Power Under Unequal Group Sizes

Why Allocation Matters

A study can recruit the same total number of donors under different group allocations. For example, 40 donors could be divided 20 and 20, or 30 and 10. These designs have the same total recruitment, but not the same precision for a group comparison. The smaller group contributes less information, so the uncertainty of the contrast is greater when allocation is more uneven.

The relevant replication unit for a donor-level condition comparison is the donor, not the number of cells measured from that donor. More cells can improve the measurement of a donor’s outcome, but they do not turn one donor into multiple independent donors. The calculations below treat each donor as contributing one outcome, such as a suitably summarized expression measure for a cell type. If the analysis uses a more complex model, the variance and power should be obtained under that model rather than assumed to match this simple illustration.

Precision and Power Under Unequal Counts

Write the donor counts as \(n_1\) and \(n_2\). Under the model above, the estimated difference has variance \(\sigma^2/n_1+\sigma^2/n_2\). Its standard error is therefore determined by both counts, not just their sum. For a two-sided test of \(H_0:\Delta=0\) at significance level \(\alpha\), the normal-approximation test rejects when the absolute standardized difference exceeds \(z_{1-\alpha/2}\), the \(1-\alpha/2\) quantile of the standard normal distribution.

Here \(\lambda\) is the standardized effect under the specified alternative. With a positive \(\Delta\), the approximate power is the probability that a normal random variable with mean \(\lambda\) and variance 1 falls outside the null rejection region:

where \(\Phi\) is the standard normal cumulative distribution function. Power is conditional on the specified alternative: it is the probability of rejecting \(H_0\) if that alternative and the model are correct. It is not the probability that the alternative is true.

Why Equal Allocation Is Efficient When Groups Are Comparable

When group variances and per-donor recruitment costs are equal, balanced allocation minimizes the variance of a two-group difference for a fixed total number of donors. The justification follows directly from the standard error. If \(N=n_1+n_2\), then \(1/n_1+1/n_2=N/(n_1n_2)\). For fixed \(N\), the product \(n_1n_2\) is largest when the counts are equal. Thus the sum of reciprocals, and hence the variance, is smallest at balance.

This result concerns precision under the stated assumptions; it does not say every study should use equal groups. A design may need a minimum number in a group, may compare several conditions, or may face different recruitment costs or outcome variances. Allocation should follow the scientific contrast and practical constraints, not a rule applied without checking its assumptions.

When Variances or Costs Differ

Balanced allocation is not generally optimal if the groups have different outcome variances or different recruitment costs. Suppose group \(g\) has variance \(\sigma_g^2\), and recruiting one donor in that group costs \(c_g\). Under independent donor outcomes, the variance of the estimated difference is \(\sigma_1^2/n_1+\sigma_2^2/n_2\). If the total budget is fixed at \(B\), so \(c_1n_1+c_2n_2=B\), a continuous variance-minimizing allocation satisfies \(n_g\) proportional to \(\sigma_g/\sqrt{c_g}\). In practice, donor counts must be integers, so nearby feasible allocations should be compared directly.

For equal costs, this rule assigns more donors to the group with greater outcome variability. For unequal costs, it reduces allocation to a group whose donors are more costly to recruit. The rule minimizes the variance of this simple two-group contrast; it does not automatically address other goals such as estimating multiple contrasts, protecting against attrition, or ensuring enough donors in a scientifically important group.

A Short Derivation of the Cost-Aware Allocation

To see why the allocation depends on both variance and cost, minimize \(V=\sigma_1^2/n_1+\sigma_2^2/n_2\) subject to \(c_1n_1+c_2n_2=B\). Using a Lagrange multiplier \(\eta\), differentiate \(V+\eta(c_1n_1+c_2n_2-B)\) with respect to each count. Setting both derivatives to zero gives \(\sigma_g^2/n_g^2=\eta c_g\). Therefore \(n_g=\sigma_g/\sqrt{\eta c_g}\), which implies the stated proportionality. This is a continuous optimum; actual recruitment plans require rounding and rechecking the budget and variance.

Diagnostics, Alternatives, and Limitations

Before using an allocation calculation, verify that the quantities entering it correspond to the planned analysis. In particular, check that counts mean independent donors, the contrast is the one the study intends to test, and the variance estimates describe donor-to-donor variation for the relevant outcome. In single-cell studies, large cell counts do not remove uncertainty caused by a small number of donors. The earlier material on intraclass correlation and effective sample size explains why within-donor observations cannot simply be counted as independent replication.

The normal calculation is a planning approximation. With small donor counts, unknown variances, non-normal outcomes, covariate adjustment, repeated measures, multiple conditions, or complicated dependence, the reference distribution and power may differ. A two-sample \(t\)-test with a noncentral \(t\) distribution is a standard alternative for a simple normal model with estimated variance. For more complex designs, simulation under the planned model can incorporate the design features and test procedure directly. The earlier material on simulation-based power addresses that approach.

A practical allocation review should also examine whether the planned counts remain feasible after integer rounding, whether group-specific recruitment constraints are met, and whether the smaller group is large enough for the intended analysis. A mathematically efficient allocation can still be unsuitable if the model assumptions fail, an important group is underrepresented, or the recruitment plan cannot be carried out.

• Common mistake: “There are 40 donors either way, so 30/10 and 20/20 have equal power.” The total count alone does not determine the variance; the standard error includes both \(1/n_1\) and \(1/n_2\).
• Correction: Calculate precision for the actual allocation and contrast. Under equal variances and costs, equal group sizes minimize the variance of a two-group difference for a fixed total.
• Another pitfall: Optimizing a single contrast and assuming the allocation is best for every study objective. Different contrasts, costs, variances, and minimum-group requirements can lead to different feasible choices.