Why Model Counts at the Cell Level?
In a cell-type QTL analysis, gene expression may be recorded as a nonnegative integer count for each cell. These counts are often sparse: many cells have zero observed counts for a gene, even when the gene is detected in other cells of the same cell type. A count model represents the discreteness of the observations directly and can account for differences in sequencing exposure and dependence among cells from the same donor.
The unit of genetic information is still the donor: every cell from a donor has that donor’s genotype. Treating cells as independent genetic replicates would therefore exaggerate the information available. Cell-level mixed models address this dependence with donor-level random effects, as in the earlier material on cell-level mixed models. Here the focus is how the count distribution and its zeros affect the model choice.
Poisson and Negative-Binomial Mixed Models
Let \(Y_{ij}\) be the observed count for a gene in cell \(j\) from donor \(i\). Let \(g_i\) denote the donor’s genotype dosage, commonly coded as 0, 1, or 2 copies of an allele. Let \(L_{ij}\) be a positive exposure measure for that cell, such as its library size relative to a reference library size. The logarithm of exposure is included as an offset, with coefficient fixed at 1, so that cells with more sequencing exposure are expected to yield more counts.
A basic cell-type-specific mean model is \(\log(\mu_{ij})=\log(L_{ij})+\beta_0+\beta_G g_i+\boldsymbol{z}_{ij}^{T}\boldsymbol{\gamma}+b_i\). Here \(\mu_{ij}=E(Y_{ij}\mid b_i)\), \(\boldsymbol{z}_{ij}\) contains specified covariates, and \(b_i\) is a donor random intercept. A usual assumption is \(b_i\sim N(0,\tau^2)\), independently across donors and independent of genotype and included covariates. Conditional on the donor effects and covariates, cells are assumed independent under the model. This conditional independence does not mean that cells from a donor are marginally independent.
Under the Poisson version, the conditional count distribution is \(Y_{ij}\mid b_i\sim\operatorname{Poisson}(\mu_{ij})\). Its conditional mean and variance are both \(\mu_{ij}\). This equality is a substantive distributional assumption, not a requirement that every observed sample mean and sample variance match exactly.
The negative-binomial (NB) version allows additional conditional variation. We use a parameterization with dispersion \(\phi>0\):
As \(\phi\) increases, the extra variance decreases; in the limit as \(\phi\) tends to infinity, the variance approaches the Poisson variance. Thus the NB model can represent extra cell-to-cell count variability beyond the Poisson assumption, while the donor random intercept represents shared variation among cells from the same donor. These components address different sources of variation and should not be treated as interchangeable.
Under either model, \(\exp(\beta_G)\) is the conditional multiplicative change in expected count for a one-allele increase in dosage, holding exposure, covariates, and donor random effect fixed. It is a rate ratio when the offset represents exposure. The model describes an association; it does not by itself establish a causal genetic mechanism.
Worked Example: A Zero Under a Poisson Model
Suppose a hypothetical cell has relative exposure \(L_{ij}=1\) and, at its covariate values with donor effect set to zero, a fitted Poisson mean of \(\mu_{ij}=0.40\) counts. The probability of observing zero counts is \(P(Y_{ij}=0\mid b_i)=\exp(-\mu_{ij})\).
Substituting the fitted mean gives \(\exp(-0.40)=0.6703\). Thus, even with a positive expected count of 0.40, a zero is the most likely single observed value under this model. The probability of observing at least one count is \(1-0.6703=0.3297\).
If the observed count is zero, it is consistent with ordinary Poisson sampling at this low mean. It does not demonstrate that the gene is not expressed in this cell. Nor does the calculation prove the Poisson model is adequate: that judgment requires considering the distribution of counts across cells and donors, the exposure offsets, and the model’s other assumptions.
Why the Negative-Binomial Model Allows More Zeros
The NB model can be understood as a Poisson model whose mean varies across observations beyond the modeled covariates. One standard construction draws a latent rate \(\Lambda\) from a gamma distribution with mean \(\mu\) and variance \(\mu^2/\phi\), then draws \(Y\mid\Lambda\sim\operatorname{Poisson}(\Lambda)\). Applying the laws of total expectation and total variance gives the NB mean and variance:
The same construction gives the probability of a zero: \(P(Y=0)=(\phi/(\phi+\mu))^{\phi}\). This can exceed the Poisson zero probability \(\exp(-\mu)\), because the distribution includes variation in the latent rate. The comparison is about the model-implied probability, not a rule that every NB data set must contain more observed zeros than every Poisson data set.
Worked Example: Comparing Zero Probabilities
Consider a hypothetical count with conditional mean \(\mu=0.40\). For a Poisson model, the probability of zero is \(\exp(-0.40)=0.6703\). For an NB model with \(\phi=0.50\), it is \((0.50/(0.50+0.40))^{0.50}\).
The ratio inside the power is \(0.50/0.90=0.5556\), so the NB zero probability is approximately \(0.7454\). Both models assign substantial probability to zero, and the NB model assigns more in this example because it allows extra variation.
This comparison does not let us classify an individual zero as “sampling” or “structural.” It shows only that, under a specified mean and dispersion, a zero has a calculable model probability. A zero-inflated or hurdle model makes additional assumptions about a separate zero-generating process; a large number of zeros alone is not sufficient justification for that change.
