Diagnosing Unstable Deconvolution Factors

Why Diagnose Deconvolution Factors?

Deconvolution uses information from groups of cells to estimate cell-level size factors when individual cells have many zero counts. A pooled count can be less sparse than its component counts, but the cell-level estimates still depend on the pooled ratios being informative and on the chosen pools providing enough independent equations. A numerical answer is not necessarily a reliable answer: some factors may be weakly determined, unusually sensitive to a few genes, or dependent on a particular pool construction.

This tutorial focuses on recognizing those problems. It does not prescribe a universal acceptable range of factor values. Factors are relative quantities, and their overall scale is arbitrary; what matters for these diagnostics is whether their relative pattern is credible and supported by the pools. The introductory material on pooling and the earlier derivation of cell-level factors provide the basic construction used here.

Definition: Let \(f_i\) denote the positive deconvolution size factor for cell \(i\), and let \(A_{pi}=1\) if cell \(i\) belongs to pool \(p\), and \(0\) otherwise. If \(q_p\) is the estimated size factor for pool \(p\), the deconvolution equations are \(q_p \approx \sum_i A_{pi}f_i\). In matrix form, \(q \approx Af\). A factor estimate is unstable when modest, defensible changes in pooled information or pool construction produce substantial changes in one or more relative cell factors.

These equations are useful only under assumptions. The pooled ratios must provide reasonable estimates of relative pooled exposure: there must be enough informative genes, and systematic expression-composition differences must not dominate the ratios. Pool membership must also give enough independent information to distinguish the cell factors. Finally, the counts and pool definitions must be correctly matched; a mistaken cell assignment can create a mathematically solvable but incorrect system. Diagnostics can reveal warning signs, but cannot establish that all assumptions hold.

Read the Factor Distribution as a Diagnostic

Inspect the positive factors and their logarithms. A long tail, one or two extreme values, a gap between groups, or a strong association between factor size and a technical feature such as the number of detected genes can flag a problem. The log scale is helpful because multiplicative differences become additive: a factor twice as large differs by \(\log(2)\) on that scale. Compare factors with cell-level library sizes and other available technical summaries, but do not assume that the factor distribution must be symmetric or unimodal.

An unusual pattern has several possible explanations. It may reflect an unstable estimate, a genuine difference in RNA content, a mixture of cell types, or confounding between biology and technical depth. Therefore, an extreme factor is a prompt to investigate its supporting pools and counts, not a reason by itself to remove a cell. Conversely, a smooth-looking distribution does not prove that the estimates are reliable: a poorly identified system can return plausible-looking values.

Worked Example: A Consistent Set of Pooled Equations

Suppose three hypothetical cells have pooled size-factor estimates \(q_{12}=3.02\), \(q_{23}=5.08\), and \(q_{13}=4.00\) for pools containing cells \(\{1,2\}\), \(\{2,3\}\), and \(\{1,3\}\), respectively. Assume the pooled ratios are informative and the factor scale is fixed consistently across these pools. The equations are \(f_1+f_2=3.02\), \(f_2+f_3=5.08\), and \(f_1+f_3=4.00\).

Adding the first and third equations and subtracting the second gives \(2f_1=3.02+4.00-5.08=1.94\), so \(f_1=0.970\). Similarly, \(2f_2=3.02+5.08-4.00=4.10\), giving \(f_2=2.05\); and \(2f_3=5.08+4.00-3.02=6.06\), giving \(f_3=3.03\).

The estimates are positive and satisfy all three pooled equations. This is an internal consistency check, not a guarantee of accuracy: these data provide exactly enough independent equations to solve for three factors, leaving no extra equation to reveal disagreement. Additional pools would provide useful redundancy for assessing whether pooled estimates agree.

Pool Coverage, Rank and Conditioning

Coverage asks how often, and in what combinations, each cell appears in the pool set. A cell absent from every pool cannot be estimated from those equations. But simple inclusion counts are not sufficient: two cells can appear equally often while their factors remain indistinguishable if they always appear together. The relevant mathematical question is whether the rows of \(A\) provide independent information about the columns, that is, whether \(A\) has full column rank for the cell factors being estimated.

