Why Sparse Counts Make Cell-Level Ratios Unstable
Relative scaling compares the observed counts of genes across cells to estimate differences in sequencing exposure. The difficulty is that, in a sparse single-cell count matrix, a gene used as a reference may have a zero count in many cells. A cell-level ratio with that gene in the denominator is then undefined, and when the denominator is merely small, the ratio can be extremely variable.
Pooling addresses this problem by adding counts across multiple cells before forming a ratio. The pooled count contains more information about the average count pattern than any single sparse observation. This can make a denominator less likely to be zero and reduce sampling variability. Pooling does not, by itself, remove biological differences between cells or guarantee that the chosen genes provide a valid reference for relative scaling.
A useful starting model is that counts for gene \(g\) in cell \(i\) have mean \(\mathbb{E}(Y_{gi})=s_iq_g\). Here \(s_i\) is a cell-specific exposure, such as sequencing depth, and \(q_g\) is the relative abundance of gene \(g\) under the normalization target. This simple model assumes that the gene’s relative abundance is shared across the cells being pooled. For the calculations below, assume counts are independent Poisson variables conditional on their means. The Poisson assumption is a convenient baseline, not a claim that every single-cell dataset has Poisson variation.
For a reference gene \(r\), the expected counts in one cell are \(s_iq_g\) and \(s_iq_r\). Their ratio is \(q_g/q_r\). This is a ratio of expected counts, not necessarily the expected value of the random ratio \(Y_{gi}/Y_{ri}\). If \(Y_{ri}=0\), the random ratio is undefined. Even when it is defined, a ratio with a small random denominator can be unstable.
Worked Example: A Cell-Level Ratio With a Zero Denominator
Suppose, hypothetically, that counts for a target gene \(G\) and reference gene \(J\) in one cell are independent, with \(G\sim\operatorname{Poisson}(2)\) and \(J\sim\operatorname{Poisson}(1)\). The ratio of the expected counts is \(2/1=2\). This is not the expected value of the random ratio \(G/J\), which is undefined when \(J=0\).
For a Poisson random variable with mean \(\lambda\), the probability of zero is \(e^{-\lambda}\). Thus,
In this model, the cell-level ratio is undefined in about 36.8% of cells. The calculation does not imply that the ratio equals 2 in the remaining cells; those ratios also vary because both counts are random. It shows why substituting a ratio of means for an observed random ratio would be misleading.
If instead we sum counts for these genes across five independent cells with the same means, the pooled reference count has a Poisson distribution with mean \(5\). Its probability of being zero is
Pooling makes a zero denominator much less likely in this example. The pooled ratio can still be variable, and it is still undefined if the pooled reference count is zero.
What Pooling Changes Under a Count Model
Under the independent Poisson model, sums of independent Poisson counts are Poisson. For a pool \(S\), let \(T_S=\sum_{i\in S}s_i\). Then the pooled counts have means \(q_gT_S\) and \(q_rT_S\). The ratio of those means is
The shared exposure term cancels, which explains why a pooled comparison can carry information about relative abundance. This statement concerns the ratio of the pooled expected counts. The observed ratio \(Y_{gS}/Y_{rS}\) is random and is undefined if \(Y_{rS}=0\); it is not generally an unbiased estimate just because the ratio of expected counts equals \(q_g/q_r\).
Pooling also changes the denominator’s zero probability. Under this model, \(Y_{rS}\sim\operatorname{Poisson}(q_rT_S)\), so \(P(Y_{rS}=0)=e^{-q_rT_S}\). Increasing the pool’s total exposure \(T_S\) decreases this probability. When cells have equal exposure \(s\), pooling \(m\) cells gives \(T_S=ms\): the expected pooled count grows in proportion to \(m\), while its Poisson standard deviation grows in proportion to \(\sqrt{m}\). Its relative sampling variation therefore decreases as the pool grows.
Worked Example: Pooling Counts Instead of Averaging Ratios
Consider four hypothetical cells with the following counts for a target gene \(G\) and a reference gene \(J\).
| Cell | \(G\) count | \(J\) count | Cell-level ratio \(G/J\) |
|---|---|---|---|
| 1 | 0 | 1 | 0 |
| 2 | 4 | 0 | Undefined |
| 3 | 0 | 0 | Undefined |
| 4 | 2 | 1 | 2 |
The pooled target count is \(0+4+0+2=6\), and the pooled reference count is \(1+0+0+1=2\). The pooled ratio is therefore
Two of the four cell-level ratios are undefined, so their ordinary arithmetic mean cannot be calculated. The pooled ratio is defined because the aggregate reference count is positive. It is not the average of the defined cell-level ratios, which would be \((0+2)/2=1\). The pooled ratio weights the aggregate counts and thus uses the cells’ counts differently from an average that gives each defined cell ratio equal weight.
This small example illustrates a computational and statistical advantage, not proof that 3 is the true relative abundance ratio. One observed pool is noisy, and the cells may not share the same relative abundance. A ratio of aggregate counts is most interpretable as a relative-scaling quantity when the model’s shared-relative-abundance assumption is reasonable.
