Rescaling and Centering Deconvolution Size Factors

Why Center Deconvolution Size Factors?

Deconvolution estimates cell-specific size factors by combining information from pooled counts. The resulting factors are intended to describe relative differences in effective library scale across cells. Their numerical level, however, can depend on the scale convention used to report them. Centering chooses a reference level—commonly a geometric mean or median of one—so the factors are easier to interpret and use.

Centering is a rescaling, not a new estimate of each cell’s relative depth. It does not make an unstable factor stable, remove biological shifts, or establish that the normalization assumptions hold. The earlier material on deconvolution and unstable factors motivates checking factor quality first; this tutorial focuses on the consequences of choosing a common scale for factors that are usable and positive.

Definition: Let \(\tilde{s}_i>0\) be an estimated deconvolution size factor for cell \(i\), for \(i=1,\ldots,n\). Geometric-mean centering divides every factor by \(G=(\prod_{i=1}^n\tilde{s}_i)^{1/n}\), producing \(s_i^{G}=\tilde{s}_i/G\). Median centering divides every factor by \(M=\operatorname{median}(\tilde{s}_1,\ldots,\tilde{s}_n)\), producing \(s_i^{M}=\tilde{s}_i/M\). Each convention gives the set of factors a reference level of one.

The factors are positive and dimensionless in this setting. If the factors are used to scale counts, a normalized count for gene \(g\) in cell \(i\) is \(Y_{gi}/s_i\), where \(Y_{gi}\) is the observed count. Centering can therefore change the numerical scale of normalized counts even though it preserves the relative factors among cells.

The Invariance Behind Centering

The key fact is that dividing every factor by the same positive constant leaves any pairwise ratio unchanged. This is why centering changes the reference level without changing the factor-based comparison between two cells.

\frac{s_i^*}{s_j^*} = \frac{\tilde{s}_i/c}{\tilde{s}_j/c} = \frac{\tilde{s}_i}{\tilde{s}_j}, \qquad c>0.

The same common scaling has a predictable effect on normalized counts. If \(s_i^*=\tilde{s}_i/c\), then

\frac{Y_{gi}}{s_i^*} = \frac{Y_{gi}}{\tilde{s}_i/c} = c\,\frac{Y_{gi}}{\tilde{s}_i}.

Thus every normalized value is multiplied by the same \(c\), regardless of the cell or gene. Within a given gene, ratios of normalized values between cells are unchanged. Their absolute numerical level is not. In an offset-based count model, using \(\log s_i^*=\log\tilde{s}_i-\log c\) changes every log offset by the same constant; that shift can be absorbed by the model intercept when the model includes one. The choice of factor scale should still be reported because fitted intercepts and displayed normalized values depend on the chosen convention.

Geometric-Mean and Median Centering

Geometric-mean centering is natural when multiplicative differences are the focus. Taking logarithms turns the geometric mean into an arithmetic mean on the log-factor scale: \(\log G=n^{-1}\sum_i\log\tilde{s}_i\). Therefore the centered log factors have mean zero, and the centered factors have geometric mean one.

Median centering sets the middle factor, or the average of the two middle factors when there are an even number of cells, to one. Because the median is less influenced by an extreme value than the geometric mean, it can be a useful reference when a few factors are unusually large or small. Neither choice repairs an outlying factor; it only determines how the factor set is scaled.

Worked Example: Centering Three Deconvolution Factors

Suppose a hypothetical analysis gives raw factors \(1,\ 2,\ 8\). Calculate the geometric-mean-centered and median-centered values, then compare the relative factor for the third cell versus the first.

The geometric mean is \(G=(1\cdot2\cdot8)^{1/3}=\sqrt[3]{16}=2.519842\ldots\). Dividing each original factor by this unrounded value gives

(s_1^G,s_2^G,s_3^G) = \left( \frac{1}{2.519842\ldots}, \frac{2}{2.519842\ldots}, \frac{8}{2.519842\ldots} \right) \approx (0.3969,\ 0.7937,\ 3.175).

The geometric mean of these centered factors is one. The median of the raw factors is \(M=2\), so median centering gives \((0.5,\ 1,\ 4)\), whose median is one. Under either convention, the third-to-first factor ratio is \(8/1=8\). For example, geometric-mean centering gives \(3.175/0.3969\approx 8\), with the small displayed difference due only to rounding.

Use full precision for the centering constant when calculating centered values, and round only the reported results. Rounding the geometric mean too early can change a final digit even when the formula is correct.

Worked Example: How an Extreme Factor Affects the Reference

Consider hypothetical raw factors \(0.5,\ 1,\ 1,\ 2,\ 20\). Find the two centers and explain what each does with the large factor.

The median is \(M=1\), so median-centered factors remain \((0.5,\ 1,\ 1,\ 2,\ 20)\). The median is one, while the largest factor remains twenty times the reference. Median centering has not corrected or removed that large value.

The product of the raw factors is \(20\), so \(G=20^{1/5}\approx1.82056\). Geometric-mean-centered factors are approximately \((0.2746,\ 0.5493,\ 0.5493,\ 1.099,\ 10.99)\). Their geometric mean is one. The large factor pulls the geometric-mean reference upward compared with the median reference, so the other centered values are smaller. Yet the ratio between the largest and smallest remains \(20/0.5=40\) under either centering.

