Sources Of Overdispersion In Single-Cell Counts

Why Counts Can Vary More Than Poisson Sampling Predicts

For a fixed gene and a population of cells, the Poisson model provides a useful reference: if each cell has the same expected count and counts arise through independent sampling, the variance equals the mean. Real single-cell data often depart from that reference. Cells can differ in their expression state, capture efficiency can vary, and measurements can reflect unrecorded differences in library exposure or experimental conditions. These differences can make counts across cells more variable than a homogeneous Poisson model predicts.

The reason for the excess matters. A variance greater than the mean does not, by itself, show whether cells differ biologically, capture varies technically, or the model omits a relevant factor. The law of total variance, introduced in the earlier material on that subject, gives a direct way to separate variation in counts conditional on a cell’s expected rate from variation in that expected rate across cells.

A Poisson distribution is equidispersed: if \(Y\sim\operatorname{Pois}(\lambda)\), then \(E(Y)=\operatorname{Var}(Y)=\lambda\). That statement applies to the distribution under the specified model. If cells with different values of \(\lambda\) are pooled, their overall count distribution need not be Poisson, even if each cell’s count follows a Poisson model conditional on its own rate.

Separating Conditional Noise From Variation in Rates

Let \(Y\) be the count for a particular gene in a randomly selected cell, and let \(Z\) represent a cell characteristic or latent factor that affects its expected count. The law of total variance states that

The first term is the average count variance among cells with the same value of \(Z\). The second is the variance in conditional means across values of \(Z\). If counts are conditionally Poisson, \(Y\mid Z\sim\operatorname{Pois}(\lambda_Z)\), then \(E(Y\mid Z)=\operatorname{Var}(Y\mid Z)=\lambda_Z\), giving

Thus, variation in cell-specific Poisson rates adds to the marginal variance. The added term is nonnegative and is positive whenever the rates vary across cells. The conclusion requires finite means and variances, and the conditional Poisson assumption. It describes variation across the specified cell population; it does not establish that the conditional Poisson model is appropriate for every cell.

Variable Capture and Variable Exposure

Capture variability is another source of differences in expected observed counts. Consider a simple model in which the number of molecules before capture in a cell is Poisson with mean \(\lambda\). Each molecule is captured independently with probability \(q\), and \(q\) may vary across cells. Conditional on \(q\), Poisson thinning gives \(Y\mid q\sim\operatorname{Pois}(q\lambda)\). This model assumes a Poisson precursor, independent capture of molecules given \(q\), and a capture probability that applies to the molecules considered.

The conditional mean and variance are both \(q\lambda\). Applying the law of total variance,

The first term in the variance equals the mean. The second is the excess arising from cell-to-cell variation in capture probability under these assumptions. If capture is constant across cells, this added term is zero. This result depends on the Poisson precursor: with a fixed number of precursor molecules, independent capture instead has a binomial conditional distribution, whose variance is below its mean. It is therefore inaccurate to say that every form of variable capture must produce overdispersion relative to a Poisson model.

A related issue is varying exposure, such as differences in effective library size. To illustrate, let \(s\) be a dimensionless cell-specific exposure and suppose \(Y\mid s\sim\operatorname{Pois}(s\mu)\). If \(s\) is not accounted for, cells with larger \(s\) have larger conditional means. A known exposure can be included when comparing cells; if it is omitted, its variability contributes to the pooled variance.

Other Sources and What a Variance Excess Can Show

Not every source of excess variation is simply a random Poisson rate. Transcriptional dynamics may produce counts that vary more than a Poisson distribution even after conditioning on measured cell characteristics. Unrecorded batch conditions, cell cycle, activation state, or differences in sample handling can also leave cells with different expected counts. Correlated measurements or dependence between molecules can violate the independent-event assumptions behind a Poisson model. In general, if \(Z\) describes measured or unmeasured conditions, then

This identity separates two broad contributions: excess or deficit variance within conditions, and variation in conditional means between conditions. If each conditional distribution is Poisson, the first term is zero; if conditional counts are themselves overdispersed, that term can be positive as well. The formula does not identify which biological or technical mechanism generated either contribution.

The earlier material on diagnosing the Poisson mean-variance assumption provides the context for comparing observed variability with a Poisson reference. In practice, a sample variance above a sample mean is only a warning sign, not a mechanism test. Sample sizes are finite, and estimated means and variances fluctuate. A useful investigation asks whether the excess persists after comparing cells with similar measured characteristics, and whether it changes with plausible exposure or batch variables.

Diagnostics, Comparisons, and Limitations

The homogeneous Poisson model is a useful baseline when cells under comparison have a common expected count and the conditional count process is plausibly Poisson. A stratified or covariate-conditional Poisson comparison is more appropriate when measured states or exposures differ. Explicitly accounting for a known exposure addresses that source of mean variation, while adding state or batch information can reveal whether pooled heterogeneity explains some of the excess. These approaches do not guarantee that the remaining conditional counts are Poisson.

Diagnostics should be tied to the proposed mechanism. Examine whether mean and variance comparisons change within relevant cell groups, whether counts vary systematically with library exposure, and whether differences align with recorded batch or state information. Check that groups have enough cells for their variance estimates to be informative. If apparent groups are defined after inspecting the same noisy counts, treat the resulting comparison cautiously. A single gene’s observed variance, especially in a small group, is not a precise estimate of its underlying variance.

The main limitation is identifiability: the same marginal mean and variance can result from distinct mixtures or from extra conditional noise. Counts alone generally cannot establish whether the source was biological heterogeneity, capture variability, exposure differences, or a misspecified conditional model. Additional measurements, experimental design information, or replicated observations may be needed to distinguish causes. Overdispersion is a property of the count distribution relative to a stated model; it is not a diagnosis of its cause.

Common Mistake

Wrong reasoning: “The variance across all cells exceeds the mean, so the Poisson model fails within every cell type and the extra variation must be technical.” This treats a pooled comparison as if all cells shared one conditional mean.

Correction: A mixture of cell populations with different Poisson means can have a variance larger than its pooled mean, as the first worked example shows. Compare like with like where possible, and distinguish variation between conditional means from extra variation within groups. Even after the latter is found, variance alone does not assign a biological or technical cause.