Why Variability Can Decrease
A short approximation makes the reduction in sampling variability explicit. Suppose \(A\) and \(B\) are independent pooled Poisson counts with means \(ma\) and \(mb\), where \(m\) is the number of equally exposed cells and \(a,b>0\) are their per-cell mean counts. For \(f(A,B)=A/B\), a first-order Taylor approximation around \((ma,mb)\) uses the derivatives \(\partial f/\partial A=1/B\) and \(\partial f/\partial B=-A/B^2\). Evaluated at the means, these are \(1/(mb)\) and \(-a/(mb^2)\).
The first-order variance approximation is the sum of the squared derivatives multiplied by the respective variances. Since \(\operatorname{Var}(A)=ma\), \(\operatorname{Var}(B)=mb\), and \(\operatorname{Cov}(A,B)=0\), this gives
Thus, when the denominator is sufficiently far from zero for the approximation to be useful, the approximate variance decreases in proportion to \(1/m\). This is a large-count approximation: it does not define the ratio when \(B=0\), and it should not be trusted when zero denominators are common. Pooling helps partly by making such failures rarer and partly by increasing the counts available to estimate a ratio.
Worked Example: Comparing Approximate Ratio Variability
Suppose, hypothetically, that each cell contributes independent Poisson counts with means \(a=2\) for a target gene and \(b=1\) for a reference gene. For a pool of \(m\) cells, the ratio of the pooled expected counts is \(2/1=2\). Using the approximation above, the variance of the observed pooled ratio is
For \(m=10\), the approximate variance is \(6/10=0.6000\), giving an approximate standard deviation of \(\sqrt{0.6000}\approx 0.7746\). For \(m=40\), the approximate variance is \(6/40=0.1500\), with standard deviation \(\sqrt{0.1500}\approx 0.3873\). The approximation suggests lower sampling variability for the larger pool. It does not say either observed ratio must equal 2, and it relies on denominators large enough that the first-order approximation is informative.
The probability of a zero pooled denominator also changes. The reference sum has means 10 and 40 in the two pools, so the zero probabilities are \(e^{-10}\approx 0.00004540\) and \(e^{-40}\approx 4.248\times 10^{-18}\), respectively. In this example, both pools have a low probability of a zero denominator, and the larger pool has still lower approximate ratio variability.
Assumptions, Diagnostics, and Alternatives
The benefits just derived depend on assumptions. Conditional independence and the Poisson variance equaling the mean make the algebra simple. In real data, counts can be overdispersed relative to Poisson, and cells may be correlated or belong to distinct biological states. If the reference gene’s relative abundance changes across the cells, a pooled ratio combines those differing abundances rather than recovering one common ratio. Likewise, pooling cannot resolve a global RNA shift that violates the identifiability assumptions of endogenous size factors; it only aggregates counts.
Diagnostics should therefore ask both whether pooling has reduced sparsity and whether it has obscured important structure. Useful checks include:
- Compare the fraction of pools with zero reference counts to the fraction of individual cells with zero reference counts. A high pooled zero fraction means the chosen reference remains too sparse for reliable ratios.
- Check pooled reference counts and ratios for extreme values. A few small denominators can still produce unstable ratios, even if most pools are adequately counted.
- Examine whether results differ markedly across cell groups or pool definitions. Large differences can reflect biological heterogeneity, changing reference abundance, or inadequate sampling rather than a simple exposure effect.
- Check whether a few highly expressed cells dominate the pooled totals. A large pool count is not automatically representative of every cell in that pool.
Pooling is different from adding a pseudocount to every cell-level denominator. A pseudocount makes a numerical ratio calculable, but the resulting value depends on the chosen constant and is not additional observed information. Pooling instead sums observed counts; it improves the signal available for an aggregate comparison when its assumptions are plausible. Filtering very sparse genes can avoid some unusable ratios, but discards information and does not create counts for retained genes. Median-ratio and TMM approaches use multiple genes and robust summaries rather than relying on one cell-level reference ratio; pooling is a distinct strategy for strengthening information in sparse counts, not a substitute for checking the normalization target and assumptions.
The main limitation is interpretive: a pooled ratio describes an aggregate. It does not provide a separate reliable ratio for each cell, nor does it reveal which cells supplied the counts. Large pools can improve counting precision while concealing cell-to-cell variation. Pooling is most useful when the goal is a more informative aggregate relative comparison, the cells are sufficiently comparable for the intended target, and the resulting denominator is appreciably more stable.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- If a reference count is Poisson with mean \(0.5\) per cell, what is the probability that the reference count summed across four independent cells is zero?
- For independent Poisson counts \(G\) and \(J\) with means 3 and 1, respectively, is the ratio of expected counts the expected value of \(G/J\)? Explain.
- Two pools have the same per-cell count model, but one contains four times as many independent cells as the other. Under the large-count approximation, how does the ratio’s variance change?
- Name one reason a pooled ratio might fail to represent a common relative abundance across the cells in the pool.
Solutions. (1) The pooled reference count has mean \(4(0.5)=2\), so its zero probability is \(e^{-2}\approx 0.1353\). (2) No. The ratio of expected counts is \(3/1=3\), but \(G/J\) is undefined when \(J=0\), an event with probability \(e^{-1}\approx 0.3679\). (3) The approximation gives variance proportional to \(1/m\); multiplying the pool size by four divides the approximate variance by four, provided the approximation and independence assumptions are reasonable. (4) For example, the reference gene’s relative abundance may differ across cell states, so the pooled ratio combines distinct abundances rather than estimating one shared value.