Pseudobulk Estimands for Cell-Type-Specific Expression

What Is the Target of a Pseudobulk Comparison?

A differential-expression result is meaningful only after the quantity being compared has been specified. For a particular gene and cell type, one possible question is whether a typical donor has higher average expression per cell in one condition than another. A different question is whether the tissue contains more cells of that type, or whether more total transcript is contributed by that type. These targets can change independently.

An estimand is the precisely defined population quantity an analysis aims to learn. It is not the calculation used to estimate that quantity. Here, the central target is a donor-population contrast in expression conditional on belonging to a specified cell type. The donor, rather than each captured cell, is the unit receiving equal weight in that target.

The definition requires a clear target population: which donors, which conditions, which cell-type definition, and what expression scale. The scale matters. If expression is defined as a normalized value, the estimand is a contrast in that normalized value, not automatically a contrast in absolute transcript molecules per cell. The target cell type must also be specified consistently enough for “expression within cell type \(k\)” to be interpretable across conditions.

For a positive expression mean, a ratio-scale target is also possible: \(\theta_{gk}(1)/\theta_{gk}(0)\), or its base-2 logarithm. A ratio of donor-population means is generally not the same quantity as the average of donor-specific log ratios. The estimand must state which is intended. If either population mean is zero, the log ratio is undefined; one should not conceal that by silently changing the target.

Donors, Cells, and the Meaning of the Average

The cell mean \(\mu_{gdk}(c)\) describes expression among cells of the specified type from one donor. The population quantity \(\theta_{gk}(c)\) then averages those donor means. This two-stage definition is important: donors with many captured cells do not automatically represent more donors. More cells may improve how accurately a particular donor’s mean is measured, but do not change the equal-donor target.

Pooling all cells instead targets a cell-weighted average. If donor \(d\) contributes \(n_d\) cells with mean expression \(\mu_d\), that pooled average is \(\sum_d n_d\mu_d/\sum_d n_d\). It gives greater weight to donors contributing more cells. It equals the equal-donor average only under special conditions, such as equal cell counts or no relationship between cell count and donor mean. The two answers can differ even when both are calculated correctly.

This distinction is related to biological replication. Cells from one donor are not new independent donors. A cell-level analysis can give a small standard error simply because many cells were measured, even when the number of independent donors is small. Earlier material on biological replication and donor-aware inference develops that concern; the key point here is that the estimand and the uncertainty calculation must both respect the donor-level target.

Expression and Abundance Are Separate Targets

Let \(N_{dk}(c)\) be the number of cells of type \(k\) contributed by donor \(d\) under condition \(c\), on a clearly defined tissue or sampling basis. For a fixed donor and condition, total expression across those cells is the sum of their expression values. By the definition of a cell mean, that total is the number of cells times their mean expression:

Here \(\bar{Y}_{gdk}(c)\) is the average expression among the cells included in that total. The identity shows why a total can change when per-cell expression does not, and why per-cell expression can change while cell abundance does not. In population summaries, the mean of the product \(N\mu\) need not equal the product of the mean abundance and mean expression: abundance and expression may vary together across donors.

Abundance itself needs a defined denominator. It might mean an absolute number of cells per tissue amount, or a proportion of captured cells among a specified set of cell types. These are different estimands. A change in a cell-type proportion can occur because that type changes, because other types change, or both. Neither proportion nor absolute number is the same as expression per cell within the type.

Conditions for Interpretation

For a descriptive comparison, \(\theta_{gk}(c)\) refers to the target donor population represented by the donors observed in condition \(c\). Interpreting the observed donor means as informative about that population requires donors to be sampled, or otherwise selected, in a way that supports the intended generalization. Donor sampling and cell sampling must also provide adequate information about each donor’s cell-type-specific mean. A large number of cells from a few donors does not compensate for having little information about the donor population.

If condition is an intervention and the goal is a causal effect, define \(\mu_{gdk}(c)\) as the donor’s mean expression that would occur under condition \(c\). The contrast \(\tau_{gk}\) then averages within-donor potential-outcome differences over the target donor population. Causal interpretation requires conditions that justify comparing those potential outcomes, such as random assignment or adequate control of confounding in an observational design. It also requires a meaningful, consistently defined cell-type target under both conditions. If condition changes which cells qualify as that type, the conditional contrast may partly reflect changes in the cells being included.

In observational data, a difference in donor-population means is generally an association unless the design and assumptions support a causal interpretation. Covariates measured on cells do not create additional independent donors, and donor-level confounding cannot be resolved merely by collecting more cells from the same donors.

Worked Example: Defining a Donor-Population Contrast

A different target would average donor-specific condition changes. For paired donors, for example, the average of \(\mu_{gdk}(1)-\mu_{gdk}(0)\) gives each donor’s change equal weight. For unpaired groups, the difference between condition-specific population means instead compares the two donor populations. The design determines which contrasts are identifiable; it does not make the estimand interchangeable with a cell-level pooled contrast.

Choosing and Checking the Estimand

Before interpreting a pseudobulk differential-expression result, state whether the target is a mean difference, a ratio of mean expression, or another precisely defined contrast. Also state the target donor population and cell-type definition, the expression scale, the donor weighting, and whether the interpretation is descriptive or causal. This makes clear what a later statistical model must estimate; the model and aggregation choices are separate from the definition of the target.

Useful diagnostics concern whether the target is supported by the data. Check how many donors contribute usable cells in each condition, whether cell counts are highly uneven across donors, whether cell-type assignment is comparable across conditions, and whether one or two donors dominate the observed contrast. Earlier material on donor influence and residual diagnostics addresses model checks in more detail. Here, these checks matter because weak or condition-dependent cell recovery can make the observed donor summaries poor representations of the intended cell population.

A very low or zero estimated mean creates an additional interpretive problem for ratio-scale estimands: a log ratio may be undefined or unstable. A transformation or adjustment may be part of a later model, but it changes how the result is represented and must not be mistaken for a definition of the original biological target. Likewise, a proportion based on captured cells may reflect capture and sampling as well as tissue composition.

Common Mistakes and Alternatives

• Calling a total-count change a per-cell expression change. Total expression can rise because cell abundance rises. Separate the abundance target from the conditional per-cell mean.
• Giving every cell equal weight while claiming an average donor effect. Cell pooling weights donors by their cell contributions. Use an explicitly donor-weighted target when the scientific question is about the average donor.
• Treating a cell-type proportion as expression. A proportion measures relative abundance under its denominator, not the expression level of a gene within that type.
• Leaving the contrast scale implicit. A difference in units per cell and a ratio of means answer different questions; report which one is intended.
• Interpreting an observational contrast causally. A condition-associated expression difference is not necessarily a condition effect without suitable design and assumptions.

The principal alternatives therefore have different targets: an equal-donor mean describes the average donor’s cell-type-specific expression; a pooled-cell mean describes the average contributed cell; a total-expression summary combines cell number and per-cell expression; and an abundance summary concerns cell counts or proportions. None is universally correct. The scientific question decides which target is relevant, and the analysis should be judged against that target.