Why the Choice of Reference Matters
A size factor is not a direct measurement of “technical variation” simply because it is used to normalize counts. Its interpretation depends on what was measured and on assumptions connecting that measurement to the counts of interest. Endogenous factors use the observed genes themselves as a reference; spike-in-based factors use control molecules introduced into the experiment. These references can behave differently when cells vary in total RNA content or when the gene-expression composition shifts.
The central question is therefore not which factor is universally better, but which source of variation each factor can identify under the experimental design. Spike-ins can anchor counts to a known input, while endogenous methods rely on enough genes being stable, or changing in balanced ways, to define a meaningful reference. Neither approach automatically separates technical and biological effects.
Two Sources of Information
Let \(Y_{ig}\) be the observed count for endogenous gene \(i\) in cell \(g\). An endogenous size-factor method estimates a relative factor \(f_g\) from the pattern of these counts across genes and cells. Median-ratio, TMM, and deconvolution methods differ in how they construct the reference, but all require assumptions about the endogenous expression pattern. The earlier material on identifiability explains why a common multiplicative change in all endogenous expression cannot, by itself, be distinguished from a change in sampling scale.
For spike-ins, let \(Z_{kg}\) be the observed count for spike-in species \(k\) in cell \(g\), and let \(A_{kg}\) be its known input amount. A simple model treats the count as Poisson with mean proportional to both the input amount and a cell-specific sampling factor \(s_g\).
The factor \(s_g\) is only relative to the measurement scale in this model. A common multiplier applied to every cell’s factors can be absorbed into the units of the inputs or the sequencing scale. As with endogenous size factors, comparisons generally concern ratios between cells or factors centered to a specified reference.
Estimating a Factor from Spike-Ins
Write \(Z_{+g}=\sum_k Z_{kg}\) for the total spike-in count and \(A_{+g}=\sum_k A_{kg}\) for the total input amount in cell \(g\). Under the independent Poisson model, the likelihood for \(s_g\), apart from terms that do not depend on \(s_g\), is proportional to \(s_g^{Z_{+g}}\exp(-s_g A_{+g})\). When \(A_{+g}>0\) and \(Z_{+g}>0\), the log-likelihood derivative gives an interior maximum:
The second derivative is \(-Z_{+g}/s_g^2<0\), so this stationary point is a maximum. The estimate is the total observed spike-in count divided by the total known input amount. This derivation is specific to the stated model; a different count distribution or a more elaborate recovery model can yield a different estimator.
If \(A_{+g}=0\), no spike-in was supplied for that cell. The likelihood then contains no information about \(s_g\), so the factor cannot be estimated from spike-ins. When the Poisson model holds, the total count has mean \(s_gA_{+g}\), and the approximate relative standard error of the positive estimate is \(1/\sqrt{Z_{+g}}\). Thus, few total spike-in reads can produce a noisy factor even when the model is appropriate.
For positive estimates, factors can be placed on a relative scale by dividing by their geometric mean across the cells being compared. This preserves pairwise ratios and makes the geometric mean one. That operation is defined only when every factor being centered is positive; it does not solve the all-zero case.
Worked Example: Estimating and Centering Positive Spike-In Factors
Suppose two spike-in species are added to each of two hypothetical cells, with input amounts \(A_{1g}=50\) and \(A_{2g}=50\) units for each cell. The observed counts are \(Z_{11}=18,\ Z_{21}=22\) for cell 1 and \(Z_{12}=9,\ Z_{22}=11\) for cell 2. Assume the inputs are known, both controls undergo the same proportional sampling within each cell, and the independent Poisson model is a reasonable approximation. Both cells have positive total input and positive total counts.
For cell 1, \(Z_{+1}=18+22=40\) and \(A_{+1}=50+50=100\), so \(\widehat{s}_1=40/100=0.400\). For cell 2, \(Z_{+2}=9+11=20\) and \(A_{+2}=100\), so \(\widehat{s}_2=20/100=0.200\).
