Why Cell-Type Proportions Are Compositional

Why proportions must be read as parts of a whole

A cell-type proportion is not an isolated measurement. It describes how a total number of cells is divided among categories. If the categories account for every cell exactly once, raising one category’s share necessarily changes the shares available to the others. This is a mathematical constraint, not evidence that cell types biologically compete or that one type caused another to change.

That distinction matters in differential-abundance analysis. A proportion can change because the number of cells of that type changed, because the total number of cells used as its denominator changed, or because both changed. Reading each percentage as if it varied independently can therefore lead to misleading descriptions of what changed and misleading combined evidence across cell types.

The definition relies on the categories being mutually exclusive and exhaustive for the total being used: each included cell belongs to one category, and every cell in that total belongs to one of the categories. The denominator must also be defined consistently for the comparison at hand. If categories overlap, or if some cells are omitted from the denominator without being accounted for, the listed proportions need not sum to one and do not describe the stated composition.

Deriving the closure constraint

The sum-to-one constraint follows directly from the definition. Each cell count is divided by the same total, so summing the proportions recovers the total count divided by itself:

This is called closure: the components are constrained to sum to a fixed total. If proportions are reported as percentages, the corresponding total is 100%, rather than 1. Closure also means there are only \(K-1\) freely varying components: once \(K-1\) proportions are specified, the last must equal one minus their sum.

The same constraint applies to changes. If the proportions for a sample change from \(p_{ij}\) to \(p'_{ij}\), then \(\sum_j(p'_{ij}-p_{ij})=0\). Thus, an increase in one or more components must be balanced by a decrease in at least one other component, unless all components are unchanged. This statement concerns relative shares; it does not say that any cell type’s absolute count must fall.

How a change in counts moves every proportion

The effect of changing one count can be calculated explicitly. Write the original total as \(N\) and the original proportion of type \(j\) as \(p_j=y_j/N\). If \(\delta\) cells of type \(j\) are added and all other counts are held fixed, the new total is \(N+\delta\). For any other type \(k\), its count stays \(y_k\), but its proportion changes from \(y_k/N\) to \(y_k/(N+\delta)\):

For \(\delta>0\), this difference is negative whenever \(y_k>0\). The changed type’s own proportion increases by \[ \frac{y_j+\delta}{N+\delta}-\frac{y_j}{N} =\frac{\delta(N-y_j)}{N(N+\delta)}, \] which is nonnegative when there is at least one cell outside type \(j\). The changes across all types still sum to zero. If \(\delta<0\), the direction reverses when the resulting counts remain valid. These calculations describe a change in the composition caused by changing counts; they do not identify why the counts changed.

More generally, if several counts change by small amounts \(dy_j\), and \(dN=\sum_jdy_j\), then the corresponding first-order change in proportion \(p_j\) is \[ dp_j=(dy_j-p_jdN)/N. \] The term \(p_jdN\) captures the effect of the changing denominator. A count can rise while its proportion falls if the total rises sufficiently faster.

Dependence among components

Because proportions in a sample sum to one, their variation is constrained. If one component varies while the others are fixed in number or share, the remaining components must collectively compensate. This creates dependence in the observed proportions, even when there is no biological interaction among the cell types.

The constraint can be expressed as an exact covariance identity. Suppose proportions for \(K\) types are measured on the same \(n\) donors, with each donor’s vector summing to one. For components \(j\) and \(k\), define their sample covariance by \[ s_{jk}=\frac{1}{n-1}\sum_{i=1}^{n}(p_{ij}-\bar p_j)(p_{ik}-\bar p_k). \] For each fixed component \(j\), the covariances with all components sum to zero:

because \(\sum_k p_{ik}=1\) for every donor and \(\sum_k\bar p_k=1\). In particular, the variance of component \(j\), which is \(s_{jj}\), must be balanced by its covariances with the other components. If \(j\) varies across donors, those covariances cannot all be zero. This is a mathematical dependence among the measured shares; its presence alone does not establish any biological mechanism.

Why ordinary independent tests can mislead

A common approach is to test each cell-type proportion separately and then interpret the results as though each test described an unrelated outcome. That interpretation can fail for two reasons. First, the proportions from the same sample share a denominator and a fixed sum; they are not independent measurements. Second, a change in one component’s share can be induced by changes elsewhere, even if that component’s own count is unchanged.

Consequently, a set of separate results may obscure the joint change in the composition. One type’s share can increase while several others decrease, and those decreases do not necessarily represent losses of cells. Treating the component-wise results as statistically independent can also make combined statements about evidence or error rates incorrect: the tests’ outcomes are dependent, so an independence-based calculation for their joint behavior need not apply.

This does not mean every test of one proportion is automatically invalid. A well-specified marginal analysis can address a clearly defined question about one cell type’s proportion, with its sampling unit and uncertainty handled appropriately. The caution is against treating all components as unrelated outcomes, or interpreting a component’s percentage change as an isolated absolute change. The choice of estimand, denominator, and inferential model determines what a test can support.

Diagnostics, alternatives, and limits

Before interpreting cell-type proportions, check the structure that makes them compositional:

• Check the sum. Confirm that the included proportions sum to one, allowing for rounding. A substantial discrepancy may signal omitted categories, overlapping labels, inconsistent denominators, or a data-processing error.
• Name the denominator. State which cells are included in the total. Proportions among all captured cells answer a different question from proportions among a selected subset.
• Report the scale. When relevant, distinguish cell counts from proportions and give the denominator for each proportion. A higher count and a lower share can occur together.
• Keep the observational unit clear. Proportions calculated for different donors are observations across donors; the cell types within one donor’s composition are not independent replicates.
• Look for impossible interpretations. A claim that every component’s proportion increased in the same sample contradicts closure. A claim that a falling proportion proves a falling count does not follow from the proportion alone.

There are several useful ways to describe a change, depending on the scientific question. Absolute counts address how many cells of a type were observed, although they do not by themselves resolve sampling and study-design issues. Component-wise proportions describe each type’s share of a stated total. Joint composition-aware analyses are designed to respect the fact that components belong together. A one-type-versus-all-others contrast can be useful for a specific marginal question, but it still compares that type with a denominator made from the remainder; it does not make the full set of components independent.

No representation is universally sufficient. Proportions discard information about the total number of cells, while counts depend on how cells were collected and counted. A composition also depends on the categories chosen: merging, splitting, or omitting cell types changes the represented parts and can change the interpretation. Closure is therefore a necessary feature of proportion data, not a complete model of biological abundance or of the process that generated the observations.