Defining Target Comparisons and Estimands

Why Define the Target Before Choosing an Analysis?

A multi-sample single-cell study can produce many summaries from the same assay: cell-type proportions, expression values within a cell type, or contrasts on a log-ratio scale, for example. These summaries answer different questions. Even after choosing an outcome, a comparison can change meaning depending on which donors the study targets and how subgroup results are combined. Defining the target quantity first makes the scientific question explicit; the analysis can then be judged by whether it estimates that quantity.

The central object is the estimand: the precisely defined population quantity a study seeks to learn. It is not the numerical result calculated from the observed donors, and it is not a test statistic or p-value. The earlier material on experimental units and sampling frames motivates specifying who contributes independent biological information; here the focus is the target comparison itself.

The notation \(Y_i(c)\) denotes the outcome for donor \(i\) under condition \(c\). It is a potential-outcome notation: for each donor and condition there is a conceptually defined outcome, though typically only the outcome under the donor’s observed condition is available. The population \(P\) must be described in scientific terms, such as eligible adults with a specified disease, rather than merely as “the donors in the data.” The outcome definition should specify the cell type, assay-derived quantity, and scale whenever these matter.

An estimand may be descriptive or causal. A descriptive contrast compares outcome distributions or averages among populations observed under different conditions. A causal contrast compares potential outcomes under alternative conditions for the same target population. Calling a contrast causal requires a defensible interpretation of the conditions as interventions and assumptions that connect observed data to those potential outcomes, including appropriate assignment or confounding control. A condition label alone does not establish causation.

Choosing the Outcome Summary and Contrast

In single-cell studies, one essential decision is the level at which the outcome is summarized. A donor-level cell-type proportion gives each donor a single outcome, so the population mean describes the average donor’s proportion. Pooling all cells instead gives more influence to donors contributing more captured cells; that is a different target quantity. Similarly, a mean expression summary within a cell type can be defined donor by donor, then averaged over donors, or calculated by pooling cells. The intended weighting must be explicit.

The contrast scale also matters. If \(\theta_a\) and \(\theta_b\) are mean proportions, \(\theta_b-\theta_a\) is an absolute difference in proportion, often reported in percentage points. A ratio \(\theta_b/\theta_a\) is relative. For log-expression outcomes, a difference of mean log2 values is measured in log2 units; exponentiating that difference gives a ratio of geometric means under the stated log-scale summary. These are not interchangeable descriptions of one number.

A useful specification states the outcome and contrast in one sentence. For example: “Among eligible adults with condition X, estimate the difference between conditions B and A in the population mean donor-level proportion of captured cells assigned to cell type T.” This wording identifies a population quantity and an assay-defined outcome. It does not claim to estimate absolute cell numbers unless the measurement procedure supports that claim.

Population Weighting and Marginal Contrasts

When a target population contains meaningful subgroups, an overall estimand depends on their weights. For example, the mean under a condition in a population comprising age groups \(s\) can be written as a weighted mean of subgroup means. The weights should describe the target population, not be chosen simply because they match the observed sample.

Here \(S_i\) is a subgroup label and \(w_s=P(S_i=s)\) is the target-population proportion in subgroup \(s\). The formula is a direct application of the law of total expectation. It clarifies why the same subgroup-specific results can yield different overall contrasts for different target populations.

Scale, Interpretation, and Assumptions

An estimand is defined before a particular estimator or statistical model is chosen. Nevertheless, it should be possible to identify the assumptions needed for the planned estimate to inform that target. At minimum, the target population and outcome must be operationally clear, the observed donors must provide information relevant to that population, and the comparison must be supported by the data. A causal interpretation additionally requires conditions and assignment assumptions that justify interpreting observed contrasts as contrasts between potential outcomes.

For expression outcomes, the summary scale should be stated precisely. Suppose each donor has a normalized log2 expression summary for a chosen cell type. A mean difference of 0.5 log2 units is not an increase of 0.5 expression units on the original scale. If the estimand is a difference in mean log2 expression, exponentiating gives \(2^{0.5}\approx1.414\), a ratio of geometric means under that definition. It is not generally the ratio of arithmetic mean expression values.

Observed single-cell outcomes also depend on measurement. Capture and classification can affect observed cell-type proportions; library normalization and filtering affect expression summaries. The estimand should name what the assay measures, and its interpretation should not silently replace an observed assay quantity with a biological quantity the assay does not identify.

Diagnostics, Alternatives, and Common Mistakes

A well-defined estimand makes design and analysis problems easier to recognize. Before fitting a model, check whether the condition labels and outcome summary match the scientific question; whether the target-population weights are stated when subgroup aggregation matters; and whether the observed data include relevant donors under the conditions being contrasted. Large differences between donor-weighted and cell-weighted summaries are a warning that the two analyses answer different questions, not simply a technical disagreement. Very sparse subgroups, absent condition observations in a subgroup, or an outcome definition that changes across conditions can also prevent the intended contrast from being supported.

Common alternatives answer distinct questions. A donor-average contrast gives equal weight to donors; a cell-weighted contrast gives greater weight to donors with more measured cells. A marginal contrast averages over a specified target population; a conditional contrast reports comparisons within specified subgroups. A difference gives an absolute change, while a ratio or log-ratio gives a relative change. Choosing among them is a scientific decision about the target, not a way to select whichever result is most favorable.

A frequent mistake is to write, “The study estimates the effect of condition B,” before specifying the population, outcome, and scale. The correction is to state the complete contrast—for example, the difference in target-population mean donor-level proportion between B and A—and then separately justify any causal language. Another mistake is to treat a large number of cells as if it automatically defines a population of cells as the target. If the intended question concerns the average donor, the donor is the unit being averaged; cell count does not change that estimand.

No estimand can repair a mismatch between the target and the data. A carefully computed estimate may still be imprecise, biased for the target, or based on an assay quantity that only partially represents the biological quantity of interest. Define the target transparently, report the operational outcome and contrast scale, and state the limits of interpretation.