Why the Denominator Is Part of the Question
A cell-type abundance is not specified by naming the cell type alone. It also depends on the set of cells against which that type is counted. The fraction of captured cells assigned to a target type answers a different question from the fraction of cells in a particular lineage assigned to that type. Restricting to cells that pass a quality filter changes the question again: the result describes composition among the eligible cells, not among all captured cells.
This distinction matters when comparing conditions. One denominator may show an increase while another shows a decrease, without either calculation being arithmetically wrong. They describe different estimands. The goal is therefore not to find a universally correct denominator, but to select and report the one that corresponds to the scientific question and the intended population of cells.
Here, “captured cells” means the cells retained in the analysis dataset under the stated capture and processing rules. It does not automatically mean every cell present in the tissue. A proportion among captured cells describes the composition of that observed set and may be affected by which cells were sampled, captured, retained, and classified.
Three Common Denominator Choices
Let \(N_i\) be the number of all captured cells for donor \(i\), \(L_i\) the number in a specified lineage, and \(M_i\) the number passing a prespecified cell-level filter. Let \(A_i\) count target-type cells. The intended numerator must be defined consistently with the denominator: for example, the filtered numerator \(A_{i,F}\) counts target cells that pass the filter, whereas \(A_i\) may count all target cells in the captured dataset.
| Denominator | Proportion | Question answered |
|---|---|---|
| All captured cells | \(A_i/N_i\) | What fraction of captured cells are the target type? |
| Cells in a specified lineage | \(A_i/L_i\), when the target is included in that lineage | What fraction of cells in this lineage are the target type? |
| Cells passing a prespecified filter | \(A_{i,F}/M_i\) | What fraction of eligible, filter-passing cells are the target type? |
A lineage-restricted denominator is useful when the scientific question concerns differentiation or subtype composition within that lineage. It does not measure the target type’s share of the entire captured cell population. Likewise, a filter denominator can be appropriate when the target population is explicitly defined as cells meeting specified eligibility criteria. It answers a conditional question about that eligible set.
A filter should be specified independently of the observed condition difference whenever possible. If the rule is changed after inspecting which choice produces a preferred result, the estimand becomes data-dependent and the comparison is harder to interpret. Even a prespecified filter can affect the result if passing it is related to condition or cell type; the estimate still describes the retained set, not necessarily the unfiltered population.
How Denominator Choice Changes the Estimand
Suppose the target type \(A\) is a subset of lineage \(L\), and \(L\) is a subset of all captured cells. For a donor with \(N_i>0\) and \(L_i>0\), the target’s all-cell proportion can be factored into the lineage’s share of all cells and the target’s share within the lineage:
This identity follows by multiplying the two fractions: \((L_i/N_i)(A_i/L_i)=A_i/N_i\), since \(L_i\) cancels. Thus, the target’s overall abundance can change because the lineage itself changes in size, because the target’s share within the lineage changes, or because both change. The two proportions are not alternative estimates of one unchanged quantity; they separate different components of cell composition.
Worked Example: One Target, Two Denominators
Consider hypothetical counts for one donor: 500 captured cells, of which 50 are in lineage \(L\), including 10 target-type cells \(A\). Calculate the target’s abundance using all captured cells and using the lineage-restricted denominator.
All captured cells. The target’s share is \(A/N=10/500=0.0200\), or 2.00% of captured cells.
Lineage-restricted cells. The target’s share is \(A/L=10/50=0.200\), or 20.0% of lineage cells.
The 2.00% value answers “what fraction of all captured cells are target cells?” The 20.0% value answers “what fraction of lineage cells are target cells?” The identity also checks: \(L/N=50/500=0.100\), and \(0.100\times0.200=0.0200\). Because the target is included in the lineage, the numerator is compatible with both denominators.
For a condition comparison, the estimand must also specify how donor-level proportions are summarized. For example, the difference between the mean donor-level proportions in two groups gives equal weight to each donor’s proportion. A ratio formed by pooling all cell counts across donors instead gives more influence to donors with more cells. Those summaries need not be equal. The choice of denominator and the choice of donor weighting are separate decisions; both should be stated. In abundance tests, donors—not individual cells—are the independent units, as established in the earlier material on donors as independent units in abundance tests.
Worked Example: A Condition Contrast Changes With the Denominator
Suppose two hypothetical donors in each condition have the same counts within their respective groups. For each control donor, there are 500 captured cells, 50 cells in lineage \(L\), and 10 target cells \(A\). For each treated donor, there are 500 captured cells, 150 cells in \(L\), and 15 target cells \(A\). The target type is included in the lineage. Compare the group proportions under both denominators using the mean of the donor-level proportions.
All-cell denominator. For each control donor, \(A/N=10/500=0.0200\). For each treated donor, \(A/N=15/500=0.0300\). Therefore the treated-minus-control difference is \(0.0300-0.0200=0.0100\), or an increase of 1.00 percentage point in the fraction of captured cells that are target cells.
Lineage denominator. For each control donor, \(A/L=10/50=0.200\). For each treated donor, \(A/L=15/150=0.100\). The treated-minus-control difference is \(0.100-0.200=-0.100\), or a decrease of 10.0 percentage points in the target’s fraction of lineage cells.
