bioRxivBioinformaticsPreprint
Overcoming systematic errors caused by log-transformation of normalized single-cell RNA sequencing data
Abstract
Applying a log-transformation to normalized expression values is one of the most common procedures in exploratory analyses of single-cell RNA sequencing (scRNA-seq) data. Normalization removes systematic biases in sequencing coverage between cells, while the log-transformation ensures that downstream computational procedures operate on relative rather than absolute differences in expression. We show that the log-transformation can introduce systematic errors when cells vary in sequencing coverage, leading to spurious non-zero differences in expression and artificial population structure in simulations. We observe similar effects in real scRNA-seq data where the difference in transformed values between groups of cells is not an accurate proxy for the log-fold change. We provide some practical recommendations to overcome this effect and analytically derive an expression for a larger pseudo-count that controls the transformation-induced error to a specified threshold.
- organism: mouse
- assay: scRNA-seq
- research focus: clustering
- research focus: trajectory inference
- research focus: differential expression
- research focus: benchmarking
- approach: statistical model
- approach: simulation
In-depth summary
Claude Opus 5.5 · xhigh effort · 8 Nov 2026 · from the full text of version 1 (31 Aug 2018) · CC BY 4.0
Log-transforming normalized scRNA-seq counts with a pseudo-count of 1 creates spurious expression differences and artificial clusters or trajectories when size factors vary; a derived larger pseudo-count bounds this error.
Differences between log-normalized values are widely used as log-fold-change proxies and as input to clustering and trajectory methods, most of which, unlike count-based models such as BASiCS or zinbwave, need transformed data. The paper shows how the gap between mean-log and log-mean values turns size-factor variation into spurious signal, consistent with the detection-rate correlations reported by Hicks et al., and supplies a pseudo-count that bounds it per gene.
Key results
ShownWith the usual pseudo-count of 1, genes that are not DE acquire non-zero differences in mean log-normalized expression between cells with different size factors, largest when the factors differ and one or both are below 1.
Taylor approximation and negative binomial simulations (two groups of 10,000 cells, 10 iterations; Figure 1): a 10-fold size-factor gap can give a difference of 1 in mean log2 values, an artificial 2-fold change. In 10X ERCC spike-in data, top vs bottom 20% of droplets by size factor differ systematically in mean log-values but not in the log-fold change of means (Figure 2a, b). · 2 Quantifying the discrepancy; 7.1 Simulated log-fold changes for non-DE genes; 7.2 Log-fold changes in real data sets; Figure legends
ShownIn real cell clusters, the difference in mean log-values departs from the log-fold change computed from mean normalized expression, so log-scale effect sizes can mislead comparisons between groups.
Brain and PBMC data processed with scater and deconvolution size factors, then clustered by shared nearest neighbours on PCs; for the clusters with the highest and lowest median size factor, the two effect-size estimates disagree (Figure 2c, d). The comparison is shown graphically, with no summary statistic. · 2 Quantifying the discrepancy; 7.2 Log-fold changes in real data sets; Figure legends
SuggestedSize-factor differences alone, with no true structure, can produce apparent clusters or trajectories in PCA of log-normalized values, because spurious per-gene differences accumulate in cell-to-cell distances.
Poisson counts for 1000 non-DE genes in 500 cells: cells with means of 0.1 vs 10 separate in PCA (Figure 3a); cell means drawn from Uniform(0.1, 5) give a continuous, trajectory-like spread (Figure 3b). · 2 Quantifying the discrepancy; 7.3 Simulations with artifical structures; Figure legends
ShownA larger pseudo-count derived from the size factors (Equation 3) caps the spurious log-difference for non-DE genes at a threshold τ = 1/(8ρ); with ρ = 1 the cap is 0.125 (about 0.18 on the log2 scale), and the resulting ĉ stays modest in real data.
