Why the Scope of Factor Estimation Matters
A size factor is meaningful only relative to the observations and assumptions used to estimate it. If cells from several experimental samples are normalized together, their factors may be placed on one common scale. If each sample is normalized separately, each sample may instead receive its own relative scale. Those choices can affect whether normalized expression values are comparable between samples, even when the same factor-estimation method is used.
The central question is not simply whether one sample has larger factors than another. It is whether differences in the factors represent technical exposure, such as sequencing depth, or biological differences that the analysis should preserve. Cell composition, global RNA output, and the set of detected genes can all complicate that distinction.
The earlier material on deconvolution factors and depth-normalization models supplies several ways to estimate cell-level factors. Here the concern is the scope of estimation and the conditions for cross-sample interpretation, not the mechanics of fitting each estimator. Joint estimation uses information from multiple samples under a shared reference or scale. Within-sample estimation uses cells from one sample at a time. A third option estimates factors within meaningful strata, such as sample and cell type, then explicitly aligns those strata if cross-stratum comparisons are needed.
What Makes a Shared Scale Possible?
A simple mean model clarifies the issue. Suppose the expected count is the product of a cell-specific technical exposure \(f_{is}\) and the expression level \(q_{gs}\) for that gene and sample:
The factor captures differences such as sequencing depth only to the extent that \(q_{gs}\) represents the underlying biological expression being modeled. A factor estimator needs information that separates these two components. For cross-sample factor ratios, a common basis is a set of reference genes whose expression is stable across samples, or a broader assumption that most genes do not undergo a coordinated shift that would be mistaken for exposure.
This is an identification issue, not merely a computational detail. Multiplying every factor by a positive constant and dividing every expression level by that constant leaves the expected counts unchanged. More importantly, if expression for all genes in sample \(s\) is multiplied by \(a_s\), then the model contains the product \(a_s f_{is}\). Counts alone cannot determine which part of that product is a biological global shift and which part is technical exposure without additional assumptions or information.
Joint Estimation and Within-Sample Estimation
Joint estimation is generally appropriate when samples share a defensible reference for expression and the scientific goal requires a common scale. It can preserve differences in estimated exposure between samples rather than forcing each sample to have the same center. Its vulnerability is that a shared estimator can be pulled by sample-specific composition or broad biological changes, especially when sample sizes or cell numbers are unbalanced.
Within-sample estimation can be useful when the reference assumptions are more credible among cells from the same sample than across samples. For example, samples might differ substantially in the mixture of cell types, while the aim is to compare cells within each sample. However, if factors are independently centered within each sample, their sample-level centers no longer encode relative exposure. The resulting normalized values are not automatically on a scale for comparing absolute expression between samples.
Stratification can help when there are known, relevant groups of cells: estimate relative factors within each sample-by-cell-type stratum, then align strata using an explicit shared reference. This is not a free correction. It depends on adequate cells and shared reference information within strata, and it can remove genuine biological differences if the chosen alignment forces them to match.
| Estimation scope | Potential advantage | Main comparability concern |
|---|---|---|
| Joint across samples | Can place factors on a shared scale and retain sample-level exposure differences | Composition or broad expression shifts can influence the shared estimate |
| Separately within each sample | Uses a reference drawn from cells within that sample | Independent centering removes the relative scale between samples |
| Within defined strata, then aligned | Can address known differences in cell composition | Requires enough shared information and a defensible alignment reference |
Worked Example: Centering Factors Jointly or Separately
Suppose two hypothetical samples each contain two cells. A preliminary method gives positive factors \(0.8\) and \(1.0\) in sample A, and \(1.6\) and \(2.0\) in sample B. Assume for this example that the preliminary factors estimate technical exposure on one shared scale, and that the samples have a valid common reference. Compare joint geometric-mean centering with separate geometric-mean centering.
The geometric mean of all four factors is \( (0.8\times1.0\times1.6\times2.0)^{1/4} = 2.56^{1/4} \approx 1.265 \). Dividing each factor by this common value gives joint-centered factors approximately \(0.6325,\ 0.7906,\ 1.265,\) and \(1.581\), respectively.
The ratio of the two sample geometric means before centering is \( \sqrt{1.6\times2.0}/\sqrt{0.8\times1.0} = \sqrt{3.2/0.8} = 2 \). The common centering preserves that ratio: sample B's factors remain twice sample A's corresponding factors.
If each sample is centered separately, sample A's geometric mean is \(\sqrt{0.8\times1.0}\approx0.8944\), and sample B's is \(\sqrt{1.6\times2.0}\approx1.789\). The centered factors in each sample become \(0.8944\) and \(1.118\). Both samples now have geometric mean 1, so this procedure has deliberately removed the twofold difference in their sample-level scale. Separate centering is suitable only if that loss of cross-sample scale is intended or if a later, justified alignment restores it.
Cell Composition Can Distort the Scale
A sample is a mixture of cells, and genes can differ sharply between cell types. If one sample contains more cells of a type with high RNA content or high expression of many genes, total counts can rise even when the technical sequencing exposure per cell has not changed. A factor estimator that relies on total counts or on a broad majority assumption may then interpret a biological mixture difference as a technical scale difference.
The relevant question is not whether composition differs in general, but whether the estimator's reference remains valid under that difference. A stable reference gene can remain useful across changing mixtures if its expression is genuinely stable in the cells being compared. Conversely, a gene that is stable within each cell type may not be stable in a sample-wide mixture when the proportions of those types change.