From Factors to Normalized Counts

A common rescaling changes the numeric level of normalized counts in a predictable way. The next example makes that distinction concrete: the normalized values shift together, while their ratios across cells do not.

Worked Example: Comparing Normalized Values Under Two Centers

Suppose three hypothetical cells have raw factors \(1,\ 2,\ 8\), and one gene has counts \(20,\ 40,\ 80\) in those cells. Calculate the gene’s normalized values using raw factors and then using geometric-mean-centered factors.

Using the raw factors gives \(20/1=20\), \(40/2=20\), and \(80/8=10\). For this set of factors, \(G=\sqrt[3]{16}=2.519842\ldots\), so the centered factors are approximately \(0.3968503,\ 0.7937005,\ 3.174802\). Dividing the counts by these unrounded centered factors gives approximately \(50.40,\ 50.40,\ 25.20\).

Each centered-factor normalized value is \(G\) times its raw-factor counterpart: for example, \(20G=50.40\) to four significant figures. The ratio of the first cell’s normalized value to the third’s is \(20/10=2\) before centering and \(50.40/25.20=2\) after centering. The displayed normalized counts changed scale, but their relative comparison did not.

Choosing and Checking a Centering Convention

For a single set of positive factors, geometric-mean and median centering differ only by a common positive multiplier. A practical choice can be based on the intended convention and the factor distribution, not on a claim that one convention fixes poor estimates.

Choice	Reference set to one	Useful property	Important caution
Geometric-mean centering	Geometric mean	Centers log factors to mean zero	More affected than the median by extreme factors
Median centering	Median	Less affected by a few extreme factors	Extreme factors remain and may still require investigation
No additional centering	Depends on the estimator’s scale	Retains the original reporting convention	Numerical levels may be less convenient to interpret or compare

Check that the factors are finite and positive before applying either definition. A zero makes the geometric mean zero and prevents division by that center; negative values are not valid multiplicative size factors. Do not silently add a pseudocount to factors to force a center. Instead, investigate why the estimates are invalid and return to the estimation procedure. Also inspect unusually large or small factors: centering cannot make an unstable estimate reliable.

A global center must be used when the aim is to retain relative comparisons across all cells in a set. If separate groups or batches are centered independently, each group is divided by a different constant. Within-group factor ratios remain unchanged, but between-group factor ratios are multiplied by the ratio of those constants. That can alter cross-group normalized-count comparisons. Such separate centering is not merely a harmless change of one common reference for the full data set.

Derivation: Why Both Centers Set Their Reference to One

For geometric-mean centering, substitute \(s_i^G=\tilde{s}_i/G\) into the definition of the geometric mean. Since \(G\) is positive,

\left(\prod_{i=1}^n s_i^G\right)^{1/n} = \left(\prod_{i=1}^n\frac{\tilde{s}_i}{G}\right)^{1/n} = \frac{(\prod_{i=1}^n\tilde{s}_i)^{1/n}}{G} = \frac{G}{G} = 1.

For median centering, multiplying every observation by the same positive constant preserves their order. Every ordered factor is divided by \(M\), so the middle factor—or the average of the two middle factors—is also divided by \(M\). The centered median is therefore \(M/M=1\). These arguments establish the reference-level properties; the pairwise-ratio calculation above establishes why neither operation changes relative cell factors.

Common Mistakes and Limitations

Calling centering a correction. Dividing by a reference constant does not repair unstable or biased factor estimates; it only changes their common scale.
Assuming normalized counts stay numerically identical. If factors are divided by \(c\), normalized counts are multiplied by \(c\). Their relative comparisons remain the same under a common rescaling, but their displayed values change.
Centering groups independently without considering cross-group comparisons. Different group-specific constants can change relative factor scales between groups. Use one shared center when comparisons across the full set require a common reference.
Rounding the center before calculating. Calculate with the original factor precision and round the reported factors at the end.
Treating the median as an outlier remedy. Median centering is less influenced by extremes when setting the reference; it does not remove or validate an extreme factor.

The central limitation is that centering addresses scale convention, not the validity of normalization. Whether the factors adequately capture unwanted technical variation requires separate assessment; a convenient center is not evidence that the normalization assumptions hold.

Check Your Understanding

Answer each question, then compare with the solutions below.

Raw factors are \(2,\ 6,\ 18\). What are their geometric-mean-centered values?
For any positive factors, explain why dividing all factors by one positive constant preserves every pairwise factor ratio.
If factors are divided by \(4\), how do normalized counts \(Y_{gi}/s_i\) change?
Why can separately median-centering two experimental groups change comparisons between the groups?

Solutions. (1) The geometric mean is \((2\cdot6\cdot18)^{1/3}=216^{1/3}=6\), giving centered factors \(1/3,\ 1,\ 3\), approximately \(0.3333,\ 1,\ 3\). (2) For any cells \(i,j\) and \(c>0\), \((\tilde{s}_i/c)/(\tilde{s}_j/c)=\tilde{s}_i/\tilde{s}_j\); the common divisor cancels. (3) If \(s_i^*=s_i/4\), then \(Y_{gi}/s_i^*=4Y_{gi}/s_i\), so every normalized count is multiplied by four. (4) Each group is divided by its own median, and those medians need not be equal. The resulting cross-group factor ratio includes the ratio of the group-specific medians, so it may differ from the ratio on the original shared scale.