Simulated maxima sit near the bound at low NB dispersion and below it at higher dispersion (Figure 5); ĉ also keeps the ERCC error near the bound (Figure 6b). With ρ = 1, ĉ was 1.36 (416B), 4.05 (mouse brain) and 2.22 (PBMC), with 12636, 649 and 217 genes respectively having mean normalized expression above ĉ. · 4 Choosing a larger pseudo-count; Figure legends
SuggestedVariance-stabilizing alternatives do not remove the artefact: square-root and DESeq2 variance stabilizing transformation (VST) values also differ between groups that differ only in size factor.
Simulation of 10,000 non-DE genes in 500 cells, Poisson or NB (dispersion 1), in two equal groups whose cell scaling factors were 0.1 vs 10: group means of the transformed values diverge in a mean-dependent way under each transformation (Figure 4). In the brain and PBMC data, 81% and 89% of non-zero counts fall below 5, a range where the author argues no transformation can stabilize variance. · 2 Quantifying the discrepancy; 7.4 Simulations with different transformations; Figure legends
Methods, data and reproducibility
The paper derives a second-order Taylor approximation for the expected log-normalized value to show how unequal size factors create differences for non-DE genes. It tests this with negative binomial and Poisson simulations and with real 10X ERCC, 416B, brain and PBMC data, then, assuming negative binomial counts, derives a pseudo-count ĉ that caps the spurious difference at a chosen threshold.
- 10X Genomics ERCC data set: gel emulsion beads carrying ERCC spike-in transcripts captured in droplets; libraries with total counts above 100 kept
- 416B scRNA-seq data set, processed with a published workflow
- Mouse brain scRNA-seq data set, processed with a published workflow · mouse
- Peripheral blood mononuclear cell (PBMC) scRNA-seq data set, processed with a published workflow
- Simulated negative binomial counts for non-DE genes in two groups with fixed size factors (Figures 1 and 5) · n = two groups of 10,000 cells; 10 iterations
- Simulated Poisson counts with no true structure (Figure 3) · n = 1000 genes in 500 cells
- Simulated Poisson or negative binomial counts for comparing transformations (Figure 4) · n = 10,000 genes in 500 cells
- Methods & tools
- R 3.5.0, Bioconductor, scater, DropletUtils, BiocFileCache, DESeq2 (variance stabilizing transformation), edgeR, TREAT, deconvolution size factors, shared nearest neighbour clustering, PCA, 10X Genomics droplet platform
- Software & environment
- R 3.5.0 with Bioconductor packages: scater 1.9.14, DropletUtils v1.1.8, BiocFileCache v1.5.5, DESeq2 v1.21.16.
- Reproducibility gaps
- The 416B, brain and PBMC data sets are referenced only through a published workflow, with no accessions or cell counts in the text; no hardware or runtime is stated.
Limitations (6)
- For a single pseudo-count across a whole data set with many groups or continuous trajectories, the right s1 and s2 are not obvious; the author suggests the smallest and largest size factors (or robust equivalents such as the 5th and 95th percentiles). · stated by the authors
- The theoretical bound sits close to simulated maxima only at low dispersion; at higher dispersion it is less accurate, though still conservative. · stated by the authors
- Using ĉ in the initial exploratory clustering is hard to justify, since there is no obvious acceptable threshold for error in cell-to-cell distances. · stated by the authors
- Correlations between log-expression and detection rate in real data mix technical effects with biology, such as differences in RNA content between cell types, so they need careful interpretation. · stated by the authors
- The cluster and transformation simulations use an extreme contrast (0.1 vs 10), and the trajectory simulation draws cell means from Uniform(0.1, 5). All of them use only non-DE genes, so how much artefact appears alongside real biological signal at typical size-factor spreads is not measured. · noted in this summary
- Real-data evidence for brain and PBMC is graphical with no summary statistic, and the proposed check of clusters with count-based DE (edgeR) or TREAT is recommended but not demonstrated on any data set. · noted in this summary
Read this if you cluster, order or compare cells using log-normalized scRNA-seq values and need to know when uneven size factors create artefacts and how to pick a pseudo-count that limits them.
- Citations
- 53
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