When Total-Count Scaling Creates Compositional Bias

Why a Larger Total Can Make an Unchanged Gene Look Smaller

Total-count scaling is useful for adjusting for differences in the number of counts recorded per cell. But its denominator is not an external measure of sequencing effort: it is the sum of the observed gene counts in that cell. A few highly expressed genes, or a broad change in the relative abundance of many genes, can therefore change the denominator enough to distort comparisons of other genes.

The key point is that counts per million (CPM) report a gene’s share of a cell’s total counts, multiplied by one million. They do not, by themselves, report the number of RNA molecules per cell. The distinction matters whenever the composition of the RNA pool differs across cells or conditions.

This scaling corrects a simple exposure difference under a useful model. Suppose counts are independent Poisson variables with expected counts \(s_c a_{gc}\), where \(s_c\) is a cell-specific capture or sequencing multiplier and \(a_{gc}\) represents the amount of gene \(g\) in the cell’s RNA composition. The expected total is \(s_c\sum_h a_{hc}\), and, conditional on the total, the counts follow a multinomial distribution with probabilities \(a_{gc}/\sum_h a_{hc}\). The multiplier \(s_c\) cancels from these probabilities.

That cancellation is helpful if differences in total counts mainly reflect differences in \(s_c\), while the composition \(a_{gc}/\sum_h a_{hc}\) is comparable. It does not mean that total-count scaling removes all technical and biological differences. In particular, the normalized value still depends on the relative abundance of every gene in the denominator.

The Denominator Determines the Comparison

For two cells, \(c=1\) and \(c=2\), the ratio of the CPM values for a gene is

when both gene counts are nonzero. A change in the total \(L_c\) therefore affects the normalized comparison even if the gene’s observed count does not change. In a count model, the same issue appears through the expected proportion: if \(a_{gc}\) stays fixed while \(\sum_h a_{hc}\) rises, the gene’s expected share falls.

Broad Composition Shifts Have the Same Consequence

Compositional bias need not be caused by one exceptional gene. If a large group of genes becomes more abundant relative to the rest, genes outside that group can lose a share of the total even when their own absolute abundance is unchanged. Total-count scaling cannot tell whether that share changed because one gene rose, many genes rose, or the rest of the RNA pool fell.

What the Scaling Assumption Buys—and What It Does Not

The Poisson model provides a concise justification for the relative interpretation. If \(Y_{gc}\) are independent Poisson counts with means \(s_ca_{gc}\), then conditional on \(L_c=\sum_gY_{gc}\), the vector of counts has a multinomial distribution with total \(L_c\) and probabilities \(p_{gc}=a_{gc}/\sum_h a_{hc}\). Therefore,

for \(L_c>0\). The exposure multiplier \(s_c\) disappears, but so does any information about a common multiplier on all the \(a_{gc}\) values: multiplying every gene amount in a cell by the same positive constant leaves every \(p_{gc}\) unchanged. This is why CPM can adjust for a shared sampling-depth multiplier while being unable to identify absolute total RNA from the composition alone.

The model is a justification for understanding the denominator, not a guarantee that real single-cell counts are independent Poisson observations. Capture variation, overdispersion, and non-detection can complicate count behavior. Even if the count model is more elaborate, however, the arithmetic of CPM remains the same: each value is a fraction of the observed total.

Diagnostics for Potential Distortion

No single diagnostic proves that total-count scaling is biased. Instead, inspect whether the patterns needed for a simple exposure interpretation are plausible and whether conclusions depend heavily on the denominator.

• Compare library totals. Summarize \(L_c\) across cells and groups. Large differences may reflect depth, composition, or both; the totals alone do not identify which explanation is correct.
• Measure dominant-gene contributions. For each cell, calculate the largest gene’s fraction \(Y_{gc}/L_c\), or the fraction contributed by a selected set of highly expressed genes. A large or condition-dependent fraction signals that a small part of the transcriptome may strongly influence the denominator.
• Compare a gene’s raw counts and scaled values. A stable count paired with a substantial CPM change is a direct indication that the denominator is contributing to the normalized change. Raw counts are not a complete solution because they can retain differences in sequencing depth.
• Check sensitivity to dominant genes. Recalculate diagnostic fractions or gene-wise comparisons after excluding a suspected dominant gene from the denominator, for diagnostic purposes only. If conclusions change sharply, they depend on that gene’s contribution. Such an exclusion is not automatically a valid normalization method; it changes the reference composition and needs justification.
• Look across genes and cells. If many genes shift in the same direction after scaling, consider whether a broad compositional change could explain the pattern. Do not infer a global absolute RNA increase or decrease from normalized values alone.

These checks are most informative when cells are compared within appropriate biological groups and when technical factors are considered alongside biological context. A difference in total counts could be due to sequencing depth, capture efficiency, composition, or a combination. A diagnostic flags a concern; it does not assign the cause by itself.

Alternatives and Their Trade-Offs

The appropriate response depends on the scientific target. Total-count scaling and CPM are simple, transparent summaries of relative composition. They can be useful when the aim is to compare relative abundance and when no few genes dominate the total. They become difficult to interpret when a small number of genes or broad shifts strongly alter the composition.

Alternative endogenous size-factor methods use different summaries of the count matrix rather than simply dividing each cell by its total. They can reduce sensitivity to some dominant genes when their assumptions are reasonable, but they do not automatically recover absolute RNA abundance or solve every global shift. Choosing and diagnosing those methods requires separate consideration.

Common Mistake and Practical Interpretation

Incorrect reasoning: “Gene B has half as many CPM in condition 2, so its expression per cell must have halved.” In the worked example, B had 100 counts in both cells; the CPM difference arose because gene A increased the total from 1,000 to 1,900.

Correction: CPM is a relative quantity. State that the gene’s share of the observed total decreased, then examine the raw counts, the total and its dominant contributors, and the experimental information available about capture or absolute RNA. Do not equate a compositional change with an absolute per-cell change without an appropriate reference.

A second mistake is to assume that equal totals guarantee comparable composition. Two cells can have identical total counts while the same gene occupies very different fractions of those totals. Conversely, different totals do not by themselves prove a compositional problem: they can arise from different sequencing exposures. The interpretation must consider both the denominator and the measurement process.