Worked Example: Total Counts and a Composition Shift
Consider two hypothetical samples, each with two cells and two genes: a reference gene H and a cell-type marker M. In sample A, the cells have counts \((H,M)=(100,20)\) and \((100,20)\), for a sample total of \(240\). In sample B, a twofold technical exposure increase doubles the reference-gene counts to \(200\) per cell. Its two cells have counts \((200,20)\) and \((200,180)\), for a total of \(600\). The higher marker count in the second cell represents a composition-related biological difference, not a further increase in technical exposure.
The sample-wide total-count ratio is \(600/240=2.5\). If treated as the technical exposure ratio, it would suggest that sample B has 2.5 times the exposure of sample A. But the stable reference-gene counts indicate a ratio of \(200/100=2\), the stated technical ratio. The extra factor \(2.5/2=1.25\) in the total-count comparison reflects the change in marker counts.
This calculation does not prove that H is a valid reference in a real experiment; that is an assumption requiring biological and technical support. It shows why totals and factor estimates based on broad expression summaries need scrutiny when cell composition differs. If no credible reference exists, the data alone may not separate exposure from a global or mixture-driven expression shift.
Detection Differences and Shared Information
Sparse counts create a second comparability problem: samples may not share enough observed genes to support the same reference. Many factor estimators rely, directly or indirectly, on ratios or summaries across genes. If the usable genes differ substantially between samples, each factor may be driven by a different subset with a different expression profile. A shared numerical output can then conceal an unstable or biased comparison.
A low-depth sample may have many zeros simply because molecules were not observed. Zeros do not all have the same meaning: absence of a count is not, by itself, evidence that the gene is biologically absent. At the same time, treating every zero as if it supplied an ordinary expression ratio can be invalid for methods whose ratios require positive values. The estimator's treatment of zeros and the genes that remain informative must be examined.
Worked Example: A Factor Ratio Based on Too Few Shared Genes
Suppose two hypothetical samples have four candidate reference genes with expected counts in sample A of \(10, 20, 40,\) and \(80\). Under a twofold exposure increase and stable expression, sample B would be expected to have \(20, 40, 80,\) and \(160\). Imagine that only the fourth gene is detected in both samples, with counts 80 in A and 160 in B; the other candidate genes are zero in B.
The observed ratio for the one shared detected gene is \(160/80=2\), consistent with the assumed exposure ratio. But this agreement is not strong evidence of reliable cross-sample estimation: the estimate rests on one gene, and the example presumes that gene is stable. If that gene were instead biologically doubled in B as well as subject to the twofold exposure increase, its count ratio would be \(4\), and a one-gene estimate would confound the biological and technical changes.
A practical conclusion is not to replace the factor estimate with the number 2 from this hypothetical setup. It is to inspect how many genes actually support the comparison, whether they are representative and plausibly stable, and whether estimates are sensitive to removing individual genes. If shared information is inadequate, cross-sample comparability should be treated as uncertain rather than inferred from a numerical factor alone.
Diagnostics, Alternatives, and Limitations
Before choosing a scope, state the intended comparison. Within-sample cell comparisons may tolerate sample-specific centering; comparing expression levels across experimental samples requires a common reference or a clearly stated alternative scale. Then evaluate whether the assumptions behind that scale are credible. Useful checks include:
- Compare factor distributions by sample and examine whether their relative centers align with independent indicators of sequencing exposure.
- Check whether factor estimates track cell-type proportions, marker abundance, or total counts in ways that suggest composition is driving the scale.
- Assess the number and expression range of genes contributing information across samples, and whether a small subset dominates.
- Re-estimate factors with plausible reference sets or estimation scopes. Large changes indicate that the claimed scale depends heavily on the chosen assumptions.
- Inspect normalized expression for remaining association with sequencing depth and for suspicious loss of known sample-level or cell-type differences.
These are diagnostics, not proofs that normalization has recovered a uniquely correct scale. For example, weak association between normalized counts and depth is reassuring about one technical pattern, but it does not establish that a biological global shift has been preserved. Likewise, stable factors across several estimators do not guarantee validity if those estimators share the same incorrect reference assumption.
The main alternatives answer different questions. Joint factor estimation is preferable when a common reference is credible and cross-sample scale matters. Separate estimation is defensible for within-sample analyses when independently centered scales are acceptable, but it should not be presented as preserving between-sample exposure differences. Stratified estimation can address known mixture structure, but requires careful alignment. External spike-ins can provide additional information about technical exposure when their use and behavior are suitable; they do not automatically resolve every design or measurement problem.
A Practical Decision Sequence
Decide whether the analysis compares cells within samples, or requires expression on a common scale across samples.
Specify which genes, spike-ins, or distributional assumption allows technical exposure to be distinguished from biological expression.
Ask whether changing cell mixtures or sparse, non-overlapping detection makes the reference less credible or less informative in some samples.
Estimate jointly when a shared scale is justified; otherwise consider within-sample or stratified estimation, and test sensitivity to reasonable alternatives.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Why can independently centering factors within each sample make between-sample comparisons invalid?
- Two samples have a total-count ratio of 3, while a plausible stable reference set suggests a ratio of 2. Give one explanation for the discrepancy and one diagnostic.
- In a sparse sample, only two genes contribute to a cross-sample factor estimate. What does that imply about the interpretation of the factor?
- Under what condition is joint estimation a defensible choice for a cross-sample comparison?
Solutions. (1) Separate centering forces each sample's factor center to the same value, discarding its relative sample-level scale; a common alignment is needed for between-sample interpretation. (2) A composition shift or broad biological expression change may inflate totals relative to technical exposure. Check cell-type proportions and whether factors track marker or total-count differences. (3) The estimate is weakly supported and sensitive to the behavior of those genes; inspect their stability and assess sensitivity to excluding either gene. (4) Joint estimation is defensible when adequate shared information supports a credible reference across samples and the estimator is not confounded by composition or global biological shifts.