Why Cell-Type Resolution Is Part of the Analysis
A differential-abundance result depends partly on what counts as a cell type. A broad label such as “T cells” may be divided into related subtypes, while several fine labels may be combined into a parent population. These choices change the hypotheses being tested: a comparison of total T-cell abundance is not the same question as a comparison of CD4 and CD8 abundance separately.
Sensitivity analysis makes this dependence visible. Repeat the abundance analysis at biologically reasonable merged and split resolutions, then compare the estimated condition effects, their uncertainty and the resulting discoveries. The purpose is not to search for the resolution that produces the smallest p-value. It is to determine which conclusions are stable, which depend on a particular grouping, and which level answers the biological question.
As in earlier material on differential-abundance analysis, the independent replication is at the sample or donor level, not the individual-cell level. A resolution comparison should therefore use the same donors, covariates, model specification and inferential procedure wherever possible. Otherwise, a difference in results could reflect a changed analysis rather than the changed cell labels.
Define the Aggregation and the Contrast
The aggregation identity requires a consistent hierarchy: each cell contributing to the parent must belong to exactly one of its child labels, and the child labels must cover the parent population. A mapping that overlaps labels, omits cells, or changes between samples will not yield a valid sum. The denominator also matters. The proportions above use the total number of cells in each sample; an analysis with a different denominator must define and preserve it across resolutions.
For a simple two-condition comparison, let \(\bar p_{1k}\) and \(\bar p_{0k}\) denote the mean sample proportions in the two conditions. The estimated difference for label \(k\), measured in proportion units, is \(\widehat\Delta_k=\bar p_{1k}-\bar p_{0k}\). Multiplying by 100 expresses it in percentage points. More complex donor-level models can adjust for design features, but the same label aggregation should be applied to the sample-level data before refitting the model.
Why Merging Can Change Uncertainty
For sample-level proportions, and for contrasts formed by applying the same linear operation to them, the parent estimate and condition difference are sums of the corresponding child quantities. Separately refitted model contrasts need not be additive; this identity requires an estimator that preserves additivity, and otherwise the parent should be formed from aggregated sample-level data and modeled directly without assuming its estimated contrast equals the sum of child-model contrasts. The variance, however, is not generally the sum of the child variances alone. It also depends on covariance between child proportions.
This follows from the variance-of-a-sum identity. For two child labels \(A\) and \(B\), expand the variance of their sum:
Multiplying out the square gives the two individual variances and twice the covariance, which yields the stated formula. Within a sample, proportions of mutually exclusive cell types often compete for a fixed total and may be negatively associated. Consequently, merging can reduce uncertainty in some settings, but it is not guaranteed to do so. Likewise, child effects in opposite directions can cancel in the parent estimate. A parent-level result can therefore conceal meaningful subtype changes, while a subtype-level result can be noisier because fewer cells or less donor-level signal support each estimate.
Worked Example: Opposing Child Effects Cancel in the Parent
Suppose four hypothetical control donors and four hypothetical treated donors have the following sample-level proportions for two mutually exclusive subtypes, A and B. Values are percentages of all cells in the sample.
| Condition | Subtype A across donors | Subtype B across donors |
|---|---|---|
| Control | 10%, 11%, 9%, 10% | 7%, 8%, 9%, 8% |
| Treated | 12%, 14%, 13%, 13% | 6%, 4%, 5%, 5% |
The control means are \(\bar p_{0A}=(10+11+9+10)/4=10\%\) and \(\bar p_{0B}=(7+8+9+8)/4=8\%\). The treated means are \(\bar p_{1A}=(12+14+13+13)/4=13\%\) and \(\bar p_{1B}=(6+4+5+5)/4=5\%\). Thus the estimated differences are \(13\%-10\%=+3\) percentage points for A and \(5\%-8\%=-3\) percentage points for B.
For the parent containing both subtypes, the mean is \(10\%+8\%=18\%\) in controls and \(13\%+5\%=18\%\) in treated donors. Its estimated difference is \(18\%-18\%=0\) percentage points. The parent result does not contradict the child estimates: it answers a different question, and the two opposing subtype shifts cancel in the sum. These means describe the hypothetical data; they do not by themselves establish statistical significance.