Choosing Between the Models
The Poisson model is a reasonable starting point when its conditional mean and variance assumptions are credible after accounting for exposure, covariates, and donor effects. The NB model is useful when counts show residual variation beyond that Poisson structure. In either case, fitted-model assessment should respect donor clustering: apparent excess variation among all cells may arise from donor-to-donor differences, not just from extra variation among cells within donors.
A practical comparison therefore asks whether the model includes relevant predictors and exposure, whether the donor random effect is appropriate, and whether the remaining count distribution is compatible with the chosen variance structure. The NB dispersion parameter is not a substitute for checking the donor effect. Conversely, a donor random intercept does not guarantee that the conditional Poisson variance is adequate.
Poisson and NB models are often compared using fit summaries and diagnostics, while considering the scientific design and the intended estimate. Because the Poisson model is a limiting case of the NB model as \(\phi\) tends to infinity, a formal likelihood-ratio comparison involves a boundary case; ordinary reference distributions for nested-model tests should not be applied without checking their validity. Model choice should not be made solely by selecting whichever model produces a smaller association p-value.
Worked Example: Checking for Extra-Poisson Variation
Suppose, for illustration, that a set of hypothetical cells has the same modeled conditional mean of 1.50 counts after accounting for exposure, covariates, and donor effects. The observed conditional variance across comparable cells is estimated as 4.50 counts squared. This estimate is greater than the Poisson variance of 1.50, suggesting that a Poisson variance may be too small.
A rough method-of-moments calculation for the NB dispersion equates the observed variance to \(\mu+\mu^2/\phi\): \(4.50=1.50+1.50^2/\phi\). Thus \(3.00=2.25/\phi\), giving \(\phi=2.25/3.00=0.750\). At \(\mu=1.50\), this dispersion implies variance \(1.50+2.25/0.750=4.50\), as intended.
This arithmetic illustrates the role of \(\phi\); it is not a complete fitted mixed-model analysis. The variance estimate would need to be based on appropriate conditional comparisons, and its uncertainty matters. If donor effects or omitted covariates were not adequately handled, this apparent extra-Poisson variation could reflect those shortcomings instead of residual NB dispersion.
Interpreting a Genetic Effect in a Count Model
The genotype coefficient is on the log expected-count scale because the model uses a logarithmic link. Exponentiating it yields a ratio of conditional expected counts for a one-unit increase in dosage. Because the offset is included, this is a ratio of expected counts at the same exposure; it can be interpreted as a relative expression rate when exposure is a sequencing-depth measure. Absolute expected counts also depend on exposure, covariates, and the donor effect.
Worked Example: Translating a Genotype Coefficient
In a hypothetical fitted count-level QTL model, suppose the conditional expected count for dosage \(g_i=0\), at relative exposure \(L_{ij}=1\), specified reference covariates, and \(b_i=0\), is 0.80. Suppose the fitted genotype coefficient is \(\beta_G=0.30\) per allele.
The per-allele count ratio is \(\exp(0.30)=1.350\). The corresponding conditional mean for dosage 1 is \(0.80\times1.350=1.080\) counts. For dosage 2, relative to dosage 0, the ratio is \(\exp(2\times0.30)=\exp(0.60)=1.822\), giving a mean of \(0.80\times1.822=1.458\) counts under the same stated conditions.
The fitted association is a 35.0% higher conditional expected count per additional allele, not an increase of 0.30 counts and not evidence that the allele causes the expression difference. A statistical test or interval for \(\beta_G\) is needed to describe its uncertainty; the point estimate alone does not establish that the association differs from zero.
When Count-Level Modeling Is Useful—and Its Limits
Count-level modeling can be useful when cell-level counts and cell-level covariates are scientifically important, when library sizes vary and need an exposure offset, or when retaining cell-level observations helps describe within-donor variability. It also makes the discreteness and sparsity of the outcome explicit. It is not automatically superior simply because the original measurements are counts: model assumptions, donor replication, and the target of inference still matter.
The donor supplies the genetic dosage, so having many cells from a small number of donors does not create the same genetic information as having many independent donors. A mixed model accounts for within-donor dependence under its assumptions; it cannot manufacture additional independent genotype observations. The analysis should also use a well-defined cell type and a defensible set of covariates and exposure measures.
The key distinction is between the mean, which describes expected count at specified exposure and covariates, and the variance structure, which describes how counts vary around that mean. Poisson and NB mixed models differ mainly in their conditional variance assumptions; both require careful treatment of donor dependence. Count-level modeling is most informative when these assumptions align with the study design and the scientific question.
Check Your Understanding
Use the model definitions and interpretations above to answer these questions.
- Under a Poisson model with conditional mean 0.20, why can an observed zero occur even if the expected count is positive?
- What does the NB dispersion parameter \(\phi\) control, and what happens to the variance as \(\phi\) increases?
- Why does a donor random intercept not make the genetic information equivalent to having one independent observation per cell?
- In a log-link model with genotype coefficient \(\beta_G\), what is the interpretation of \(\exp(\beta_G)\) for a one-allele increase in dosage?
- Why is an excess of observed zeros alone not enough to conclude that a zero-inflated model is needed?