These results are compatible. The lineage occupies 10.0% of captured cells in control donors and 30.0% in treated donors, while the target’s share within the lineage falls. The all-cell proportion is the product of those two components: \(0.100\times0.200=0.0200\) in control and \(0.300\times0.100=0.0300\) in treatment. The condition contrast depends on whether the question concerns the target’s overall share or its share within the lineage. These descriptive differences alone do not establish a statistically significant effect.
Using a Prespecified Filter
A filter-based proportion must use the same eligibility rule in its numerator and denominator. If \(M_i\) is the number of cells passing the filter, then \(A_{i,F}\) must count target-type cells among those same passing cells. Dividing all target cells by only the passing-cell denominator can produce a number that is not a proportion of the stated set; it may even exceed 1.
Worked Example: Filtering Defines a Conditional Abundance
Imagine hypothetical counts for one donor in each condition. In control, 500 cells are captured; 200 pass a prespecified filter, including 10 target cells. Another 10 target cells fail the filter. In treatment, 500 cells are captured; 180 pass the filter, including 9 target cells, while 21 target cells fail. Calculate the all-cell and filter-based proportions.
All captured cells. Control has \(20\) target cells in total, so \(20/500=0.0400\), or 4.00%. Treatment has \(30\) target cells in total, so \(30/500=0.0600\), or 6.00%. The all-cell contrast is an increase of 2.00 percentage points.
Among filter-passing cells. The matching numerators are 10 and 9. Control has \(10/200=0.0500\), or 5.00%; treatment has \(9/180=0.0500\), also 5.00%. The filter-based contrast is zero percentage points.
The filter-based result does not contradict the all-cell result. It describes target abundance among passing cells, whereas the all-cell calculation includes both passing and failing cells. In this example, the target cells excluded by the filter differ across conditions. If the scientific target is abundance among all captured cells, the filtered calculation does not answer it. If eligibility is central to the question, the filter-based calculation is relevant, but its conditional interpretation must be reported.
Assumptions, Diagnostics, and Common Errors
Before interpreting any denominator-based comparison, check that the cell-type labels and eligibility rules have the same meaning across donors and conditions. The target type must be defined consistently, and lineage membership must be specified rather than inferred differently for different groups. A donor with a zero denominator has no defined proportion for that denominator; it should not be assigned a value of zero, which would assert that the target was absent from a nonempty reference set.
Useful checks include tabulating, for each donor, the target count, denominator count, and excluded-cell count; confirming that the target count does not exceed its denominator; and verifying subset relationships such as \(A_i\leq L_i\leq N_i\) when they are intended. For a filter-based proportion, verify \(A_{i,F}\leq M_i\), and confirm that numerator and denominator were produced by the identical filter. Plotting or tabulating donor-level proportions can reveal whether a group summary is driven by one donor or by unequal donor cell yields.
A common mistake is to say, “The target type increased, so its abundance increased,” after observing a larger \(A_i/N_i\), without specifying that the claim concerns all captured cells. The correction is to name the estimand: for example, “the target type’s fraction of captured cells increased,” and separately report whether its within-lineage fraction changed. Another mistake is to interpret a filtered fraction as an estimate for all cells. It is conditional on filter passage unless additional assumptions justify extending it beyond that set.
The main alternative denominator choices are not competing estimators of a single parameter. All-captured-cell proportions describe overall representation in the captured sample; lineage-restricted proportions describe composition within a chosen lineage; filter-restricted proportions describe composition among eligible cells. Each is limited by what the captured and classified cells represent. None alone establishes absolute cell numbers in the tissue, because capture and recovery can vary. A proportion can also shift because other cell populations change, even when the target count does not. These are consequences of the denominator-defined estimand, not arithmetic defects.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A target type has 12 cells among 300 captured cells and 12 among 60 cells in its lineage. What do the two proportions describe?
- In a condition comparison, the target’s all-cell proportion rises while its within-lineage proportion falls. Is this necessarily a contradiction? Explain.
- A filter-passing denominator is 80 cells, including 8 target cells; 5 additional target cells fail the filter. What is the filter-based proportion, and why is \(13/80\) inappropriate?
- Why can the mean of donor-level proportions differ from a proportion calculated by pooling cell counts across donors?
- For each donor, the target is a subset of lineage \(L\), which is a subset of all captured cells. Write the identity connecting the target’s all-cell proportion to the two nested proportions.
Solutions. (1) The all-cell proportion is \(12/300=0.0400\), or 4.00% of captured cells. The lineage-restricted proportion is \(12/60=0.200\), or 20.0% of lineage cells. They answer different questions. (2) No. The overall proportion depends on both the lineage’s share of all captured cells and the target’s share within the lineage. One can rise while the other falls. (3) The filter-based proportion is \(8/80=0.100\), or 10.0%. The five cells that fail the filter are not in the denominator set, so they must not be included in its numerator. (4) The mean donor-level proportion weights donors equally, while pooling counts gives greater weight to donors contributing more cells. (5) \(A_i/N_i=(L_i/N_i)(A_i/L_i)\), because \(A_i\) is included in \(L_i\), allowing \(L_i\) to cancel.