Marker Genes As Descriptive And Inferential Claims

Two Different Questions About Marker Genes

A marker-gene table can answer a useful descriptive question: which genes differ most across the partition of cells we observed? It does not automatically answer the inferential question: which genes differ in a target population, with uncertainty calibrated to the process that produced the groups? These questions are related, but their evidence requirements are not the same.

The distinction matters in single-cell analysis because a partition is often discovered from expression measurements. A gene may rank highly because its observed values differ between the groups, because it helped determine which cells were grouped together, or both. A ranking can still be informative for describing an exploratory result. The difficulty arises when that ranking is treated as if it were an independently planned test of a gene-level population difference.

For example, a mean difference between cluster \(k\) and the other observed cells can be written as

where \(n_k\) and \(n_{-k}\) are the numbers of observed cells in the cluster and its complement. Once the measurements and assignments are fixed, this is a numerical summary of those cells. It has no sampling uncertainty unless a target population and a sampling model are specified. A ranking based on \(\widehat{\Delta}_{g,k}\), a fold change, or another score therefore describes the observed partition; it is not, by itself, a test of a population-level null hypothesis.

What a Descriptive Ranking Can and Cannot Say

Marker rankings depend on the chosen summary. A mean difference emphasizes expression magnitude on the scale being analyzed. A fraction-expressing summary emphasizes how often values meet a chosen detection criterion. A ranking based on one measure need not match a ranking based on another. In all cases, state the scale, the group contrast, and the ranking criterion so that readers know what the table summarizes.

A descriptive statement can be precise without being inferential. For instance: “In this observed partition, gene G had the largest mean normalized-expression difference between cluster A and the remaining cells.” That statement concerns the data at hand. By contrast, “gene G is differentially expressed in the population from which these cells were sampled” goes beyond that finite-data summary. It needs a design and analysis that justify the population-level uncertainty being reported.

When a Gene-Level Test Has a Defensible Interpretation

A familiar two-sample test illustrates what inferential claims require. Suppose group labels are fixed independently of the gene values being tested, observations are independent sampling units, and the values in each group are normally distributed with a common variance. Under the null hypothesis of equal group means, the pooled two-sample \(t\)-statistic has a \(t\) distribution with \(n_1+n_2-2\) degrees of freedom. This result also requires that the test and contrast were specified without selecting them because they showed a large difference in these same observations.

In single-cell data, independence must be considered at the level of sampling. Cells from one donor are generally not independent biological replicates. If the target claim concerns differences across donors, the analysis should use a defensible donor-level design, rather than treating many cells from a few donors as if they were independent donors. A valid test also does not by itself address testing many genes; a broad set of gene-level claims requires an appropriate approach to multiplicity.

Why Cluster Discovery Changes the Claim

If the same expression data are used both to form a partition and to rank genes that separate that partition, the assignments are data-adaptive: the observed values helped determine the groups. Ordinary tests that treat those groups as fixed in advance do not account for that selection. As a result, their usual p-values or confidence intervals may not have their advertised calibration for the post-discovery claim.

This does not mean that the descriptive contrast is false or that a discovered cluster is useless. It means that the statistic and the inferential interpretation must not be conflated. The contrast remains a correct summary of the assigned cells. What is not justified by that calculation alone is a claim that the gene would show the same evidence under repeated sampling and the same discovery process. The appropriate inferential strategy depends on the target claim and on how clustering and marker selection were performed.

Justifying the Distinction

The distinction follows from what is held fixed. For a descriptive summary, fix the observed expression values and assignments. The group means, their difference, and the resulting rank are then determined calculations. There is no random variable left in that finite-data calculation.

Inference instead considers what could vary under repeated sampling. For a prespecified contrast with independent observations, the sampling distribution of the test statistic under \(H_0\) supplies the reference for a p-value. The calculation in the preceding donor example uses the pooled two-sample model, so its calibration is conditional on the groups having been specified independently of the tested gene. If the tested gene helped determine the labels or was selected because it ranked highly, the observed statistic has also passed through a selection process. A reference distribution that ignores that process need not give the intended error rate. This is why a descriptive marker table and an inferential result should be labeled and interpreted separately.

Diagnostics, Alternatives, and Reporting

Before making an inferential claim from marker analysis, check the link between the data and the proposed target. Ask whether group labels were externally defined or discovered from the tested expression values; whether the independent units are cells, donors, or another sampling unit; whether the assumed model is plausible; and whether the analysis involves many gene tests. Large cell counts do not substitute for independent donors. Strongly skewed values, outliers, unequal variances, or donor-level dependence can also make a simple pooled \(t\)-test inappropriate.

The main alternatives serve different purposes. A descriptive ranking is appropriate for summarizing and exploring an observed partition, and should be reported with its effect measure and scale. A conventional inferential test can address a prespecified comparison when its sampling and model assumptions are defensible. When labels or genes were selected from the same data, an inferential analysis must additionally account for the discovery process; independent confirmation or a design that separates discovery from testing are examples of broad strategies, not automatic guarantees. The specific method must match the claim and its assumptions.

A useful report makes the status of the result explicit:

• For a descriptive result, name the observed groups, expression scale, summary statistic, and observed effect size.
• For an inferential result, identify the target population, independent sampling unit, model, contrast, uncertainty measure, and multiplicity handling when many genes are assessed.
• If the same data helped discover the groups or select the gene, state that fact and do not present an unadjusted fixed-group p-value as if the discovery step had not occurred.

The limitation is substantive: a strong observed marker contrast can be valuable for describing a dataset, but its magnitude alone does not determine reproducibility, generalizability, or causal relevance. Conversely, failure to justify a particular p-value does not erase the observed contrast. Keeping those conclusions separate makes marker reports both informative and appropriately cautious.