What Does “No Change in Composition” Mean?
A cell-type composition is a vector of proportions whose components correspond to cell types and sum to one. For a condition \(g\), write its population composition as \(\mathbf{p}^{(g)}=(p_{g1},\ldots,p_{gK})\), where \(p_{gj}\) is the proportion in cell type \(j\), \(p_{gj}\geq 0\), and \(\sum_{j=1}^{K}p_{gj}=1\). The meaning of a comparison depends on which parts of these vectors the null hypothesis equates.
An omnibus test asks whether the composition as a whole differs across conditions. A component-wise test asks whether one specified cell type has the same proportion across conditions. A pairwise test may compare two conditions for the full composition, or compare a specified pair of cell types. These are distinct questions; the word “pairwise” should always be accompanied by a statement of what is being paired.
These hypotheses concern population compositions, not merely the observed proportions in a particular data set. In practice, a test uses estimates and a sampling model to evaluate evidence about the corresponding population null. Its validity depends on an appropriate estimand, comparable cell-type definitions and denominators, and a model that accounts for the study design. The hypotheses can be defined before choosing that model.
The Global Null and Its Component Restrictions
The global null is a single scientific claim about a vector: every cell-type proportion is unchanged across the conditions being compared. Rejecting it supports evidence that at least one component differs, but does not identify which component differs or establish the direction of change. A global test is therefore an omnibus test, not a collection of automatically interpreted component results.
The component-wise nulls specify where a difference might occur. For two conditions, the null for cell type \(j\) is \(H_0:p_{1j}=p_{2j}\); its alternative is \(H_a:p_{1j}\ne p_{2j}\) for a two-sided question. For several conditions, equality across all conditions is one component-wise null, while an individual pairwise comparison tests equality for a selected pair. Neither one component test nor one pairwise comparison answers the omnibus question.
This identity explains an important constraint. If one component proportion rises between two conditions, at least one other component must fall, unless the proportions are unchanged. The component differences cannot vary independently: they sum to zero. Consequently, with \(K\) cell types, only \(K-1\) component differences are free to vary for a two-condition comparison.
More generally, each composition has \(K-1\) free dimensions because its components sum to one. Comparing \(G\) unrestricted compositions with a common composition imposes \((G-1)(K-1)\) restrictions: the unrestricted collection has \(G(K-1)\) free dimensions, while the common composition has \(K-1\). Under a suitable regular model, this count informs the degrees of freedom of an omnibus test. The exact test and its calibration depend on the sampling model and design; the restriction count alone does not provide a valid test statistic.
Worked Example: A Global Difference With One Unchanged Component
Suppose hypothetical estimated compositions for two conditions are \((0.50,0.30,0.20)\) and \((0.50,0.25,0.25)\), in the order type A, type B, type C. What do the component contrasts show, and which null is contradicted by these estimates?
For each type, subtract the condition 2 proportion from the condition 1 proportion. The contrasts are \(0.50-0.50=0\) for A, \(0.30-0.25=0.05\) for B, and \(0.20-0.25=-0.05\) for C. Their sum is \(0+0.05-0.05=0\), as required for compositions.
The estimated proportions for A are equal, while those for B and C differ. Therefore, the estimated vectors are not equal, so these estimates do not satisfy the global equality restriction. The example illustrates why “type A did not change” is not the same claim as “the composition did not change.”
These are descriptive estimates, not a significance test. Their differences alone do not quantify sampling uncertainty or establish that the population composition changed. A test of the global null would require an appropriate sampling model and study design.
Pairwise Comparisons: Specify Both Sides
A pairwise condition comparison can address either the full composition or one component. For conditions \(g\) and \(h\), the full-composition null is \(H_0:\mathbf{p}^{(g)}=\mathbf{p}^{(h)}\); rejecting it supports a difference somewhere in the composition for that pair. The component-specific null \(H_0:p_{gj}=p_{hj}\) instead concerns one named cell type. A component-specific result does not establish that every other component differs.
A different pairwise question compares two cell types within each condition and asks whether their relative proportions are stable across conditions. For types \(j\) and \(k\), when the relevant proportions are positive, one such null is \(H_0:p_{gj}/p_{gk}=p_{hj}/p_{hk}\). This asks whether the relative representation of the two types is the same across the selected conditions. It is not the same as asking whether either type’s proportion is individually unchanged. For example, both proportions may change while their ratio remains constant. If a proportion is zero, this ratio is undefined, so the hypothesis needs a different formulation.
Worked Example: Three Conditions and a Component-Wise Question
Consider hypothetical estimated compositions for conditions A, B, and C, with cell types X, Y, and Z: A has \((0.40,0.35,0.25)\), B has \((0.30,0.45,0.25)\), and C has \((0.40,0.30,0.30)\). Calculate the type Z contrasts and distinguish their scope from the global question.
For type Z, the A-versus-B contrast is \(0.25-0.25=0\). The A-versus-C contrast is \(0.25-0.30=-0.05\), and the B-versus-C contrast is also \(0.25-0.30=-0.05\). Thus the estimated Z proportion is the same in A and B, and is 0.05 lower in each of those conditions than in C.
The global null across all three conditions requires all three vectors to be equal. They are not: for example, the estimated X proportion is 0.40 in A and 0.30 in B. The component-wise null for Z across all three conditions also fails descriptively because 0.25 differs from 0.30. But a pairwise component null for Z in A versus B is consistent with the estimates. None of these descriptive comparisons supplies a p-value or establishes a population difference.
