Why the Estimand Comes First
A statement such as “cell type \(k\) increased after treatment” is incomplete until “increased” has a defined statistical meaning. It might mean that the cell type forms a larger share of all cells in a sample, that a typical donor has a larger share, or that there are more cells of that type per unit of tissue. These are different targets and need not give the same result.
An estimand is the precisely defined population quantity an analysis aims to learn about. For differential abundance, stating the estimand means specifying at least the cell type, the quantity being compared, the conditions, and the population of donors to which the comparison refers. The calculation or model comes later: it cannot resolve an ambiguity in the target itself.
The denominator must be named, not left implicit. It might be all cells passing a stated quality-control rule, all cells in a specified tissue compartment, or another defined collection. Changing the denominator changes the proportion being targeted. The proportion is a donor-level quantity: each donor has one value for the chosen cell type and condition.
The positive-denominator condition is essential. If \(N_i(c)=0\), then \(A_{ik}(c)/N_i(c)\) is undefined; assigning it a value of zero would silently change the estimand. If zero denominators occur for some donors in the intended population, one must instead define a target population for which denominators are positive, or specify and justify another handling rule. Merely dropping those donors after observing their condition-specific denominator can change which population the result describes.
State the Population and Condition Contrast
For a target population \(\mathcal{T}\) of eligible donors, the mean donor-level proportion in condition \(c\) is
The expectation weights donors equally: it describes the average of their proportions, not the proportion among all pooled cells. For two conditions \(c_0\) and \(c_1\), the difference in mean donor-level proportions is
This difference is in proportion units. For example, \(\Delta_{P,k}=0.02\) means a two-percentage-point higher average donor-level proportion in \(c_1\) than in \(c_0\), for the stated population and denominator. It does not mean a two-percent relative increase. A relative change would be a different contrast and would require its own definition.
For a causal interpretation, \(A_{ik}(c)\), \(N_i(c)\), and \(P_{ik}(c)\) can be viewed as potential outcomes under condition \(c\). The same target population must be used for both conditions, and each donor’s proportions must be defined under both. A randomized assignment, along with consistency and appropriate handling of missing outcomes, can support causal estimation. In an observational comparison, one may instead define condition-specific means after standardizing to a stated target population; causal interpretation then requires additional assumptions about confounding and selection. The estimand’s definition alone does not establish those assumptions.
An absolute-number estimand targets a different quantity. If \(A_{ik}(c)\) is measured per a fixed unit, such as cells per cubic millimetre of tissue under a defined measurement protocol, its target-population mean can be written
The unit of \(\Delta_{A,k}\) is the unit of \(A\), such as cells per cubic millimetre. A raw number of cells recovered in a library is not automatically an absolute tissue abundance: it can depend on how much tissue was processed and on cell recovery. The measurement protocol and scale must support the absolute interpretation.
Worked Example: A Paired Difference in Mean Donor Proportions
Suppose a hypothetical study defines the denominator as all eligible profiled cells and includes three donors, each with a positive denominator under both conditions. The cell-type counts are:
| Donor | Condition \(c_0\): type \(k\) / total | Condition \(c_1\): type \(k\) / total |
|---|---|---|
| 1 | 100 / 1,000 | 180 / 1,200 |
| 2 | 80 / 800 | 100 / 1,000 |
| 3 | 240 / 1,200 | 180 / 1,000 |
The donor proportions in \(c_0\) are \(0.10\), \(0.10\), and \(0.20\); their mean is \( (0.10+0.10+0.20)/3=0.1333\). In \(c_1\), the proportions are \(0.15\), \(0.10\), and \(0.18\); their mean is \( (0.15+0.10+0.18)/3=0.1433\). Thus the estimated contrast in this hypothetical three-donor population is
That is an increase of 0.0100 in proportion, or 1.00 percentage point, in the mean donor-level proportion. The paired donor differences are \(0.05\), \(0\), and \(-0.02\), whose mean is \((0.05+0-0.02)/3=0.0100\), confirming the same contrast. This descriptive calculation does not by itself establish a causal treatment effect or quantify uncertainty.
Donor Averages and Pooled Proportions Are Different Targets
A common alternative is to add the cell-type counts across donors and divide by the total number of cells across donors. For a finite set of donors, that pooled proportion is
This is a denominator-weighted average of donor proportions: donors contributing more denominator cells receive more weight. By contrast, the donor-average proportion is \(\sum_i P_{ik}/m\), where \(m\) is the number of donors. The two quantities answer different questions.
Worked Example: The Pooled Contrast Can Be Smaller
Consider two hypothetical donors in each condition. Under \(c_0\), donor 1 has 10 type-\(k\) cells among 100 cells, and donor 2 has 90 among 900. Both donor proportions are \(0.10\), so the mean donor proportion is \(0.10\), and the pooled proportion is \((10+90)/(100+900)=0.10\).
Under \(c_1\), donor 1 has 20 type-\(k\) cells among 100, while donor 2 has 90 among 900. The mean donor proportion is \((0.20+0.10)/2=0.15\), giving a donor-average contrast of \(0.15-0.10=0.05\), or 5 percentage points. The pooled proportion in \(c_1\), however, is \((20+90)/(100+900)=0.11\), so its contrast from \(c_0\) is \(0.11-0.10=0.01\), or 1 percentage point.
The difference occurs because the pooled calculation gives the large-denominator donor more weight. Neither answer is an arithmetic error: the first targets the average donor proportion, while the second targets the share among the pooled cells. The appropriate choice depends on the scientific question.
