Defining the Normalization Target in Sparse Count Matrices

Start by Naming the Quantity You Want

A cell-by-gene count matrix can look like a direct table of gene expression: each row is a cell, each column is a gene, and each entry is a count. But a count is first an observation produced by a measurement process. Its magnitude can depend both on the RNA in the cell and on how much of that RNA was captured and sequenced. Normalization is meaningful only after specifying which part of that process should be adjusted and what quantity the adjusted values are intended to represent.

Three quantities must be kept distinct. Raw counts are the recorded integers. Relative expression describes a gene’s contribution relative to a defined total or reference. Absolute RNA abundance refers to a molecule amount, such as transcripts of a gene per cell. These quantities are related by measurement assumptions, but they are not interchangeable.

The definition of \(T_i\) matters. It could mean all RNA molecules in a cell, or only the molecules from genes represented and measurable in a particular assay. Those totals need not be the same. In practice, a count matrix alone generally does not enumerate every RNA molecule in the cell, so the phrase “fraction of total RNA” must be interpreted with respect to the measurement model and its gene universe.

Counts, Relative Expression, and Absolute Abundance

A simple sequencing model illustrates the distinction. Suppose that, conditional on a cell’s total sequenced count \(N_i\), the counts across genes follow a multinomial distribution with probabilities \(p_{ij}\). This model assumes that each sequenced molecule is assigned to one measured gene and that the probability of assignment to gene \(j\) is proportional to its abundance in the defined RNA pool. Under this model, the count total \(N_i\) governs the number of observations, while \(p_{ij}\) governs their allocation across genes.

This model makes clear why a larger count for one cell does not by itself establish that the cell had more molecules of every gene. The cell may simply have had a larger observed library total. Conversely, the proportions \(Y_{ij}/N_i\) estimate the composition of the observed library under the model, but they do not give molecules per cell. The distinction is between how many observations were obtained and how those observations were distributed.

A different but related formulation is a multiplicative mean model. Let \(\theta_{ij}\) be an expression target on a specified scale and let \(s_i\) represent a cell-specific exposure. If the conditional mean is \(E[Y_{ij}\mid s_i]=s_i\theta_{ij}\), the expected count combines exposure and target. The model alone does not say that \(\theta_{ij}\) is an absolute molecule count. That interpretation requires a known mapping from exposure to molecules and a scale that is anchored to physical units.

If \(s_i\) is known, fixed, and positive, then linearity of expectation gives \(E[Y_{ij}/s_i\mid s_i]=\theta_{ij}\). This is a statement about a fixed denominator under the stated mean model. If \(s_i\) is estimated from the data, the normalized value \(Y_{ij}/\hat{s}_i\) is a plug-in value. Its expectation depends on the estimation procedure and on any dependence between \(\hat{s}_i\) and \(Y_{ij}\); it is not generally equal to \(\theta_{ij}\).

Why the Absolute Scale Is Not Determined by Counts Alone

The ambiguity can be shown directly. In the multinomial model, the likelihood for the gene allocation depends on the proportions \(p_{ij}\), not on the total abundance \(T_i\). Multiplying every abundance in a cell by the same positive constant leaves those proportions unchanged:

Thus, under this model, the count allocation can inform relative abundance but cannot identify the absolute total RNA scale. This is not a defect in a particular normalization formula; it is a limit of what the specified observations and model determine. To estimate molecules per cell, one needs additional information that connects observed counts to a physical abundance scale, such as suitable calibration or a measurement design with known exposure. Merely dividing counts by a cell-level factor does not supply that information.

The same issue appears in a multiplicative mean model. If the expected count is \(s_i\theta_{ij}\), replacing \(s_i\) by \(c s_i\) and \(\theta_{ij}\) by \(\theta_{ij}/c\) preserves the product and hence the expected count. A convention can set a common scale for reporting relative factors and targets, but a reporting convention does not create absolute units.

What a Size-Factor Normalization Can Claim

A size-factor normalization aims to remove cell-to-cell differences in an exposure or scale component so that expression values can be compared on a chosen common scale. In an idealized model with known fixed \(s_i\), dividing by \(s_i\) puts the conditional mean on the \(\theta_{ij}\) scale. If the target is relative expression, the normalized values can be interpreted relative to that defined target, subject to the model. If the target is absolute RNA abundance, the exposure must be calibrated so that the scale corresponds to molecules per cell.

In ordinary use, factors are estimated rather than known. The resulting values \(Y_{ij}/\hat{s}_i\) should then be described as normalized, plug-in values on the chosen scale. Their expected value is not automatically \(\theta_{ij}\), and the uncertainty in \(\hat{s}_i\) is not automatically represented by treating the factor as fixed. A claim about unbiasedness or exact estimation requires additional conditions on how the factor is obtained and how it relates to the counts.

A factor may be useful for relative comparisons even when it has no physical unit. In that case, its common scale is a convention used to make cells comparable, and the interpretation is correspondingly relative. The precise estimator and its assumptions are method-specific; the essential question at this stage is not which formula to apply, but what target that formula could support.

Diagnostics, Alternatives, and Limitations

Before interpreting normalized values, make the target explicit and check whether the data and design support it. Useful questions include:

• Are the entries raw observed counts, or values already transformed or scaled?
• Is the intended target a library composition, a relative expression scale, or molecules per cell?
• What does the factor represent in the assumed count model, and is its scale physically calibrated or only relative?
• Could variation in capture, sequencing, or the measured RNA pool affect counts separately from the biological quantity of interest?
• Are zeros or very sparse counts making a cell-level ratio unstable or uninformative?

A mismatch between the biological question and the normalization target is a failure even if the arithmetic is correct. For example, a normalization designed to compare relative profiles cannot, without further calibration, answer a question about whether total RNA per cell doubled. Likewise, a normalized value should not be reported with units such as “molecules per cell” unless the measurement and normalization establish that unit.

The main alternative to an endogenous relative scale is to use an experimental calibration or other external information that ties counts to absolute abundance. That approach may support a physical-scale claim when its assumptions hold, but it requires additional measurements and calibration assumptions. A relative normalization instead relies on the information in the count matrix and a chosen scale; it is often appropriate for comparing profiles, but does not recover a cell’s absolute total RNA by itself.

These are distinct goals, not competing formulas for the same estimand. Relative expression can be scientifically useful, and absolute abundance can be scientifically important, but conclusions should match the quantity the design actually identifies. Detailed choices among size-factor estimators address how to estimate a relative scale; they do not change the need to define the target first.