Two Ways a Count Can Be Zero
A zero in a single-cell count matrix records that no molecules were observed for a gene in a cell. It does not, by itself, say why the observed count is zero. One possibility is that the gene or measurement process was in a distinct inactive state. Another is that the cell was in the ordinary count-generating state, but the random count happened to be zero. These explanations are different descriptions of the data-generating process, even though both produce the same recorded value.
The distinction matters because interpreting every zero as evidence of an inactive state can overstate what the observations establish. Count models such as the Poisson and negative binomial already assign positive probability to zero. A separate inactive state is an additional assumption about how the data were generated, not a label directly observed in the count matrix.
A Latent-State Model
Let \(Y\) be the observed count for one gene in one cell. Let \(S=1\) indicate the structural-zero state and \(S=0\) indicate the active state. Suppose \(P(S=1)=\pi\), where \(0\leq\pi\leq1\). In the active state, assume \(Y\) follows a count distribution \(F_\theta\), with probability mass function \(f_\theta(y)\) for nonnegative integers \(y\). In the structural state, \(Y=0\) with probability one.
The two terms in the zero probability correspond to disjoint routes: the cell is structural with probability \(\pi\), or it is active with probability \(1-\pi\) and its ordinary count is zero with probability \(f_\theta(0)\). Thus, among the structural model's zeros, some are structural and some are sampling zeros.
For the simple calculations below, assume cells are independent draws from the same model, each has the same structural probability \(\pi\), and active-state counts share the same distribution \(F_\theta\). These assumptions can fail if cells differ systematically or observations are dependent. The formulas for a single cell still describe its marginal probabilities when those probabilities are correctly specified, but treating the observed cells as independent and identically distributed would then require justification.
What Is the Probability That a Zero Is Structural?
The model gives a probability for the latent source conditional on observing a zero. By the definition of conditional probability, this is the probability of being in the structural state divided by the probability of observing any zero.
Similarly, the conditional probability that a zero is a sampling zero is \(P(S=0\mid Y=0)=(1-\pi)f_\theta(0)/P(Y=0)\). The two conditional probabilities sum to one. Neither says which state produced a particular observed zero; they describe model-based uncertainty about its latent source.
A brief derivation also shows why the count mean alone cannot separate the two parts of the model. Let the active-state mean be \(\mu_F=E(Y\mid S=0)\). Since a structural-state count is zero,
A low observed mean could therefore reflect a lower active-state mean, a larger structural probability, or both. Additional information or assumptions are needed to distinguish these explanations.
Worked Example: Expected Structural and Sampling Zeros
Suppose, hypothetically, that counts for a gene in 100 independent cells follow a model with structural probability \(\pi=0.20\), and that the active-state count is Poisson with mean \(\lambda=1.5\). For a Poisson count, \(P(Y=0\mid S=0)=e^{-\lambda}\). Find the overall zero probability and the expected numbers of each kind of zero.
The active-state sampling-zero probability is \(e^{-1.5}=0.2231\). The total probability of an observed zero is therefore
The expected number of observed zeros among 100 cells is \(100(0.3785)=37.85\), or about 38. The expected number generated in the structural state is \(100(0.20)=20\). The expected number of sampling zeros is \(100(0.80)(0.2231)=17.85\). These add to \(37.85\), as they should.
Given that a zero was observed, the model-based probability that it is structural is \(0.20/0.3785=0.5284\), about 52.8%. This is a probability across zeros under the assumed model, not a way to determine the origin of any particular cell's zero.
What Counts Can Identify Depends on the Assumptions
There are two different questions: can the model's population parameters be determined from its probability distribution, and can the latent state for an individual zero be determined from its observed count? These questions have different answers. Even if model parameters are identifiable under specified assumptions, a single observed zero may still have either source.
For example, if the active-state distribution is known and \(f_\theta(0)<1\), the population zero probability \(q_0=P(Y=0)\) determines \(\pi\) through
This follows by rearranging \(q_0=\pi+(1-\pi)f_\theta(0)\). The result is meaningful only when the assumed model is compatible with the probabilities: in particular, \(f_\theta(0)\leq q_0\leq1\). If the active distribution is unknown, its zero probability and the structural probability can trade off. Estimation from a finite sample also has sampling uncertainty, so replacing \(q_0\) with the observed fraction of zeros does not reveal the exact population value.
Worked Example: Estimating a Structural Fraction When the Active Model Is Fixed
In a hypothetical set of 100 independent cells, 40 have an observed count of zero. Suppose external calibration justifies fixing the active-state distribution as Poisson with mean \(\lambda=1\), and suppose the common structural probability model is appropriate. Use the observed zero fraction to calculate a plug-in estimate of \(\pi\).
The observed zero fraction is \(\hat q_0=40/100=0.40\). Under the fixed active model, \(f_\theta(0)=e^{-1}=0.3679\). Substituting into the rearranged equation gives
Under these assumptions, the plug-in estimate of the structural probability is 0.0508, or about 5.08%. The active model alone predicts a zero probability of 0.3679, so the observed fraction of 0.40 is accommodated by a relatively small additional structural probability. This calculation does not establish that any of the 40 particular zeros was structural. It also depends on the active Poisson mean being correctly fixed and on the shared-probability model being appropriate.