Even full rank may not be enough in practice. If columns of \(A\) are nearly redundant, small errors in the pooled estimates can cause large changes in the solved factors. This is poor conditioning. Pool coverage should therefore be checked both for missing or unevenly represented cells and for whether the pool combinations separate cells sufficiently.

Diagnostic principle: If the estimated pooled vector is \(q+\varepsilon\), where \(\varepsilon\) represents pooled-ratio error, then the factor error is amplified when the pool system is weakly identified. For a full-column-rank matrix, the least-squares solution is \( \hat f=(A^\mathsf{T}A)^{-1}A^\mathsf{T}(q+\varepsilon)\), so its deviation from the solution for \(q\) is \( (A^\mathsf{T}A)^{-1}A^\mathsf{T}\varepsilon\). Redundant, well-conditioned pools can reduce the effect of noise; nearly dependent pool patterns can magnify it.

This gives a practical justification for inspecting both coverage and the fit of the pooled equations. If a system has more pools than cells, residuals \(q-A\hat f\) show whether the estimated factors reproduce the pooled estimates. Large, structured residuals suggest inconsistent pooled ratios, an unsuitable model for the pool sums, or errors in pool construction. A small residual is reassuring only in combination with informative pools: an exactly solvable but underdetermined system can fit perfectly while leaving factors ambiguous.

Worked Example: Complete Coverage but Insufficient Information

Consider four hypothetical cells and three pools: \(\{1,2\}\), \(\{2,3\}\), and \(\{3,4\}\), with pooled estimates \(q_{12}=3\), \(q_{23}=5\), and \(q_{34}=7\). Every cell is covered, but there are only three equations for four unknown factors. Let \(f_2=t\). The equations require \(f_1=3-t\), \(f_3=5-t\), and \(f_4=2+t\). Every \(t\) between 0 and 3 yields nonnegative factors and exactly reproduces all three pooled estimates. Thus coverage alone does not identify the four factors.

Now add a pool \(\{1,3\}\) with \(q_{13}=4\). Its equation gives \((3-t)+(5-t)=4\), so \(8-2t=4\), and \(t=2\). The resulting factors are \(f_1=1\), \(f_2=2\), \(f_3=3\), and \(f_4=4\), which satisfy the original equations as well. The additional pool supplies an independent constraint. In real data, the diagnostic is not merely whether an extra pool exists, but whether its membership pattern contributes independent and sufficiently precise information.

Sensitivity to Pool Construction

A direct stability check is to repeat estimation using defensible alternative pool constructions—for example, different pool groupings or sizes while keeping the underlying cells and analysis choices comparable. Compare relative factors across constructions, allowing for an arbitrary common multiplicative scale. For two cells \(i\) and \(j\), the ratio \(f_i/f_j\) is unchanged by a common rescaling, so a change in that ratio is evidence of a change in their estimated relative factors. Large shifts localized to cells that appear in few pools, or that occur in pools with weak pooled counts, are especially informative.

Sensitivity is not a contest to select whichever construction gives the most attractive distribution. Alternative constructions should remain valid for the data and the pooling assumptions. If only one particular design gives a plausible answer, that dependence is itself a warning. When estimates differ, examine which pools changed, which cells they share, and whether a small set of genes or a composition difference could have driven the pooled ratios.

Worked Example: A Cell Factor Changes Across Pool Designs

For three hypothetical cells, one pool design gives \(q_{12}=2.70\), \(q_{23}=4.50\), and \(q_{13}=3.60\). Applying the three-cell equations gives \(f_1=(2.70+3.60-4.50)/2=0.900\), \(f_2=(2.70+4.50-3.60)/2=1.80\), and \(f_3=(4.50+3.60-2.70)/2=2.70\). A second reasonable design gives pooled estimates \(3.00\), \(6.00\), and \(5.00\), respectively. It yields \(f_1=(3.00+5.00-6.00)/2=1.00\), \(f_2=(3.00+6.00-5.00)/2=2.00\), and \(f_3=(6.00+5.00-3.00)/2=4.00\).

