Power From Independent Donor Counts

Why Donor Counts Drive Power

A single-cell experiment may measure thousands of cells, but the biological comparison is usually about differences among donors or samples assigned to conditions. Cells from one donor share biological and technical features, so they are not independent biological replicates. For planning a comparison across donors, the central quantities are the number of independent donors, the variability of the donor-level outcome, and the effect size worth detecting.

The outcome used for planning might be a donor-level expression summary for a cell type, such as a normalized pseudobulk measure, or another prespecified donor-level quantity. The calculations below do not prescribe a particular expression model. They show how to plan the donor count once a target contrast and a plausible donor-level standard deviation have been specified.

In a two-condition study, let \(Y\) denote the donor-level outcome, and let \(\delta\) be the target difference in population means, measured in the outcome’s units. For example, \(\delta=0.3\) might represent a planned difference of 0.3 units on a transformed expression scale. Planning requires a scientifically meaningful target: a smaller effect generally requires more donors to detect with the same power.

Two Independent Groups: Approximate Power

Suppose \(n\) independent donors are assigned to each of two conditions. Assume donor outcomes are independent within and between groups, have a common population standard deviation \(\sigma\), and are approximately normal or sufficiently well behaved for a normal approximation to the difference in sample means. Assume also that the contrast, test direction, and significance level are specified in advance. Under \(H_0\), the mean difference is zero; under the planning alternative, it is \(\delta\).

With a known planning value for \(\sigma\), the standard error of the difference in group means is \(\sigma\sqrt{2/n}\). The standardized alternative mean, often called the noncentrality parameter for this normal approximation, is therefore \(\lambda=\delta/(\sigma\sqrt{2/n})\). For a two-sided test at level \(\alpha\), the null rejection cutoffs are \(-z_{1-\alpha/2}\) and \(z_{1-\alpha/2}\), where \(z_q\) is the \(q\)-quantile of the standard normal distribution and \(\Phi\) its cumulative distribution function.

The first probability is the chance that the statistic falls above the upper rejection cutoff under the alternative; the second is the chance it falls below the lower cutoff. For a positive target effect, the second probability is often small, but it belongs in a two-sided power calculation.

Solving for the Number of Donors

For equal group sizes, the same calculation can be rearranged to estimate a donor count that achieves a desired power. For a two-sided test, a common planning approximation is to choose \(n\) so that \(\lambda\) is about \(z_{1-\alpha/2}+z_{1-\beta}\), where \(1-\beta\) is the target power. This approximation treats the probability in the far, opposite tail as negligible; the final power should be checked after rounding \(n\) up to a whole donor.

The factor of 2 arises because the variance of the difference between two independent group means is \(\sigma^2/n+\sigma^2/n=2\sigma^2/n\). Solving \(\delta/(\sigma\sqrt{2/n})=z_{1-\alpha/2}+z_{1-\beta}\) for \(n\) gives the displayed expression. The result is the approximate number of donors per condition, not the total.

Paired Designs: Count Donors, Not Measurements

When each donor contributes measurements under both conditions, the natural analysis may be based on within-donor differences. Let \(D_i\) be the difference for donor \(i\), with target mean difference \(\delta\) and standard deviation \(\sigma_D\). If the \(n\) donors are independent of one another and the differences are approximately normal, the standard error of the mean difference is \(\sigma_D/\sqrt{n}\). The paired design’s power depends on the variation of these differences, not on treating the two measurements from a donor as independent observations.

This setup is appropriate only when the conditions can genuinely be compared within the same donor and the planned analysis uses that pairing. The earlier material on paired designs emphasizes that pairing controls stable between-donor differences; it does not create additional independent donors.

Assumptions, Diagnostics, and Alternatives

A power calculation is conditional on its design and model assumptions. Before relying on a donor-count result, check that the unit counted as a replicate is an independent donor, the target contrast matches the eventual analysis, the estimated variability is on the same outcome scale, and the target effect is scientifically meaningful. If the final analysis adjusts for planned covariates or blocks, the relevant variance and degrees of freedom may differ from those in the simple formulas. A chosen \(\alpha\) should also reflect the planned testing burden; a stricter threshold generally requires more donors for the same power.

The formulas use a normal approximation with a planning standard deviation. In practice, \(\sigma\) is usually estimated rather than known. When donor counts are small, tests that estimate variance have heavier-tailed reference distributions than a standard normal test, so the displayed approximation can overstate power. A t-based calculation using the appropriate degrees of freedom is preferable when the analysis is a t test; its power can be found from the noncentral t distribution and sample size selected iteratively. Likewise, uncertain variance estimates should be examined over a plausible range rather than treated as exact.

Diagnostics for planning include checking donor-level outcome distributions and variability in pilot data, reviewing whether donors are plausibly independent, and confirming that the expected number of usable donors is the number used in the calculation. A few extreme donor outcomes can make a normal-theory calculation sensitive to the assumed distribution. A large number of cells per donor does not resolve a shortage of independent donors: more cells may improve the measurement of that donor’s outcome, but they do not supply new independent biological replicates. Detailed modeling of within-donor cell dependence requires additional methods.

The simplest analytical calculation is most useful for a single, prespecified contrast with reasonably balanced groups and a credible variance estimate. Paired calculations may be more efficient when within-donor differences are less variable than between-donor outcomes, but they require both conditions to be observed for each donor. More complex designs, non-normal count outcomes, several interacting factors, or substantial missingness may call for design-specific methods rather than these formulas. In every case, the comparison should be made at the level of independent donors, with the analysis model and estimand aligned to the design.

Common Mistake: Counting Cells as Replicates

A common but incorrect argument is: “We have 10 donors per condition and 500 cells per donor, so we have 5,000 replicates per condition.” This counts observations that share a donor as though each were an independent draw from the population of donors. The correction is to plan donor-level replication using the number of independent donors. Cell yield remains important for measurement quality and cell-type coverage, but does not convert one donor into many independent donors. Power calculated from cell counts as if they were independent can therefore be seriously overstated.