More generally, the same observed distribution can sometimes be described without asserting a separate latent inactive state. If the active distribution is unrestricted, define a count distribution \(G\) by
These probabilities are nonnegative and sum to one, because \(g(0)+\sum_{y\geq1}g(y)=\pi+(1-\pi)\sum_{y\geq0}f_\theta(y)=1\). Thus, the observed count distribution from the latent-state model is exactly the distribution \(G\). Without constraints on the competing count distribution or additional information about the state, counts alone cannot distinguish these descriptions.
Worked Example: The Same Count Distribution, Two Interpretations
Consider a hypothetical latent-state model with \(\pi=0.30\) and an active Poisson distribution with mean 1. The active probabilities at counts 0, 1, 2, and 3 are approximately 0.3679, 0.3679, 0.1839, and 0.06131. The observed probabilities under the latent-state model are
One interpretation assigns 30% of cells to a separate state that forces a zero, with ordinary Poisson counts in the remaining cells. A second interpretation uses a single count distribution whose probabilities are exactly the observed probabilities above, with no separate state in its definition. The probability for counts above 3 is also given by \(0.70\) times the corresponding active Poisson probability, so the distributions agree for every nonnegative count.
No sample of counts can distinguish these interpretations if both are allowed as models of the same distribution. A more restrictive model comparison could distinguish some probability distributions, but that conclusion would depend on the restrictions imposed; it would not directly reveal the biological or technical cause of an individual zero.
Distinguishing the Concepts in Practice
A sampling-only model, such as a Poisson or negative binomial model, allows zero counts as part of its ordinary count distribution. A zero-inflated model adds a separate state that forces a zero while retaining the active distribution's own possibility of a zero. A hurdle model uses a different construction: one process determines whether the count is zero or positive, and the positive counts follow a distribution conditioned to be positive. These models make different assumptions about how zeros and positive counts arise; none makes the latent cause directly observable.
| Model description | How zero counts arise | What a zero establishes |
|---|---|---|
| Sampling-only count model | From the ordinary count distribution | Only that the recorded count is zero |
| Zero-inflated latent-state model | From a structural state or from the active distribution | Not which of the two routes occurred |
| Hurdle model | From the zero-versus-positive process | That the hurdle was not crossed, under the model |
The word structural describes a state in a statistical model; it does not, by itself, establish a biological gene-off state. Nor does an observed zero prove technical dropout. Those interpretations require evidence and assumptions beyond the recorded count. For a particular zero, the conditional probability formula can summarize uncertainty if a latent-state model is justified, but it cannot convert an unobserved state into a known fact.
Common Mistakes and Limitations
- Calling every zero structural. An active Poisson or negative binomial count can be zero. Check the ordinary model's zero probability before attributing zeros to another state.
- Treating a fitted structural probability as a count of known cells. The parameter describes a probability under a model. It does not identify which observed zeros came from that state.
- Interpreting model labels as mechanisms. A structural component is a statistical assumption, not proof of a biological or technical process.
- Ignoring uncertainty in the active distribution. The calculation \(\pi=(q_0-f_\theta(0))/(1-f_\theta(0))\) relies on \(f_\theta(0)\). If that probability is uncertain or misspecified, the resulting structural estimate can be misleading.
- Confusing model-based identification with certainty about a cell. A model may determine population parameters under its assumptions while leaving the latent source of an individual zero uncertain.
The practical limitation is therefore fundamental: an observed count matrix records outcomes, not the latent path that generated each outcome. Constrained distributions, experimental design, and external biological or technical information can make some explanations more plausible, but the strength of that conclusion depends on the assumptions and evidence used.
Check Your Understanding
Answer each question using the model definitions and calculations in this tutorial.
- If \(\pi=0.10\) and the active distribution has \(f_\theta(0)=0.25\), what is the overall zero probability?
- For the same model, what is \(P(S=1\mid Y=0)\)?
- In an independent sample of 200 cells under that model, what are the expected numbers of structural zeros, sampling zeros, and total zeros?
- Why can a known active-state distribution make \(\pi\) estimable from a population zero probability, while an observed zero still cannot be assigned to a source with certainty?
Solutions. 1. \(P(Y=0)=0.10+0.90(0.25)=0.325\). 2. \(P(S=1\mid Y=0)=0.10/0.325=0.3077\), or about 30.8%; this is conditional uncertainty, not a label for a specific zero. 3. The expected structural-zero count is \(200(0.10)=20\); the expected sampling-zero count is \(200(0.90)(0.25)=45\); the expected total is \(20+45=65\), also \(200(0.325)=65\). 4. If \(f_\theta(0)\) is known, \(q_0=\pi+(1-\pi)f_\theta(0)\) can be rearranged to determine \(\pi\) at the population level. But both latent routes still produce \(Y=0\), so observing that value alone does not reveal which route occurred.