Why Endogenous Size Factors Need Assumptions
An endogenous size factor is estimated from the genes measured in the sample rather than from a known external quantity. The data can reveal that one cell has counts consistently larger than another for a useful set of genes. But counts alone do not explain why: the difference could reflect technical exposure, such as sequencing depth or capture efficiency, or a biological difference in RNA abundance. Endogenous normalization therefore depends on assumptions that separate these explanations.
The central assumption is that a sufficiently large and informative set of genes has comparable underlying expression across the cells being normalized. These genes provide an internal reference. A second assumption is that sampling acts comparably across genes: for a given cell, technical exposure multiplies counts by a common cell-specific factor rather than changing each gene by a different amount. Under these conditions, differences shared across the reference genes can identify relative cell-specific scaling.
A Mean Model and What It Can Identify
Here, \(s_i\) is an effective sampling scale, not necessarily a pure measure of sequencing depth. It can combine technical influences that act multiplicatively on counts. The abundance \(\theta_g\) is the part assumed stable for that gene over the cells under comparison. A reference gene need not have the same count in every cell: its expected count changes with \(s_i\). Stability means that its underlying \(\theta_g\) does not systematically change between those cells.
The model has an unavoidable scale ambiguity. For any constant \(c>0\), replacing every \(s_i\) with \(c s_i\) and every \(\theta_g\) with \(\theta_g/c\) leaves all expected counts unchanged:
Thus, endogenous counts cannot determine absolute exposure and absolute abundance separately. They can identify relative size factors, such as \(s_i/s_j\), when the assumptions hold. To report one numerical set of factors, an arbitrary convention is needed, such as setting their geometric mean to one. That convention chooses a scale; it does not add biological information.
Why Stable Genes Anchor Relative Factors
For a stable gene \(g\), the ratio of its expected counts in cells \(i\) and \(j\) is
The unknown gene abundance cancels. If several stable genes provide similar information, their shared relative scaling can be estimated even though their separate \(\theta_g\) values are unknown. In observed sparse counts, a ratio for one gene may be unavailable when a count is zero, and random variation can make ratios differ. Estimation procedures therefore combine information across genes rather than treating every observed ratio as exact.
The assumption that the stable set is sufficiently large is practical as well as conceptual. A single anchor gene may be zero in many cells, unusually noisy, or mistakenly assumed stable. A broader set gives the estimate a chance to be robust to some genes changing or behaving atypically. However, “large” is not enough by itself: if most candidate genes share a biological shift, their agreement can still point to the wrong interpretation.
Worked Example: Relative Scaling From Stable Anchors
Suppose two hypothetical cells have observed counts of 60, 40, and 20 for three genes in cell A, and 120, 80, and 40 for those genes in cell B. Assume these three genes are stable and that sampling is comparable across them. Estimate the relative factor for B versus A from the gene-wise ratios.
The ratios of B's counts to A's counts are \(120/60=2\), \(80/40=2\), and \(40/20=2\). Under the stable-gene model, each ratio estimates \(s_B/s_A\), so the common relative factor is 2. If we set \(s_A=1\), this gives \(s_B=2\). Dividing B's counts by 2 gives 60, 40, and 20, matching A's counts for these reference genes.
This arithmetic illustrates the identifying information in the assumptions; it does not prove that the genes are truly stable. In real data, the observed counts also include sampling variation, and agreement across a reference set is evidence to assess rather than a guarantee.
Assumptions for an Endogenous Reference
The stable-gene argument relies on several linked conditions. They should be stated for the cells and genes being compared, not treated as universal properties of a normalization algorithm.
- A stable reference set exists. For the genes serving as anchors, underlying abundance is sufficiently comparable across the cells. The assumption is about abundance relevant to the normalization target, not merely that a gene is commonly detected.
- The set is large and informative enough. There must be enough usable genes, with adequate signal across the cells, that a common relative scale can be estimated despite random count variation and a limited number of atypical genes. A gene that is almost always zero supplies little direct ratio information.
- Sampling is comparable across genes. Within each cell, the effective exposure acts approximately as one multiplier across the reference genes. A cell-specific technical effect that raises some genes and lowers others cannot be represented by one \(s_i\).
- Cells are connected through informative measurements. Comparisons must share enough expressed reference genes, directly or through overlapping sets, to link the relative scales. If two groups have no informative reference genes in common, their factors are not linked by this model.
- Biological and technical changes are not perfectly confounded. If all endogenous genes change together between groups, their counts alone cannot determine whether that common change is biological or technical. Calling the change a size-factor difference requires an identifying assumption or information beyond those endogenous counts.
The mean model does not require counts to be free of noise. It does require that noise and missing detections do not create a systematic gene-by-cell distortion that defeats the shared scaling structure. Zeros are compatible with a positive expected count: an observed zero is not, by itself, evidence that the underlying abundance is exactly zero.
Worked Example: A Stable Majority and One Changing Gene
Consider five hypothetical genes. Cell A has counts \((100,80,40,20,10)\), and cell B has counts \((200,160,120,40,20)\). Suppose the first, second, fourth, and fifth genes are stable, while the third gene has a biological increase. Calculate the ratios and a robust relative factor based on the majority of stable genes.
The B-to-A ratios are \(200/100=2\), \(160/80=2\), \(120/40=3\), \(40/20=2\), and \(20/10=2\). The median of the five ratios is 2. It estimates \(s_B/s_A\) because four of the five genes follow the stable-gene ratio. Taking \(s_A=1\), the estimated factor for B is 2.
