Deriving Cell-Level Factors from Pooled Ratios

From Pool-Level Ratios to Cell-Level Factors

Sparse single-cell counts make it difficult to estimate a reliable normalization factor for every cell directly. Pooling helps by combining counts from selected groups of cells before estimating relative factors. The key mathematical step is then to recover the individual cell factors from those pooled estimates. This is a deconvolution problem: each pool provides an equation involving the factors of the cells it contains.

The earlier material on constructing pool sets establishes which cells are combined in each pool. Here, represent that design explicitly and derive the equations it implies. The equations are simple, but their solution depends on important conditions: pooled factors must represent sums on a common scale, and the pool design must contain enough independent information to distinguish the cells.

The factor \(s_i\) describes the relative sequencing exposure or count scale assigned to cell \(i\), under the normalization target being used. It is not, by itself, a measure of the cell’s biological size or total RNA content. As with other endogenous size factors, its interpretation relies on the assumptions that make the target identifiable from the observed genes.

Why Pool Factors Add

Suppose \(Y_{ig}\) is the count for gene \(g\) in cell \(i\). A simple working mean model is

where \(\mu_g\) is the gene’s expression level on the chosen reference scale. For pool \(k\), the count for gene \(g\) is the sum of counts from its member cells:

By linearity of expectation,

Thus, under this model, the pool acts as though its exposure factor were the sum of its member-cell factors. When a ratio-based normalization procedure estimates relative pool factors across genes, the common gene-level reference is intended to cancel, leaving an estimate of that sum, up to a common multiplicative scale. The resulting equations are not exact identities for noisy estimated ratios; they are a working linear system.

Solving the Equations

If the pool estimates are treated as exact for the moment, deconvolution means solving \(As=p\). For a square, nonsingular \(A\), ordinary linear algebra gives a unique solution. More commonly, there are more pools than cells, and noise or estimation error means the equations cannot all hold exactly. Then the ordinary least-squares estimate minimizes the sum of squared discrepancies between observed pooled factors and the sums predicted by the cell factors:

When \(A\) has full column rank, the least-squares solution satisfies the normal equations \(A^\mathsf{T}A\hat{s}=A^\mathsf{T}p\), giving

The full-rank condition matters because otherwise \(A^\mathsf{T}A\) is singular: distinct cell-factor vectors can produce identical pool sums. No calculation can recover information the design did not record. A common multiplier is also not identified by ratio-based normalization; multiplying every cell factor and every pooled factor by the same positive constant leaves the equations intact. The solution is therefore relative to the scale chosen for the pooled estimates.

Least Squares When Pool Estimates Are Noisy

Estimated pooled factors will generally not obey every additive equation exactly. The least-squares solution finds the cell-factor vector whose predicted pool sums are collectively closest in squared Euclidean distance. This is a fitting criterion, not a claim that the discrepancies vanish or that all pool estimates have equal precision. The following calculation shows why residuals must be interpreted using the design rather than by assuming their sum is zero.

What Makes the Solution Trustworthy?

The algebra provides a cell-level solution only when the model and the pool design support it. The rank condition is a precise design check: if \(A\) lacks full column rank, the cell factors are not uniquely determined from these pools. If it has full column rank, the unconstrained least-squares solution is unique, but that does not by itself establish that the underlying normalization assumptions are appropriate.

In practice, useful checks include comparing observed pooled estimates with fitted sums \(A\hat{s}\), inspecting residuals by pool, and confirming that the pool membership and pooled-count construction agree. Large or structured discrepancies can signal noisy pool estimates, a poorly matched factor model, or a problem in the design or data handling. A negative unconstrained estimate is also a warning: size factors are expected to be positive, so such a result suggests that the estimates and design do not support a plausible additive solution without further modeling choices.

Least squares is a natural baseline when pool estimates have roughly comparable precision. If their uncertainty differs substantially, a weighted least-squares criterion can give more influence to better-estimated pools. Direct cell-level estimation avoids solving a pooled system, but can be unreliable when counts are sparse; pooling gains information by aggregating counts, at the cost of relying on the pool-additivity model and a sufficiently informative design. Neither approach removes the need for an identifiable normalization target.

The derivation also has limits. A common scalar factor across genes is an approximation, and biological differences in expression composition can interfere with ratio-based estimation. A pool factor is not automatically the sum of cell factors if each pool has been independently rescaled or estimated against an incompatible reference. Finally, a unique numerical solution is not evidence that the assumptions hold: it only means the specified equations have one solution.