Compare Discoveries Without Confusing the Testing Families
Splitting a parent creates more hypotheses. If each label is tested, the family of hypotheses changes, and multiplicity adjustment must be applied to the full set defined in advance for that resolution. A p-value is evidence against its specific null hypothesis under the fitted model; it is not an effect size, and a change in its adjusted value can reflect the changed testing family as well as a changed estimate or standard error.
Holm adjustment is one standard family-wise error procedure. Order the \(m\) raw p-values as \(p_{(1)}\le\cdots\le p_{(m)}\). The adjusted value at rank \(j\) is the running maximum of \((m-j+1)p_{(j)}\), capped at 1. The running maximum is essential: adjusted p-values must not decrease as the raw p-values increase.
Worked Example: Holm Adjustment at Fine and Merged Resolutions
Consider hypothetical outputs from the same donor-level model applied to the same samples. At the fine resolution, four labels have raw p-values A: 0.00226, B: 0.00520, C: 0.0410 and D: 0.400. Suppose the estimated condition effects for A and B are in the same direction. The family contains four tests, so the ordered Holm calculations are:
For A, \(4(0.00226)=0.00904\). For B, first calculate \(3(0.00520)=0.01560\), then take the running maximum: \(\max(0.01560,0.00904)=0.01560\). For C, \(2(0.0410)=0.0820\), giving \(\max(0.0820,0.01560)=0.0820\). For D, \(1(0.400)=0.400\), giving \(\max(0.400,0.0820)=0.400\). At \(\alpha=0.05\), A and B are discoveries in this four-test family.
Now merge A and B into their parent, retaining C and D as separate labels. Suppose the parent’s raw p-value is 0.000800, while C and D retain raw p-values of 0.0410 and 0.400. This three-test family gives adjusted values: parent \(3(0.000800)=0.00240\); C, \(\max(2(0.0410),0.00240)=0.0820\); and D, \(\max(1(0.400),0.0820)=0.400\). The parent is a discovery, but C and D are not. The fine analysis identifies two child labels; the merged analysis identifies their parent. These conclusions are not inconsistent, and their adjusted p-values should not be compared as though they came from an identical set of hypotheses.
The p-values in this example are stipulated hypothetical model outputs; the arithmetic demonstrates the adjustment, not the fitting of the abundance model. In a real analysis, report the model, the hypothesis family at each resolution, the raw and adjusted p-values, and the estimated effects. A declaration of discovery should not replace the effect estimate or its uncertainty.
A Practical Sensitivity Analysis
A defensible comparison treats resolution as a planned analysis dimension. First define a small set of plausible levels from the cell ontology and the scientific question. Then create a stable mapping between labels at those levels, aggregate sample-level counts or proportions accordingly, and refit the same chosen donor-level procedure. Preserve the sample set, covariate adjustment, contrast direction and significance threshold. When a model’s assumptions or inputs change because of aggregation, explain that difference rather than attributing every change solely to resolution.
Choose biologically interpretable parent and child groupings before examining which one produces the strongest result.
Verify that child labels are mutually exclusive, nested in the intended parent, and applied consistently across samples.
Use donor-level observations and retain the same model structure, condition contrast and included samples where possible.
Record effects in percentage points, uncertainty measures and raw p-values at each level; adjust for the tests in that level’s declared family.
State the primary resolution and why it matches the biological question, then describe important sensitivity findings at the other plausible levels.
Worked Example: Choosing a Reporting Level From Sensitivity Results
Suppose a hypothetical study compares a broad parent population with two related child labels. The estimated parent difference is \(+4.0\) percentage points. The child estimates are \(+9.0\) percentage points and \(-5.0\) percentage points, which sum to \(+4.0\) percentage points. A fitted analysis gives a parent p-value of 0.030, a child-1 p-value of 0.010 and a child-2 p-value of 0.220.