Compare relative factors using cell 1 as the reference. Cell 2 relative to cell 1 is \(1.80/0.900=2.00\) under the first design and \(2.00/1.00=2.00\) under the second. Cell 3 relative to cell 1 changes from \(2.70/0.900=3.00\) to \(4.00/1.00=4.00\), a 33.3% increase relative to the first ratio. This selective change suggests that cell 3’s estimate is more sensitive than cell 2’s. It does not identify the cause by itself; inspect the pools involving cell 3, their pooled counts, and the genes contributing to their ratios.

Locate Likely Causes and Choose a Response

Several failure modes can produce unstable factors. Very sparse pooled counts or few shared informative genes make pooled ratios noisy. A small number of highly abundant genes may dominate a ratio, particularly if their expression differs across cells or conditions. A pool set with weak coverage or near-redundant membership patterns can amplify that noise. Finally, pooling cells with markedly different expression compositions can undermine the assumption that pooled ratios estimate relative exposure in a way that supports cell-level deconvolution.

A useful diagnostic sequence is to check the data and pool definitions, inspect factor distributions and coverage, assess whether the equations are identifiable, and then compare valid alternative pool constructions. Where possible, investigate pooled-ratio support: for example, whether estimates are based on a reasonable set of genes or are driven by very few. A problem in pooled evidence calls for revisiting pool design or the ratio information; a problem in identification calls for more informative pool combinations. If factors track a biological grouping, do not automatically remove that pattern: first consider whether it reflects real RNA-content differences or a violation of the assumptions used to interpret the factors.

Other size-factor approaches, including median-ratio and TMM estimators, can serve as comparisons, but agreement does not prove that any method’s assumptions are satisfied. Disagreement can help locate a data feature that affects the methods differently; it is not, on its own, a rule for choosing a winner. Deconvolution is particularly useful when sparse cell-level counts make pooling helpful, but it relies on informative pooled ratios and a well-designed system. Diagnostics cannot recover information absent from the pools or distinguish technical and biological causes without additional evidence.

Common Mistake

Common mistake: “Every cell occurs in at least one pool, and the solver returned positive factors, so the estimates are stable.” Coverage and positivity do not establish identifiability, precision, or robustness. Check whether pool combinations provide independent information, whether pooled estimates are mutually consistent, and whether relative factors change under reasonable alternative pool constructions.

Check Your Understanding

Answer each question, then compare with the solutions below.

Can a pool design cover every cell yet fail to identify all cell factors? Explain briefly.
For three cells, pooled estimates are \(q_{12}=4\), \(q_{23}=7\), and \(q_{13}=5\). Calculate the three factors from the pairwise pool equations.
Two constructions give \(f_1=1, f_2=2, f_3=3\) and \(f_1=2, f_2=4, f_3=9\). Which relative comparison is stable, and which changes?
Name two possible causes of an extreme factor and one diagnostic that would help distinguish them.

Solutions. (1) Yes. Cells can all be included while pool patterns are redundant; the equations may not have full column rank. (2) \(f_1=(4+5-7)/2=1\), \(f_2=(4+7-5)/2=3\), and \(f_3=(7+5-4)/2=4\); these reproduce all three pooled sums. (3) The ratio \(f_2/f_1\) is 2 in both constructions, while \(f_3/f_1\) changes from 3 to \(9/2=4.5\). The relative estimate for cell 3 is sensitive. (4) Possible causes include noisy pooled ratios from sparse counts, an influential gene, weak pool identification, or a genuine RNA-content difference. Inspect pool coverage and pooled-ratio support, and compare valid alternative pool constructions; the factor alone cannot identify the cause.