrt{K})^2\sigma^2/N\). Their difference is \(V_{\mathrm{sep}}-V_{\mathrm{shared,min}}=(\sqrt{K}-1)^2\sigma^2/N\). Thus the optimised shared-control allocation has lower variance for each contrast when \(K>1\), and the variances are equal when \(K=1\).

Worked Comparisons

Worked Example: Three Comparisons With an Optimised Shared Control

Suppose \(K=3\), total recruitment is \(N=240\), and the common outcome standard deviation is \(\sigma=12\) points. Compare separate trials with an integer shared-control allocation close to the variance-minimising ratio.

Separate trials. Each trial receives \(240/3=80\) participants. Equal allocation gives 40 participants per arm, so for each difference,

V_{\mathrm{sep}}=12^2\left(\frac{1}{40}+\frac{1}{40}\right) =144(0.05)=7.20\ \text{points}^2.

The standard error is \(\sqrt{7.20}=2.683\) points.

Shared control. The target ratio is \(\sqrt{3}\approx1.732\). An allocation of 87 control participants and 51 in each treatment arm uses all 240 participants and has ratio \(87/51\approx1.706\). Each treatment–control difference has variance

V_{\mathrm{shared}}=144\left(\frac{1}{87}+\frac{1}{51}\right) =144(0.031102)\approx4.479\ \text{points}^2.

Its standard error is approximately \(\sqrt{4.479}=2.116\) points. Under the model, this particular shared allocation gives a more precise estimate of each difference than the separate-trials allocation. The calculation compares standard errors in points; it does not establish that either design has adequate power for a particular treatment effect.

Worked Example: A Shared Control Need Not Have the Optimal Ratio

Suppose \(K=2\), \(N=180\), and \(\sigma=10\) units. Consider a shared design with 60 participants in the control group and 60 in each treatment group. This uses \(60+2(60)=180\) participants, but its allocation ratio is \(r=1\), not the variance-minimising ratio \(\sqrt{2}\).

In separate trials, each comparison has 90 participants, allocated 45 to treatment and 45 to its own control. Thus

V_{\mathrm{sep}}=100\left(\frac{1}{45}+\frac{1}{45}\right) =\frac{200}{45}\approx4.444\ \text{units}^2.

For the proposed shared allocation,

V_{\mathrm{shared}}=100\left(\frac{1}{60}+\frac{1}{60}\right) =\frac{200}{60}\approx3.333\ \text{units}^2.

The shared design is more precise for each difference here, even though its ratio is not optimal. Its standard error is \(\sqrt{3.333}\approx1.826\) units, compared with \(\sqrt{4.444}\approx2.108\) units for separate trials. A non-optimal allocation can still compare favourably; that conclusion must be checked from the actual arm sizes.

Worked Example: An Extreme Allocation Can Lose Precision

Suppose \(K=3\), \(N=240\), and \(\sigma=12\) points. If a shared design allocates 210 participants to control and 10 to each treatment, the total is still 240, but the control-to-treatment ratio is 21.

For each shared-control difference,

V_{\mathrm{shared}}=144\left(\frac{1}{210}+\frac{1}{10}\right) =144(0.104762)\approx15.086\ \text{points}^2.

The standard error is approximately 3.884 points. The separate-trials allocation from the first example has variance \(7.20\ \text{points}^2\) and standard error 2.683 points. So this shared allocation is less precise for each treatment–control difference, despite using a common control and the same total recruitment. The 10-person treatment groups, rather than the large control group, limit the precision.

What Changes When Controls Are Shared?

The variance formula describes one contrast at a time. In a shared-control design, the estimated contrasts are also correlated because they contain the same estimated control mean. For two distinct treatment arms \(j\) and \(\ell\), write \(D_j=\bar{x}_j-\bar{x}_0\) and \(D_\ell=\bar{x}_\ell-\bar{x}_0\). Independence of outcomes across arms gives

\operatorname{Cov}(D_j,D_\ell) =\operatorname{Var}(\bar{x}_0) =\frac{\sigma^2}{n_0}.

The covariance is positive: a control mean that happens to be high shifts both estimated treatment–control differences downward. This does not change the variance of either individual difference, but it matters when considering the estimates jointly. Therefore, a statement about better precision for each contrast is not a statement that the contrasts are independent.

Diagnostics, Alternatives, and Limitations

Before using the formulas, check that the comparison matches the planned design. In particular, verify that the total \(N\) includes all control and treatment participants, that the separate-trial control groups are distinct, and that the shared design counts its control group only once. Check the assumed common variance and the actual arm sizes. If arm sizes are unequal, calculate each variance directly as \(\sigma^2(1/n_T+1/n_0)\) rather than substituting a balanced-allocation formula.

A useful comparison is the variance ratio \(V_{\mathrm{shared}}/V_{\mathrm{sep}}\). A value below 1 means a smaller variance for an individual difference under the stated model; a value above 1 means a larger one. When \(K>1\), the optimised shared design has a ratio below 1, but a poorly chosen allocation need not. Integer rounding can also move a planned allocation away from its optimum, so use the final whole-number counts for the final calculation.

Separate trials are the direct alternative: each comparison has its own control group, and estimates from different trials do not share a control mean. A shared-control design can use recruitment more efficiently for individual contrasts, but it creates correlated contrast estimates. Neither variance calculation accounts for unequal recruitment costs, differing outcome variances, missing outcomes, clustering, or differences in trial conduct. If these features matter, the assumptions and variance expressions need to be adapted; the simple equal-variance comparison is not sufficient.

Common pitfall: “One common control always makes every comparison more precise because it is reused.” Reuse alone does not guarantee that result. Precision depends on how the fixed total is divided: very small treatment groups can make shared-control differences less precise than separate-trial differences. Compare the variances using the actual allocation, and remember that sharing the control also correlates the contrasts.

Check Your Understanding

Work each problem using the fixed-total formulas. The solutions follow the questions.

With \(K=4\), \(N=240\), and common standard deviation \(\sigma=8\) units, what is the variance-minimising control-to-treatment ratio for a shared-control design? Give the corresponding ideal group sizes and per-contrast variance.
For \(K=2\), \(N=120\), and \(\sigma=6\) units, calculate the variance of each separate-trials difference under equal allocation.
Why is the shared-control contrast variance not enough to describe the joint behaviour of all estimated treatment effects?
State two assumptions that would need reconsideration if treatment arms had substantially different outcome variances.

Solutions. (1) The ratio is \(r=\sqrt{4}=2\). Since \(n=N/(K+r)=240/6=40\), the ideal allocation has 40 participants in each treatment arm and 80 in control. The variance is \(8^2(1/40+1/80)=64(0.0375)=2.40\ \text{units}^2\). (2) Each separate trial has \(120/2=60\) participants, or 30 per arm. Thus \(V_{\mathrm{sep}}=36(1/30+1/30)=2.40\ \text{units}^2\) for each difference. (3) The contrasts share the estimated control mean, so distinct contrasts have covariance \(\sigma^2/n_0\); they are not independent. (4) The common-variance assumption would fail, and the variance-minimising allocation derived for equal variances would no longer be justified. The arm-specific variances should be represented in the variance calculation, and the allocation should be reconsidered under that model.