The arithmetic confirms that the parent effect is the sum of the child effects: \(+9.0+(-5.0)=+4.0\) percentage points. The opposite child estimates indicate that the parent’s positive shift does not describe a uniform change across its subtypes. If the scientific question concerns total parent abundance, the parent is the direct estimand. If the question concerns which subtype changes, the child estimates are directly relevant, but their uncertainty and the multiplicity adjustment for the child-level family must also be reported. The raw p-values alone do not settle the reporting choice.
A defensible report would name the biologically motivated primary level, give the corresponding estimated effects and uncertainty, and state that the parent result masks opposing subtype estimates. If annotation evidence does not support distinguishing the children reliably, the broad estimate may be the more defensible primary result, with the split shown as a sensitivity analysis rather than a firm subtype claim.
Diagnostics, Alternatives and Limitations
Inspect the analysis at each level for reasons that apparent sensitivity may be misleading. Check that parent proportions equal the sums of the corresponding child proportions in every sample. Review per-donor proportions and cell counts: a fine label represented by very few cells in some donors can produce unstable estimates and make model diagnostics less reliable. Look for a result driven by one donor, large changes in the number of usable samples, or a condition-specific failure to assign cells to a label. Recheck model diagnostics after aggregation; acceptable fit at the parent level does not guarantee acceptable fit for each rare child.
The main alternatives are not interchangeable. Reporting only a broad parent can provide a stable, interpretable answer to a broad question, but can hide opposing subtype shifts. Reporting only fine labels can localize heterogeneous changes, but increases the number of hypotheses and often reduces precision. A hierarchical report can name a prespecified parent as primary and provide child-level estimates as secondary detail, with the testing strategy stated clearly. Whichever approach is used, avoid selecting a resolution after inspecting p-values and then presenting it as if it had been the sole planned analysis.
Sensitivity analysis cannot prove that an ontology is biologically correct, eliminate annotation errors, or establish absolute changes in cell numbers. It also cannot make results from different hypotheses identical: merging changes the estimand, and splitting changes both the estimands and the multiplicity burden. The aim is transparent interpretation, not a resolution at which every label becomes significant.
Common Mistakes
- Calling a merged and split result contradictory. A parent sum and its child components test different hypotheses; opposing child effects can cancel in the parent.
- Comparing adjusted p-values across different families as if they were directly comparable. State the number and identity of tests at each resolution and adjust within the declared family.
- Using cells as independent replicates. Abundance inference must respect sample- or donor-level replication; a large cell count does not create more independent donors.
- Choosing the resolution because it yields significance. Specify biologically defensible levels and a primary reporting level independently of the observed p-values.
- Assuming merging always improves precision. The variance of a sum includes covariance terms, so its uncertainty depends on the joint behavior of the child proportions.
- Reporting significance without magnitude. Give the estimated change in context and units, alongside uncertainty and the relevant adjusted result.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Two mutually exclusive child labels have estimated condition differences of \(+6\) and \(-2\) percentage points. What is the parent difference, and what does it conceal?
- Why is it not valid to treat a split-label analysis and a merged-label analysis as testing exactly the same hypothesis?
- For three ordered p-values \(0.01, 0.03, 0.20\), calculate the Holm-adjusted values.
- Name two checks that help identify whether a fine-resolution result is unstable.
- What should be reported when the primary conclusion is stable at the parent level but child-level estimates point in opposite directions?
Solutions. (1) The parent difference is \(+6+(-2)=+4\) percentage points. The positive parent shift conceals an increase in one child and a decrease in the other. (2) The parent tests the aggregate abundance, while the split analysis tests child-specific abundances; the estimands and, usually, the hypothesis family differ. (3) With \(m=3\), the first adjusted value is \(3(0.01)=0.03\); the second is \(\max(2(0.03),0.03)=0.06\); the third is \(\max(1(0.20),0.06)=0.20\). (4) Review per-donor cell counts and proportions, check whether one donor drives the estimate, and examine fit or missing-label patterns. (5) Report the parent effect and its uncertainty, the child estimates and their uncertainty with the appropriate multiplicity treatment, and explain that the parent-level conclusion masks heterogeneous child changes.