Why Assess Factors When Expression Truth Is Unknown?
A size factor is a cell-specific adjustment used to make counts more comparable across cells with different effective sequencing depths. In real single-cell data, the true expression level of each gene in each cell is generally unknown. There is therefore no direct way to subtract the factors from counts and compare the result with a complete ground-truth expression matrix.
Assessment instead asks whether candidate factors behave consistently with several kinds of evidence: their distribution across cells, their relationship to observed depth, the behavior of genes expected to be stable, and the preservation of plausible cell-level variation. These checks are complementary. A factor set can pass one check and fail another, and a plausible diagnostic pattern is not proof that the factors are correct.
Let \(K_{gi}\) denote the observed count for gene \(g\) in cell \(i\), let \(L_i=\sum_g K_{gi}\) be that cell's total observed count, and let \(s_i>0\) be a candidate size factor. A simple normalized value is \(K_{gi}/s_i\). Multiplying every factor by the same positive constant changes the numerical scale of these values but not their relative comparisons across cells. Factors are commonly centered to a convenient scale, such as a median of one; centering is a change of scale, not evidence that the factors are accurate.
Factor Distributions and Depth Associations
Begin by inspecting the distribution of the factors, often on the log scale. Look for extreme values, unexpectedly wide spread, clusters, or patterns aligned with batch, condition, or cell type. A boxplot or histogram can reveal cells that merit investigation. But factors need not have a particular shape: multiple cell populations, genuine depth differences, and technical variation can all produce structure. An outlier is a flag for review, not an automatic reason to remove a cell or alter its factor.
Next, compare factors with observed library sizes \(L_i\), for example in a scatterplot of \(\log(s_i)\) against \(\log(L_i)\). Some positive association is often expected because cells with more observed counts tend to have greater effective depth. However, the relationship need not be exact or have slope one. Total counts are affected by expression composition: a few abundant genes can raise \(L_i\), and broad biological changes can alter totals. This is the compositional concern emphasized in the earlier material on total-count scaling and global RNA shifts. A strong factor-depth association is not by itself proof of success, nor is a weaker-than-expected association automatically a failure.
A more targeted check uses genes expected to have stable expression across the cells being compared. If a stable gene has expected count proportional to the effective depth factor, its normalized counts should not systematically increase with that factor. The reasoning is conditional: stability must be a reasonable assumption for those genes in this population, and the candidate factor must capture the relevant count-scale differences. Sparse counts add noise, so inspect patterns across several suitable genes rather than treating one gene's zeros or fluctuations as decisive.
Worked Example: Interpreting the Factor Distribution
Suppose four hypothetical cells have candidate factors \(0.5, 1, 2,\) and \(4\). The median is \((1+2)/2=1.5\). Centering these factors to median one divides each by 1.5, giving approximately \(0.333, 0.667, 1.333,\) and \(2.667\).
The raw largest-to-smallest factor ratio is \(4/0.5=8\), and centering leaves that ratio unchanged: \(2.667/0.333\approx 8\). Thus the cells differ substantially in their estimated effective count scales. The centered values have a median of one, but this convention does not make them more or less accurate.
Now suppose a second candidate set is \(0.2, 0.8, 1.2,\) and \(6\). Its median is 1, and its range ratio is \(6/0.2=30\). The last cell is a possible outlier worth checking against its library size, batch, and count profile. The distribution alone cannot tell whether the factor reflects a real technical difference, a biological composition difference, or an estimation problem.
Worked Example: Checking Normalized Control Counts Against Depth
Consider four hypothetical cells with candidate factors \(s_i=(0.5,1,2,4)\). A gene regarded as a stable control has observed counts \(K_{gi}=(10,20,40,80)\). Its normalized values are
The raw counts rise with the factors, while the normalized values do not. This is consistent with the working relationship \(K_{gi}\approx s_iq_g\), here with \(q_g=20\).
For comparison, suppose another candidate set for the same cells is \((0.5,1,2,8)\). The normalized control counts become \((20,20,20,10)\), since \(80/8=10\). The final normalized value is lower, not stable with the others. If this pattern repeats across several suitable controls, it raises concern that the last factor is too large for those cells. A single control gene, especially one with sparse or noisy counts, would not establish that conclusion.