The geometric mean of these estimates is \(\sqrt{0.400\times0.200}=\sqrt{0.0800}=0.2828\). Dividing each estimate by it gives centered factors \(0.400/0.2828=1.414\) and \(0.200/0.2828=0.7071\). Their ratio is \(1.414/0.7071=2.000\), matching the uncentered ratio \(0.400/0.200=2.000\). Under the model, cell 1 has twice the estimated spike-in sampling factor of cell 2; the centered values are relative factors, not absolute probabilities of capture.
What Endogenous Factors Assume
An endogenous factor is informative about technical sampling only if the observed endogenous pattern contains a suitable reference. For example, a median-ratio factor depends on a sufficiently large, informative set of genes whose relative expression pattern is stable enough that systematic shifts do not dominate the reference. TMM uses trimmed weighted log ratios, and deconvolution uses pooled counts, but neither removes the basic need for an identifiable endogenous reference. If most genes truly increase together, a procedure that treats the typical gene as stable can interpret that biological shift as a larger factor.
A useful conceptual model is \(E[Y_{ig}]=s_g\mu_{ig}\), where \(s_g\) is a sampling scale and \(\mu_{ig}\) is the underlying abundance for gene \(i\) in cell \(g\). This factorization alone does not identify both quantities: multiplying all \(\mu_{ig}\) in a cell by a constant and dividing \(s_g\) by the same constant leaves the expected counts unchanged. Endogenous methods resolve this ambiguity through their reference assumptions; their factors therefore need not represent technical sampling alone.
Spike-ins replace the endogenous reference with known added material, but their interpretation also rests on assumptions. The spike-ins must be introduced at a stage that exposes them to the technical variation of interest. If added after a source of cell-specific loss, they cannot report that earlier loss. They must also behave sufficiently like the endogenous molecules during the relevant stages. Differences in length, structure, polyadenylation, or other assay-specific properties can make their recovery unrepresentative.
Worked Example: A Global Endogenous Shift
Consider two hypothetical cells with observed endogenous counts of 100 and 100 for two genes in cell A, and 200 and 200 for the same genes in cell B. Suppose spike-in inputs total 100 units per cell and spike-in counts total 40 in each cell. Assume for this example that the spike-ins and endogenous molecules share a multiplicative technical sampling factor, and that library composition does not create a cell-specific competition effect on the spike-in counts. The counts are treated as suitable summaries under that simplified model.
The spike-in estimates are \(\widehat{s}_A=40/100=0.400\) and \(\widehat{s}_B=40/100=0.400\); their ratio is 1. Under an endogenous reference in which the two genes are treated as stable in relative expression, the typical endogenous count in B is twice that in A, so the corresponding relative factor is 2. A normalization based on that factor would attribute the twofold difference to scale.
If the biological question concerns a genuine twofold increase in the abundance of both genes in cell B, then the endogenous assumption is inappropriate for that question: it absorbs the global shift into the factor. Under the stipulated model, the spike-in comparison does not show a change in technical sampling and can preserve that global difference. This conclusion depends on the example’s explicit no-competition assumption; it is not guaranteed simply because spike-ins were used.
Experimental Conditions and Diagnostics
A major practical complication is that sequencing reads are finite. With fixed-depth sequencing, a larger endogenous contribution to a library can reduce the fraction of reads assigned to spike-ins, even if the absolute spike-in recovery is unchanged. In that setting, spike-in counts can depend on endogenous library composition as well as on technical recovery. The simple model \(E[Z_{kg}]=A_{kg}s_g\) is then incomplete unless its sampling factor appropriately accounts for this effect. Spike-ins can be especially useful when their addition, processing, and measurement are designed to make the technical scale identifiable, but they do not automatically eliminate compositional effects.
Before comparing factors, check the experimental record and the observed data. Ask when spike-ins were added, whether the same known amounts were added to each cell, whether any cell has no usable spike-in reads, and whether the controls span enough molecules to estimate a stable factor. Examine total spike-in counts and estimated factors for extreme values, and compare them with library size, cell quality, and other technical indicators. A strong disagreement between spike-in and endogenous factors is a signal to investigate, not a reason to select whichever result is preferred.
The following patterns can help locate a problem:
- Very low spike-in totals or zero totals: factor estimates may be imprecise or lie at the zero boundary. Log-centering is not defined for a zero estimate.
- Spike-in factors track endogenous totals closely: this can reflect a shared technical effect, but may also indicate sequencing competition or a relation between cell RNA content and spike-in read allocation.