The distinction can also be seen at the population level. When the expectations exist and \(E_{\mathcal{T}}[N_i(c)]>0\), the ratio of expected counts is
The identity follows from \(A_{ik}(c)=N_i(c)P_{ik}(c)\) and \(E[NP]=E[N]E[P]+\operatorname{Cov}(N,P)\). Thus the ratio of expected counts equals the mean donor proportion only when the covariance term is zero. A pooled-cell target and a donor-average target can differ even when both are calculated correctly.
Proportion Change Does Not Determine Absolute Change
A proportion is relative to its denominator. It can stay constant while the number of cells of the type changes, or change while the type’s absolute number stays constant. Therefore, a proportion contrast is not a substitute for an absolute-abundance contrast. Whether absolute counts are measurable on a comparable scale is a design and measurement question, not something the proportion alone can answer.
Worked Example: Equal Proportions, Different Absolute Numbers
Suppose a hypothetical paired study measures cell numbers per cubic millimetre using a protocol that supports that scale. For three donors, the control counts for type \(k\) are 100, 120, and 80 cells per cubic millimetre, with corresponding total counts of 1,000, 1,200, and 800. Under treatment, the type-\(k\) counts are 200, 240, and 160, with totals of 2,000, 2,400, and 1,600.
Every donor has proportion \(0.10\) in both conditions. The mean donor proportion is therefore \(0.10\) under control and \(0.10\) under treatment, so \(\widehat{\Delta}_{P,k}=0.10-0.10=0\). The mean absolute count is \((100+120+80)/3=100\) cells per cubic millimetre under control and \((200+240+160)/3=200\) under treatment. The absolute contrast is \(200-100=100\) cells per cubic millimetre.
The conclusion is not that one result disproves the other. In this hypothetical example, the cell type’s share is unchanged while its absolute number per stated tissue volume is higher. If the counts were merely the numbers recovered from differently sized or processed samples, this absolute interpretation would not follow without a valid calibration to a common tissue scale.
Checks Before Reporting an Estimand
State which donors the result is intended to describe and whether both conditions refer to that same population.
Specify cell type \(k\), the denominator for \(P_{ik}(c)\), or the fixed measurement scale for \(A_{ik}(c)\).
For each proportion condition, require \(N_i(c)>0\) almost surely in the target population, or define a justified alternative population or rule.
For example, state a difference in mean donor proportions in percentage points, or a difference in cells per unit of tissue.
Describe an observed condition contrast as causal only when the design and assumptions support that interpretation.
Diagnostics begin with checking the definition against the data: confirm that the denominator matches the stated rule, identify donors with zero denominators, and verify that absolute counts are on a comparable scale if an absolute target is claimed. A condition-specific set of excluded donors, inconsistent tissue units, or changing denominator can make the reported quantity differ from the intended estimand. These are not problems that a more elaborate model can repair without further assumptions.
A useful comparison is between a mean donor proportion and a pooled proportion. The former gives each donor equal weight; the latter weights donors by their denominator size. An absolute-number contrast instead compares counts on a defined common scale. They are alternatives only in the sense that each may be scientifically relevant: they are not interchangeable estimates of one universal “abundance change.”
Common Mistakes and Limitations
- Leaving the denominator unstated. A proportion is only interpretable relative to its defined denominator. State which cells contribute to \(N_i(c)\).
- Taking an expectation over undefined proportions. If \(N_i(c)=0\) for target donors, \(P_{ik}(c)\) is undefined. Require \(N_i(c)>0\) almost surely or explicitly redefine the target or handling rule.
- Calling a pooled proportion the average donor proportion. The pooled value weights donors by their cell totals; report it as a pooled-cell share, not an equal-donor average.
- Inferring absolute growth from a higher proportion. A higher share does not by itself establish more cells per unit of tissue.
- Claiming a causal effect from a contrast alone. A numerical difference defines a target; causal interpretation also depends on assignment, confounding, selection, and measurement assumptions.
A well-defined estimand still has limitations. It may be difficult to measure absolute abundance, some donors may not have valid denominators under a condition, and the target population may not match the donors actually observed. State such restrictions rather than allowing an apparently precise estimate to imply broader applicability than the design supports.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A study defines \(P_{ik}(c)\) as a cell-type proportion. What must be true of the denominator in each condition for the target-population mean proportion to be defined?
- Two donors have proportions \(0.10\) and \(0.30\), with equal donor weight. What is their mean donor proportion? If their denominator sizes are 100 and 900, what is the pooled proportion?
- In one condition, the mean donor proportion is \(0.12\); in another, it is \(0.15\). State the difference in percentage points and say what it describes.
- What additional measurement condition is needed to interpret a difference in observed cell counts as a difference in absolute abundance per unit of tissue?
- When may a condition contrast be described as causal rather than as a condition-associated difference?
Solutions. (1) \(N_i(c)>0\) must hold almost surely in the target population for every condition in the estimand; otherwise the proportion and its expectation are not defined without a specified alternative rule. (2) The equal-donor mean is \((0.10+0.30)/2=0.20\). The pooled proportion is \((10+270)/(100+900)=280/1000=0.28\), because the second donor contributes more denominator cells. (3) The difference is \(0.15-0.12=0.03\), or 3 percentage points. It describes the difference in mean donor-level proportions for the stated population and denominator. (4) Counts must be measured or validly calibrated on a comparable fixed tissue scale, such as cells per cubic millimetre; raw recovered counts alone do not guarantee this. (5) Causal language requires a design and assumptions that support it, such as appropriate treatment assignment and consistency, with confounding and selection addressed when assignment is not randomized. A contrast by itself does not establish causation.