Controls and Preservation of Cell-Level Variation
Control genes are useful only to the extent that their stability assumption is credible for the comparison at hand. A gene stable across one cell type may differ across another; a gene affected by condition or cell state is not a neutral control for that comparison. When possible, use multiple controls chosen for the relevant biological context and check whether their normalized behavior agrees. Apparent stability in a small dataset is weak evidence because sparse counts can conceal real differences.
Normalization should also remove technical scale differences without erasing plausible biological variation. Compare known cell groups, replicates, or well-supported marker patterns before and after applying candidate factors. The aim is not to preserve every raw difference: differences caused by depth should be reduced. Instead, ask whether expected distinctions remain and whether the normalization creates group shifts in genes believed to be stable.
This check is delicate. If factors are strongly associated with a biological group, normalization can reduce a real group difference when that difference contributes to the count scale. Conversely, failing to adjust for depth can make a technical difference look biological. Without ground truth, the data alone may not distinguish these explanations. Use experimental design and established biological expectations as context, and report uncertainty rather than treating one visual pattern as definitive.
Worked Example: Checking Both a Control and a Marker
Four hypothetical cells comprise two conditions, A and B. Their candidate factors are \((0.5,1,2,4)\), with the first two cells in A. A stable control gene has counts \((10,20,40,80)\); a marker expected to be higher in B has counts \((10,20,80,160)\).
Dividing by the candidate factors gives control values \((20,20,20,20)\). The marker values are \((20,20,40,40)\), retaining a twofold difference between the condition averages: A has mean \((20+20)/2=20\), and B has mean \((40+40)/2=40\). This is consistent with reducing the common depth scale while preserving the marker difference.
Now consider alternative factors \((0.5,1,4,8)\), which are more strongly aligned with condition B. The control values become \((20,20,10,10)\), while the marker values become \((20,20,20,20)\). The marker difference disappears, and the control now has a twofold lower average in B. This combination makes the alternative factors questionable under the stated expectations. Still, the example's expectations are assumptions: in an actual dataset, the marker might not truly differ, or the control might not be stable. The arithmetic diagnoses consequences of the candidate factors; it cannot supply missing biological truth.
A Practical Assessment and Its Limits
A useful assessment combines several views rather than reducing the decision to a single correlation or cutoff. For each candidate factor set:
Confirm factors are positive, record how they were centered, and inspect their spread and extreme values across cells and relevant groups.
Plot factors against total counts and inspect cells that depart markedly from the general pattern, accounting for possible compositional and biological effects.
Check normalized counts for multiple genes expected to be stable. Look for repeated residual trends with factor, depth, batch, or condition.
Check whether plausible group or marker differences remain and whether controls acquire unexpected group patterns after normalization.
These checks complement, rather than replace, the earlier material on comparing median-ratio and TMM factors or diagnosing deconvolution-factor instability. Those approaches provide candidate estimates and related diagnostics; assessment here asks how the candidates behave against broader expectations when no full expression truth is available. A disagreement between methods is a reason to investigate sensitivity to the choice of factors, not an automatic vote for one method.
The main limitations are fundamental. Control genes may not be stable, observed library size is composition-dependent, sparse counts make cell-level patterns noisy, and biological changes can be confounded with technical depth. Centering factors cannot resolve any of these issues. Nor does a flat normalized control profile prove that all genes are properly normalized. The strongest conclusion is usually comparative: one candidate set is more consistent with the available diagnostics and assumptions than another.
Check Your Understanding
Answer each question using the stated assumptions. Brief solutions follow the questions.
- Why does multiplying every size factor by the same positive constant change normalized count units but not relative comparisons among cells?
- A control gene has counts \((6,12,24)\) and candidate factors \((0.5,1,2)\). Calculate its normalized values and describe what they suggest.
- Why is a factor-depth association that is weaker than the association between total counts and themselves not automatically evidence of failure?
- Candidate factors remove an expected marker difference but cause stable controls to differ by condition. What should be concluded, and what remains uncertain?
Solutions. (1) If each factor is multiplied by \(c>0\), then \(K_{gi}/(cs_i)=(1/c)(K_{gi}/s_i)\); every normalized value is rescaled equally, so ratios between cells are unchanged. (2) The values are \(6/0.5=12\), \(12/1=12\), and \(24/2=12\). They are consistent with counts proportional to the factors for this control, but one gene does not validate the factors. (3) Total counts depend on expression composition and biology as well as depth, so the factor need not track totals perfectly. (4) The pattern makes the factors questionable under the stated expectations. It does not prove the marker difference is real or the controls are stable; those assumptions need biological and design-based support.