A test of the global null would address whether any composition differs among the conditions. Follow-up comparisons could ask which condition pairs differ overall or which cell types differ for selected pairs. Those follow-up claims require tests designed for those hypotheses and attention to the number of comparisons made.
Worked Example: A Pairwise Cell-Type Relative Comparison
In a hypothetical two-condition comparison, estimated proportions for types X and Y are 0.40 and 0.20 in condition A, and 0.30 and 0.30 in condition B. Calculate the relative proportions and interpret the pairwise type question.
In A, the estimated X-to-Y ratio is \(0.40/0.20=2\). In B, it is \(0.30/0.30=1\). The ratios differ, so these estimates do not satisfy the pairwise relative-proportion equality \(p_{AX}/p_{AY}=p_{BX}/p_{BY}\).
This comparison supports a different descriptive statement from either component contrast: X changes by \(0.30-0.40=-0.10\), and Y changes by \(0.30-0.20=0.10\), while the X-to-Y ratio changes from 2 to 1. Again, these calculations do not by themselves test a population null. The ratios are defined here because both estimated proportions are positive.
From Hypotheses to Scientific Claims
The appropriate test follows from the intended claim. If the question is whether the overall distribution of cell types differs, specify a global null. If the question names a particular type, specify its component-wise null. If the question concerns a particular pair of conditions, state whether the target is the full composition or one component. If the question compares two types relative to each other, define the ratio or other contrast being tested.
A significant omnibus result does not name the responsible cell type. Conversely, a non-significant omnibus result does not prove exact equality; it indicates that the test did not provide sufficient evidence against its global null at the chosen significance level. A component-wise test can be significant even when a particular global test is not, but interpreting such a result requires care about the analysis plan, calibration, and multiplicity rather than treating the tests as interchangeable.
When many component or pairwise tests are performed, the chance of at least one false positive across the family can exceed the nominal significance level for an individual test. A multiplicity procedure should match the planned family of claims. For instance, a family might consist of all cell-type contrasts between two conditions, or all condition pairs for one cell type. Define that family before interpreting adjusted results. The choice of family affects which scientific claims the reported error control covers.
| Question | Null hypothesis | Claim supported by rejection |
|---|---|---|
| Is the composition unchanged across conditions? | All condition-specific composition vectors are equal | At least one component differs somewhere among the conditions |
| Is cell type \(j\) unchanged across conditions? | The proportion for type \(j\) is equal across conditions | The proportion for that specified type differs across at least two conditions |
| Does the full composition differ for conditions \(g\) and \(h\)? | The two composition vectors are equal | At least one component differs for that condition pair |
| Is the relative representation of types \(j\) and \(k\) stable? | Their proportion ratio is equal across the selected conditions | The relative proportion of the selected types differs across those conditions |
Diagnostics and Common Mistakes
Before interpreting a result, check that the cell types and the denominator define the same composition in every condition. If categories are defined differently or the denominator changes, equality of reported proportions may not represent the intended scientific comparison. Also check that the fitted test matches the experimental design and that its uncertainty estimates are credible; the number of observed cells alone does not determine the number of independent biological units.
- Mistake: treating the global null as a separate null for every type. The global null is equality of whole vectors. Correct this by writing the vector equality explicitly, then identify any component hypotheses separately.
- Mistake: interpreting a rejected global test as proof that a named type changed. Rejection only indicates that at least one component differs somewhere. Identify a type with an appropriate follow-up analysis.
- Mistake: calling an unchanged observed component proof of no population change. Equal estimates do not establish population equality. Use uncertainty and a test or interval appropriate to the component estimand.
- Mistake: reporting “pairwise difference” without naming the pair. Specify the conditions or cell types involved and the quantity compared.
- Mistake: treating all component tests as independent. Their differences are constrained by the sum-to-one condition. Use inference and multiplicity procedures suited to the joint design rather than assuming independent outcomes.
A useful reporting practice is to give the null, the comparison, the estimand, and the scope of the conclusion together. “The omnibus test found evidence against equal compositions across the three conditions” is appropriately scoped. “Cell type X changed” requires evidence from a type-specific analysis, and “X increased relative to Y” requires a relative comparison of those types.
Check Your Understanding
For each question, identify the relevant null and interpret what rejection would support.
- Two conditions have three cell types. The scientific question is whether the full composition differs. Which null applies?
- A researcher tests whether the proportion of cell type C is equal across four conditions. Is this a global or component-wise question?
- What does rejection of a global null across three conditions establish, and what does it not identify?
- In two conditions, the proportions for types A and B are positive. What null expresses that their relative representation is unchanged?
- Why does an unchanged estimated proportion for one type not prove that its population proportion is unchanged?
Solutions: (1) The global null is equality of the two composition vectors; rejection supports a difference in at least one component. (2) It is the component-wise null \(p_{1C}=p_{2C}=p_{3C}=p_{4C}\). (3) Rejection establishes evidence that at least one component differs in at least one condition comparison; it does not identify the component or direction. (4) The null is equality of the type-A-to-type-B proportion ratio across the two conditions. (5) An estimate is subject to sampling uncertainty, so equal observed values do not establish equality of the population proportions.