Between-Donor Heterogeneity

Why the Average Is Not the Whole Result

A study may estimate an average response across donors, but that average does not describe how similar the donors are to one another. Two sets of donors can have the same average outcome while one set is tightly clustered and the other contains very different responses. In single-cell studies, this distinction matters because the biological interpretation of an average can depend on whether it is typical of most donors or conceals substantial donor-to-donor variation.

The target here is between-donor heterogeneity: the spread of donor-level outcomes in the population of donors relevant to the study. This is not the same as variation among cells within a donor, nor is it the uncertainty in estimating a mean from a finite sample of donors. Earlier material on separating within- and between-donor variance and on generalizing beyond observed donors provides useful context; here the focus is on defining, estimating, and interpreting the spread itself.

For example, \(Y_i\) might be a donor-specific mean expression value for a gene in a defined cell type, or a donor-specific treatment contrast on a specified expression scale. The outcome and scale must be fixed before interpreting its spread. A standard deviation of 1 on a log expression scale does not mean the same thing as a standard deviation of 1 on a raw-count scale.

Estimating Spread Across Observed Donors

Suppose \(m\) donors are sampled from the target population and a donor-level outcome is observed for each. Write those outcomes as \(y_1,\ldots,y_m\), their sample mean as \(\bar{y}\), and their sample variance as \(s_D^2\):

When the donors are independent and representative draws from the target population, \(s_D^2\) estimates \(\tau^2\), and \(s_D\) estimates \(\tau\). The sample variance uses \(m-1\) in the denominator because the sample mean is estimated from the same donor outcomes. Its units are the squared units of \(Y_i\); the standard deviation returns to the original units.

A key distinction is that \(s_D\) describes the spread of donor outcomes, whereas the standard error of the sample mean describes uncertainty in estimating \(\mu_D\). Under independent, identically distributed donors with finite variance, the standard error is estimated by \(s_D/\sqrt{m}\). More donors can reduce uncertainty in the estimated mean without making the underlying donors more alike.

A confidence interval for the mean is not a summary of donor heterogeneity. For example, under independent normally distributed donor outcomes, a \(t\) interval for \(\mu_D\) uses \(s_D/\sqrt{m}\) and has \(m-1\) degrees of freedom. Its width concerns uncertainty in the population mean, not the range in which individual donor outcomes are expected to fall.

Uncertainty in an Estimate of the Spread

A sample standard deviation is itself uncertain, especially when the number of donors is small. If the donor outcomes are independent draws from a normal population with variance \(\tau^2\), then \((m-1)s_D^2/\tau^2\) follows a chi-square distribution with \(m-1\) degrees of freedom. This result gives a confidence interval for the variance. Let \(\chi^2_{q,df}\) be the \(q\) quantile of the chi-square distribution with \(df\) degrees of freedom. A \(100(1-\alpha)\%\) interval is:

The denominators appear in this order because the larger chi-square quantile produces the lower endpoint. Taking square roots of both endpoints gives an interval for the standard deviation. This interval relies on normality of the donor-level outcomes; with few donors, departures from normality or one unusual donor can have a substantial effect.

When Donor-Level Outcomes Are Noisy Estimates

In single-cell data, a donor-level outcome may be estimated from a finite number of sampled cells rather than observed without error. Denote the estimate for donor \(i\) by \(Z_i\), and write \(Z_i=Y_i+e_i\), where \(Y_i\) is the donor's underlying outcome and \(e_i\) is estimation error. Suppose the errors have mean zero, are independent across donors and independent of the underlying outcomes, and have known variances \(v_i\). Then:

Consequently, the across-donor variance of the estimates can reflect both true donor heterogeneity and estimation noise. If donors have unequal cell counts or different precision, their \(v_i\) values may differ. Treating the observed spread of \(Z_i\) as if it were entirely biological can therefore overstate the spread of underlying donor outcomes.

For independent estimates with a common mean and the stated error model, the expected sample variance of the \(Z_i\) values is \(\tau^2\) plus the average measurement-error variance. This motivates a method-of-moments correction:

This is a simple correction, not a universal replacement for a model that handles donor-specific precision. In particular, the corrected estimate can be negative in a finite sample. Since a variance cannot be negative, one may report a boundary estimate of zero, but should also recognize that the data then provide little evidence for resolving true heterogeneity from estimation noise. Uncertainty in the \(v_i\) values also affects the correction.

The justification follows from the definition of sample variance. If \(Z_i\) have a common mean and independent variances \(q_i=\tau^2+v_i\), then \(\sum_i(Z_i-\bar{Z})^2=\sum_i Z_i^2-m\bar{Z}^2\). Taking expectations, the common mean terms cancel, leaving \(\sum_i q_i-(1/m)\sum_i q_i\). Dividing by \(m-1\) gives:

Subtracting the average \(v_i\) therefore removes the expected contribution of estimation noise under these assumptions. The derivation does not make noisy donor estimates equivalent to error-free outcomes; it identifies what the correction estimates on average.

Comparing Heterogeneity Without Losing the Scale

A standard deviation gives an absolute measure of spread in the outcome's units. This is usually the clearest first summary. A coefficient of variation, \(s_D/\bar{y}\), expresses spread relative to the mean, but is only sensible when the outcome has a meaningful zero and the mean is sufficiently far from zero. It can be unstable or misleading for outcomes that can be negative, are centered near zero, or use an arbitrary transformed scale.

A group comparison may focus on the difference in population means, the difference in variances, or both, depending on the scientific question. These are distinct estimands. Comparing sample standard deviations is descriptive, and a formal claim about population differences requires an inferential procedure with assumptions suited to the design and distribution. A mean-only analysis can miss heterogeneity, while a spread-only comparison does not establish a difference in average outcome.

Diagnostics, Alternatives, and Limitations

Before interpreting an estimate of heterogeneity, check what contributes to the donor-level values and whether the sampling assumptions are credible. Useful checks include plotting donor outcomes, checking for influential donors, examining donor-specific precision, and verifying that the outcomes share a common definition and scale. A highly skewed distribution or an extreme donor can make the sample variance and the normal-theory variance interval particularly sensitive. The normality assumption concerns the distribution of donor-level outcomes, not the distribution of individual cells.

When donor estimates have unequal precision, reporting their raw sample standard deviation alongside the estimated \(v_i\) values makes the limitation visible. A measurement-error correction is an option when those variances are defensible and the independence assumptions are reasonable. A hierarchical or other model-based approach may be preferable when precision varies substantially or when a full uncertainty analysis is required. A simple sample variance is easier to interpret, but it does not automatically accommodate noisy estimates, complex donor sampling, or dependence between donors.

• Do not confuse spread with precision. \(s_D\) describes variation among donors; \(SE(\bar{y})\) describes uncertainty in an estimated average.
• Do not treat cells as independent donor replicates. The number of cells does not change the number of independently sampled donors.
• Do not label all observed spread biological. Noisy donor-level estimates can add variability beyond true between-donor differences.
• Do not assume a stable variance estimate from few donors. Small donor samples can yield very wide intervals and sensitivity to individual outcomes.