- Large factor changes driven by a few controls: inspect species-specific counts and recovery. A common-factor model may be implausible if different spike-ins behave very differently.
- Endogenous factors change while spike-in factors do not: a biological shift is one possible explanation, but differences in recovery, sampling, or model fit also need consideration.
The Poisson model is a useful derivation, not a guarantee of independent, equally reliable counts. Extra variability, batch-specific recovery, contamination, or control-specific behavior can make the simple estimate unstable. More spike-in molecules can improve precision only when they are measured under a valid design; they cannot repair a mismatch between the controls and endogenous molecules.
Worked Example: Handling an All-Zero Spike-In Total
Suppose a hypothetical cell receives two spike-in species, each at 50 known units, so \(A_{+g}=100\). Neither species is observed: \(Z_{1g}=0\) and \(Z_{2g}=0\). The input is positive, but the total observed count is \(Z_{+g}=0\).
The likelihood as a function of \(s_g\) is proportional to \(\exp(-100s_g)\). For every \(s_g>0\), \(\exp(-100s_g)<1\), while its value at \(s_g=0\) is 1. Therefore the maximum over \(s_g\geq0\) is at \(\widehat{s}_g=0\). The interior derivative equation would read \(0/s_g-100=0\), or \(-100=0\), which has no solution.
This estimate cannot be included in geometric-mean centering because \(\log(0)\) is undefined. A researcher could use a separately specified zero-handling method or obtain additional information, but replacing zero by an arbitrary small number and proceeding as though it were the likelihood estimate would be incorrect. The cell also warrants investigation for failed recovery or insufficient sequencing.
Comparison and Practical Choice
| Feature | Spike-in-based factor | Endogenous factor |
|---|---|---|
| Reference | Known amount of introduced control material | Observed endogenous genes or pooled endogenous counts |
| Main identifying assumption | Controls experience relevant technical sampling proportionally and inputs are known | The endogenous reference pattern is stable enough to distinguish scale from biology |
| Potential advantage | Can provide an external anchor when broad endogenous shifts are plausible | Uses the RNA molecules of interest and can be efficient when the reference assumptions hold |
| Important vulnerability | Timing, control mismatch, low counts, or library competition can distort the factor | Global or widespread expression shifts can be absorbed into the estimated scale |
There is no universal winner. When the aim is to compare per-cell RNA abundance and substantial global biological shifts are plausible, a validated external reference can address an ambiguity that endogenous factors cannot resolve alone. When spike-ins are absent, sparse, added too late, or poorly representative, an endogenous method may be more defensible if its stability assumptions are credible. In either case, report the factor’s construction and the interpretation it supports; a normalized count is not automatically an absolute abundance.
A common mistake is to say, “Spike-ins are external, so their factors must be technical and are always preferable.” That reasoning skips the assumptions. Controls added after a relevant loss cannot measure it, and a fixed-read sequencing process can make spike-in read counts depend on endogenous library composition. The correction is to specify the technical process the factor is intended to capture, verify that controls experience that process, and assess whether their measured counts are affected by other changes.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A cell has total spike-in input \(A_{+g}=200\) units and total observed spike-in count \(Z_{+g}=50\). What is the model-based estimate of \(s_g\)?
- Why is geometric-mean centering not defined if one estimated spike-in factor is zero?
- In what circumstance can endogenous factors absorb a biological change rather than only a technical sampling difference?
- Name two experimental conditions that can make spike-in factors fail to measure the technical variation of interest.
- What should be concluded if spike-in and endogenous factors disagree strongly?
Solutions. (1) \(\widehat{s}_g=50/200=0.250\). (2) The logarithm of zero is undefined, so the log-based geometric-mean calculation cannot include that factor; a specified zero-handling method is required. (3) If most or all endogenous genes undergo a common shift and the method assumes their typical expression is stable, the shift can be assigned to the size factor. (4) Examples include adding controls after the technical loss of interest, using controls with recovery unlike endogenous molecules, or having fixed-depth sequencing make spike-in read allocation depend on endogenous library composition. (5) Treat the disagreement as a diagnostic signal; examine design, composition, recovery, and factor stability before choosing an interpretation.