Capture Variation And Apparent Overdispersion

Why Capture Variation Matters

Even if cells contain molecules according to the same count process, they may not capture the same fraction of those molecules. A cell with higher capture efficiency tends to contribute larger observed counts; a cell with lower efficiency tends to contribute smaller counts and more zeros. Pooling the cells can therefore make counts look more variable than a constant-efficiency model predicts.

The key distinction is between variation within a cell at a fixed capture efficiency and variation between cells in their capture efficiencies. The first is sampling noise. The second mixes cells with different expected observed counts. This tutorial quantifies how that mixing affects the variance and the probability of observing zero counts.

A Mixed-Efficiency Capture Model

The assumptions matter. Conditional on a cell's \(C_i\), each molecule is captured independently with the same probability. The efficiency is shared by the molecules in that cell, but varies between cells. The baseline Poisson model assumes a common \(\lambda\) and independence between \(X_i\) and \(C_i\). These assumptions isolate capture variation; biological differences in molecule abundance or dependence between abundance and capture efficiency would add further variation.

By binomial thinning, \(Y_i\mid C_i\sim\operatorname{Poisson}(\lambda C_i)\). Thus, conditional on capture efficiency, the observed count is Poisson; pooled across cells, it is a mixture of Poisson distributions with different means. The marginal mean is \(\lambda\bar c\), but the marginal variance also reflects how much \(C_i\) varies.

How Efficiency Variation Inflates the Variance

To obtain the variance, apply the law of total variance: total variance is the expected conditional variance plus the variance of the conditional mean. Since both the mean and variance of a Poisson variable equal its rate, \(E[Y_i\mid C_i]=\operatorname{Var}(Y_i\mid C_i)=\lambda C_i\).

The first term, \(\lambda\bar c\), is the variance expected under a Poisson model with constant efficiency \(\bar c\). The second term is the additional variance from cell-to-cell efficiency differences. If \(\operatorname{Var}(C_i)>0\) and \(\lambda>0\), the pooled variance exceeds the pooled mean. The ratio of variance to mean is

This ratio equals 1 under constant-efficiency Poisson sampling and is greater than 1 when capture efficiency varies. The extra variance is not evidence, by itself, that counts within each cell follow a different distribution: conditional on \(C_i\), they still follow the specified Poisson model.

Why the Mixture Has More Zeros

For a cell with efficiency \(C_i\), the conditional probability of a zero count is \(e^{-\lambda C_i}\). Averaging over cells gives the marginal probability \(E[e^{-\lambda C_i}]\). A constant-efficiency model with the same mean efficiency instead gives \(e^{-\lambda\bar c}\). These quantities need not agree.

This inequality follows from Jensen's inequality: \(f(c)=e^{-\lambda c}\) is convex for \(\lambda>0\), since \(f''(c)=\lambda^2e^{-\lambda c}>0\). The inequality is strict when efficiency varies across cells. Low-efficiency cells contribute disproportionately many zeros, and their effect is not cancelled by high-efficiency cells when comparing with a single cell at the average efficiency. This is a change in sampling-zero probability under the mixture; it does not, on its own, identify why a particular cell has a zero count.

Using the Variance Relationship

The variance formula can be rearranged to describe how much capture heterogeneity would be needed to account for an observed variance, if \(\lambda\) and \(\bar c\) were known. The rearrangement is a method-of-moments calculation, not proof that the assumptions hold.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to ask whether counts vary systematically with information related to capture efficiency. If cells can be divided into efficiency groups, compare the count mean and zero frequency within each group with the conditional model, rather than comparing only the pooled data with a single-rate model. In the two-group example, the conditional means are 1 and 4 counts, and the corresponding zero probabilities are \(e^{-1}\) and \(e^{-4}\). Pooling hides that difference and produces the higher overall zero probability.

When efficiencies are known or credibly estimated, an alternative is to model counts conditional on each cell's efficiency, using \(\lambda C_i\) as the Poisson rate. This accounts for the expected change in counts with efficiency. If efficiency is unknown, a constant-rate Poisson model is simpler but may misrepresent pooled variance and zeros. A negative binomial model can accommodate extra marginal variance, but its dispersion parameter does not by itself establish that capture efficiency is the cause. The mixed-efficiency distribution need not be negative binomial.

Several features can make the calculation an imperfect description of real data. Capture efficiency may be estimated with error, may differ by gene, or may be associated with the number of molecules in a cell. Pre-capture abundance may also vary between cells. In those settings, the simple independence assumption fails or additional sources of variation enter the marginal distribution. A larger observed variance or zero frequency is therefore a reason to check the assumptions and relevant cell-level information, not a unique diagnosis of capture heterogeneity.