Separating Within- and Between-Donor Variance

Why Separate the Sources of Variation?

Cells from one donor are not simply interchangeable with cells from another donor. Cells from the same donor share a biological background and may also share collection, processing, and measurement conditions. Consequently, observed cell-to-cell variation combines differences among donors with differences among cells within a donor. Separating these sources helps describe the data at the level relevant to the scientific question: how much do donors differ, and how variable are cells within a donor?

The decomposition is descriptive and model-based, not a claim that every source of variation has a single biological cause. For a given gene or other cell-level outcome, the within-donor component can include cell-state heterogeneity, measurement error, and unmodelled cell-level factors. The between-donor component can reflect donor biology as well as donor-associated technical or clinical differences. Interpretation therefore depends on study design and the outcome scale.

A Nested Random-Intercept Model

Let \(Y_{ij}\) denote the outcome for cell \(j\) from donor \(i\), where \(i=1,\ldots,D\). A simple model that represents the two sources of variation is

Here, \(\mu\) is the population mean, \(b_i\) is donor \(i\)’s deviation from that mean, and \(\varepsilon_{ij}\) is the cell’s deviation from its donor’s mean. The random-effects assumptions are \(E(b_i)=0\), \(\operatorname{Var}(b_i)=\tau^2\), \(E(\varepsilon_{ij})=0\), and \(\operatorname{Var}(\varepsilon_{ij})=\sigma^2\). Donor effects are independent across donors; cell residuals are independent across cells conditional on the donor effects; and the residuals are independent of the donor effects. The simplest model also assumes a common residual variance across donors.

Under this model, \(\tau^2\) is the between-donor variance component and \(\sigma^2\) is the within-donor variance component. Normality of the random effects and residuals is an additional assumption when using normal-theory likelihood or exact small-sample procedures; it is not required for the variance identity below. The model also presumes donors are meaningfully viewed as sampled from a population. If the scientific target is only the particular donors observed, a population-level random-effects interpretation may not be appropriate.

This identity does not say that the components are equally important, nor does it identify their biological causes. It says that, under the model, the variance of a randomly selected cell outcome consists of variation in the donor mean plus variation around that donor mean.

Deriving the Decomposition

The variance identity follows from conditioning on the donor. Given donor \(i\), the donor effect is fixed, so the conditional mean is \(\mu+b_i\) and the conditional variance is \(\sigma^2\). The law of total variance states that the marginal variance equals the expected conditional variance plus the variance of the conditional mean. Applying it gives

There is also an exact sum-of-squares identity for observed data. Let \(n_i\) be the number of measured cells from donor \(i\), \(\bar{Y}_i\) the mean outcome for that donor, and \(\bar{Y}\) the mean across all observed cells. Then

To see why, write \(Y_{ij}-\bar{Y}=(Y_{ij}-\bar{Y}_i)+(\bar{Y}_i-\bar{Y})\), square, and sum. The cross-product for donor \(i\) is zero because \(\sum_j(Y_{ij}-\bar{Y}_i)=0\). The first sum on the right is the within-donor sum of squares; the second is the between-donor sum of squares, with each donor mean weighted by its cell count. This identity describes the observed data exactly. Turning these sums into estimates of population variance components requires additional assumptions and an estimation method.

Estimating Components in a Balanced Design

For a balanced design, suppose every donor contributes the same number \(m\) of cells. Define \(SS_{\text{within}}\) and \(SS_{\text{between}}\) as in the preceding decomposition. Their mean squares are

Under the random-intercept assumptions, the expected mean squares are

The reason for the second expectation is that a donor mean averages \(m\) residuals: its residual contribution has variance \(\sigma^2/m\), while its donor effect has variance \(\tau^2\). The between-donor sum of squares is based on deviations of donor means from their overall mean. Its mean square therefore has expectation \(m\tau^2+\sigma^2\). Subtracting the within mean square removes the residual contribution in expectation, and dividing by \(m\) gives the method-of-moments estimate

This simple formula applies to a balanced one-way nested design under the stated model. In unbalanced designs, unequal cell counts affect the expected mean squares; do not substitute an average cell count into this formula and assume the result is valid. A mixed-model estimation method can accommodate unequal counts, provided the design and assumptions support that model.

Biological Interpretation, Diagnostics, and Alternatives

Suppose the outcome is log-normalized expression for a gene. A substantial estimated \(\tau^2\) means donor-specific average expression differs appreciably on the log-expression scale, relative to the model’s residual variation. It does not show that donor identity caused the differences: age, treatment, disease status, cell composition, batch, and other donor-associated factors may contribute. A substantial \(\sigma^2\) means cells from a donor vary around that donor’s mean; it does not establish that this entire variation is biological. It may include technical noise, unmeasured cell states, and departures from the assumed model.

The components depend on the outcome and the design. A log transformation, a different normalization, or adjustment for covariates can change the variance split. If donors differ systematically in cell type composition, a cell-level outcome may show a large within-donor component that reflects a mixture of distinct cell states rather than random fluctuation within one homogeneous population. Define the cell population and outcome carefully before interpreting either component.

Useful diagnostics include plotting cell outcomes by donor, checking donor means and within-donor spreads, and examining residuals against fitted values and technical variables. Look for donors with markedly different spreads, outliers, strong batch alignment, or patterns across donors that a single common residual variance cannot represent. A variance estimate at zero or a negative unconstrained method-of-moments estimate is also a signal to report uncertainty and inspect the data, not to force a biological story. With few donors, the between-donor component can be imprecise even when each donor contributes many cells.

The principal alternatives answer related but distinct questions. Donor-level summaries, followed by inference across donors, put the donor at the unit of analysis and avoid treating cells as independent biological replicates; they may not directly estimate a within-donor component. A fixed-donor-effects model estimates a separate mean for each observed donor and can control for those donor-specific levels, but it does not by itself estimate a population distribution of donor effects. A random-intercept model represents that distribution and separates the two variance components, but depends on its population and variance assumptions. The appropriate choice follows the target of inference, not a preference for a more elaborate model.