Cell Sampling Uncertainty Within Donors

Two Sources of Uncertainty

A donor-level estimate is often calculated from a finite sample of cells. Even if the donors in a study were fixed in advance, repeating the cell-sampling process could produce different estimates for those donors. That is cell-sampling uncertainty within donors. If the donors are viewed as a sample from a broader population, the true donor values also vary from donor to donor. That is donor-sampling uncertainty.

These components answer different questions. For the donors actually observed, how precisely have their donor-specific quantities been measured from cells? For a target population of donors, how uncertain is the estimated population mean after accounting for both imperfect cell measurements and variation among sampled donors? A standard error that answers only one of these questions should not be described as answering the other.

The cell-level quantity may be a mean, a proportion, or another donor-specific summary. The central requirement is that \(\hat{\theta}_i\) estimate the stated \(\theta_i\) without systematic cell-sampling bias under the sampling design. For example, a sample mean is conditionally unbiased if the sampled cells represent the target cell population within that donor and the relevant mean exists. If cells were selected selectively, the sampling variance alone does not repair bias.

Conditional Uncertainty for the Observed Donors

First treat the \(D\) observed donors as fixed. Assume each \(\hat{\theta}_i\) is conditionally unbiased for \(\theta_i\), and that the cell-sampling errors are independent across donors. Conditional on these donors, the target is their finite-donor mean \(\bar{\theta}_D=D^{-1}\sum_i\theta_i\). The estimator's conditional variance is the sum of the donor-specific variances divided by \(D^2\):

The conditional standard error is therefore \(\sqrt{D^{-2}\sum_i v_i}\). This describes repeated cell sampling within these same donors. It does not include uncertainty from choosing a different set of donors from the donor population.

Adding Donor-Population Uncertainty

For population inference, assume the donors are independent draws from a target donor population, with \(\operatorname{E}(\theta_i)=\mu\) and \(\operatorname{Var}(\theta_i)=\tau^2\). For each donor, assume \(\hat{\theta}_i\) is conditionally unbiased for \(\theta_i\), and that conditional cell-sampling errors are independent across donors. The cell-sampling variances \(v_i\) may differ across donors and may themselves vary randomly in the donor population.

The law of total variance separates the variation in the true donor mean from the cell-sampling variation around it. Conditional on the sampled donors, the estimator's expectation is their mean; across repeated donor samples, that mean has variance \(\tau^2/D\). The expected conditional variance is the population expectation of the donor-specific variances, not their realized sum treated as a fixed population quantity.

Here \(\operatorname{E}(v_i)\) averages over the donor population under the allocation assigned to donor \(i\). If all donors have the same allocation and the same distribution of conditional variances, this reduces to \(\tau^2/D+\operatorname{E}(v_i)/D\). If the \(v_i\) are fixed design quantities, then \(\operatorname{E}(v_i)=v_i\); otherwise, replacing \(\operatorname{E}(v_i)\) by a realized \(v_i\) is not the population variance formula.

Why the Decomposition Holds

Write \(\hat{\theta}_i=\theta_i+e_i\), where \(e_i\) is the cell-sampling error. Conditional unbiasedness gives \(\operatorname{E}(e_i\mid\text{donor }i)=0\), and the conditional variance of \(e_i\) is \(v_i\). Let \(\mathcal{D}\) denote the sampled donors and their donor-specific quantities. Applying the law of total variance gives

The last equality uses independent, identically distributed donor values, so the variance of their mean is \(\tau^2/D\). It uses linearity of expectation for the cell-sampling term; it does not require the \(v_i\) to be nonrandom or independent of \(\theta_i\). If donors are dependent, or not sampled from a common population, the between-donor term must be modified rather than automatically set to \(\tau^2/D\).

Estimating the Components and Checking Assumptions

In practice, \(\tau^2\) and the expected cell-sampling variances are usually unknown. A within-donor sample variance can estimate cell-level variability for a sample mean under an appropriate sampling model; for independent, identically distributed cells, the estimated conditional variance is \(s_i^2/n_i\). More complex sampling, weighting, or dependence requires a variance estimate that matches that design. A donor-level variance estimate can then inform the between-donor component. Substituting estimated components into the decomposition gives an estimated variance, but it does not automatically account for uncertainty in estimating those components, especially with few donors.

Useful checks include whether the sampled cells plausibly represent the target cells within each donor; whether cell counts and within-donor variability are recorded correctly; and whether the donor sampling scheme supports treating donors as independent draws from the stated population. Large differences in cell counts or within-donor variances make a single common cell-sampling variance implausible. Strong cell dependence, batch effects, or selection of cells by expression can also invalidate a simple \(s_i^2/n_i\) calculation. When cells are sampled without replacement from a known finite cell population, the variance should reflect that design, including a finite-population correction when appropriate.

A common mistake is to report the conditional cell-sampling standard error as though it were the population standard error. The correction is to state the target first. For a fixed set of donors, use the conditional variance \(D^{-2}\sum_i v_i\). For inference to a donor population, include the between-donor term and average random conditional variances: \(\tau^2/D+D^{-2}\sum_i\operatorname{E}(v_i)\). A second mistake is to pool all cells and thereby let donors with many sampled cells dominate an estimate intended to average donors equally. That changes the weighting and generally changes the estimand.

Donor-level procedures, including donor bootstrap methods, address uncertainty from donor sampling; resampling cells within donors can represent cell-sampling uncertainty. A nested resampling scheme can represent both levels when its resampling steps match the study design. Neither method fixes biased cell selection or makes a small number of donors equivalent to a large donor sample. The decomposition is also model-dependent: it relies on unbiased donor estimates, a meaningful donor target, and appropriate dependence assumptions. It separates sources of variance but does not by itself guarantee valid inference or provide a confidence interval.