After dividing B's counts by 2, the values are \((100,80,60,20,10)\). Four genes now match cell A; the third remains higher, at 60 rather than 40. The remaining difference is consistent with the stipulated biological change. This example shows why a sufficiently broad stable set can protect a relative estimate from a minority of changing genes. If the changing genes were the majority, the same calculation could instead absorb their shared shift into the factor.
Identifiability Is Not the Same as Estimation
Identifiability asks whether different parameter values can produce the same distribution of the data under the stated model. Estimation asks how well a procedure can recover identifiable quantities from a finite, noisy data set. More cells or genes may improve precision, but they cannot resolve a structural ambiguity that the model leaves unresolved.
For example, under the stable-gene model, expected-count ratios identify relative factors because the gene abundance cancels. If the “stable” genes actually differ systematically in abundance between groups, that cancellation no longer yields a purely technical comparison. A robust estimator may limit the influence of a few unusual genes, but robustness is not a substitute for the existence of a stable reference. Likewise, a normalization convention can set the average factor to one, but cannot decide whether a shared shift is biological or technical.
Worked Example: A Global Change Cannot Be Classified From Endogenous Counts Alone
Suppose three hypothetical genes have baseline abundances \((100,50,25)\), and cell A has effective exposure \(s_A=1\). Cell B has observed counts \((200,100,50)\). Compare two explanations for these counts.
Technical-scaling explanation. Keep the abundances at \((100,50,25)\) and set \(s_B=2\). The expected counts for B are \(2(100,50,25)=(200,100,50)\).
Biological-change explanation. Set \(s_B=1\) and let the abundances in B be \((200,100,50)\). The expected counts are again \((200,100,50)\).
The two explanations give the same expected counts. If every endogenous gene follows the same twofold change, these counts do not identify which explanation is correct. An endogenous size factor can express the relative scaling under a chosen stable-reference assumption, but the numerical factor alone is not evidence that the difference was technical.
Diagnostics, Alternatives, and Limits
Diagnostics can reveal tension with the assumptions, although no diagnostic based only on the same counts can prove that a reference set is biologically stable. Useful checks include whether many candidate genes support a similar relative scale, whether the estimate changes substantially when a subset of genes is removed, and whether residual gene-wise ratios show systematic patterns rather than scattered variation. Large instability across reasonable reference choices, or markedly different scaling for different groups of genes, is a warning that one common factor may not describe the data well.
A single cell-specific factor also fails when sampling is not comparable across genes. For instance, gene-specific technical effects that vary by cell can produce inconsistent ratios even if the underlying biology is stable. In that situation, forcing one factor onto all genes may leave structured differences behind. Conversely, coherent agreement among many genes can still be misleading if those genes share a biological change.
Alternative endogenous estimators, such as median-ratio and trimmed log-ratio approaches, differ in how they combine gene-wise information and limit the influence of extreme genes. They can improve stability when some genes change, but they still need an adequate reference pattern: a sufficiently informative set whose changes do not dominate the comparison. Total-count scaling uses a different summary, but it too relies on assumptions about how the overall counts relate to the intended exposure. No choice of estimator can recover a biological-versus-technical distinction that is absent from the data and assumptions.
The practical conclusion is conditional. Endogenous factors are useful for relative scaling when stable genes and comparable sampling are credible for the comparison at hand. They do not automatically estimate absolute RNA content per cell, and their interpretation should state the reference assumption. If that assumption is doubtful, report the limitation rather than treating a normalized matrix as a uniquely determined biological scale.
Name the cells and the gene set for which a common relative factor is being considered.
Explain why a sufficiently large, informative set is assumed to have comparable underlying abundance across those cells.
Assess whether the reference genes support a common multiplier and whether the cells are linked by informative measurements.
Remember that the scale convention sets a numerical origin; it does not establish absolute exposure or resolve a shared biological shift.
Check Your Understanding
Answer each question, then compare your reasoning with the solution given immediately afterward.
- For a stable gene, expected counts are 30 in cell A and 45 in cell B. What relative factor \(s_B/s_A\) does this gene imply under comparable sampling? Solution: \(45/30=1.5\), so the implied relative factor is 1.5. This inference depends on the gene being stable and on the shared-multiplier model.
- Why can endogenous counts identify relative factors but not a unique absolute scale? Solution: Multiplying all \(s_i\) by \(c>0\) and dividing all \(\theta_g\) by \(c\) leaves every product \(s_i\theta_g\) unchanged. The data therefore do not distinguish those parameterizations.
- Five genes give B-to-A ratios \(1.8, 2.0, 2.1, 2.0,\) and \(4.5\). If the first four genes are stable and the last is atypical, what is the median ratio? Solution: In increasing order the ratios are \(1.8,2.0,2.0,2.1,4.5\). The median is 2.0, an estimate of the relative factor under the stated stable-majority assumption.
- Every endogenous gene in cell B has twice the count of its counterpart in cell A. Can the counts alone show that B had twice the technical exposure? Solution: No. A twofold technical multiplier and a twofold biological change in all gene abundances produce the same expected counts. An additional identifying assumption or information is needed.
- Different groups of genes consistently imply very different B-to-A ratios. Which model condition is in doubt? Solution: The assumption that comparable sampling acts as one common cell-specific multiplier across reference genes is in doubt. Gene-specific technical effects, biological changes, or an unsuitable reference set may explain the disagreement.