There are two major ways to use Z-score transformation with gene expression data.
- Gene-wise Z-score transformation, in which each gene is standardized across samples to show its relative changes among samples
- Sample-wise Z-score normalization, in which the distribution of gene expression values within each sample is standardized
Gene-wise Z-scores are widely used in RNA-Seq heatmaps and similar visualizations. The limitations of this approach are discussed in detail in Why Z-scores Should Not Be Used Uncritically in RNA-Seq Heatmaps. In contrast, this article focuses on sample-wise Z-score normalization.
Sample-wise Z-score normalization was used during the microarray era as a way to place gene expression data obtained from different samples onto a common scale. However, the values being standardized, and therefore the meaning of the normalization, differ between two-color and one-color microarrays. In two-color arrays, the signal ratio or log ratio between two samples is standardized. In one-color arrays, by contrast, the signal values of all genes measured from a single sample are standardized. Furthermore, as the number of measurable genes increased and large numbers of unexpressed or low-expression genes became included, the assumption that all gene values could be treated as a single distribution and standardized using one mean and one standard deviation became increasingly difficult to justify.
In this article, we examine what sample-wise Z-score normalization meant in two-color microarrays, what changed with the transition to one-color arrays, and what problems arise when it is applied to comprehensive gene expression datasets.
In Two-Color Microarrays, the Ratio between Two Samples Was Measured
In early spotted two-color microarrays, two samples were labeled with Cy5 and Cy3 and measured on the same spot. For each gene, the resulting value was the Cy5-to-Cy3 signal ratio or its logarithm.
At the same time, early spotted microarrays had many technical problems.
- Differences in the amount of DNA deposited at each spot
- Irregularities in spot shape and position
- Local background signal
- Differences in the properties of Cy5 and Cy3 dyes
- Differences in labeling efficiency
- Differences in scanning conditions
- Signal-intensity-dependent bias
Because this technical variability was large, it was difficult to compare results from different microarrays without some form of normalization. Under these conditions, adjusting the center and spread of the log-ratio distribution for each microarray had practical value, and sample-wise Z-score normalization can be understood as one method that emerged from this way of thinking.
Standardizing Technical Variation Can Also Hide Biological Variation
However, sample-wise Z-score normalization does not standardize only technical variation. Consider, for example, a time-course experiment measuring cell differentiation. At an early stage of differentiation, changes in expression may be small for many genes, and the distribution of log ratios between two samples may be concentrated around zero. As differentiation progresses, however, many genes may become activated or repressed, causing the distribution of expression changes to broaden substantially.
In this case, the increase in the width of the distribution is itself biological information indicating that the transcriptional state has changed more extensively as differentiation progresses. However, if the data at each time point are Z-score normalized separately, both the narrow distribution at the early stage and the broad distribution at the later stage are forced to have a standard deviation of 1. As a result, small early changes are relatively amplified, while larger late changes are compressed. The information that overall expression changes became larger as differentiation progressed is lost through sample-wise Z-score normalization.
Even if Z-score normalization makes the data easier to compare, this does not mean that only measurement error has been selectively removed. Technical differences and biological differences are both being standardized by the same procedure.
In One-Color Microarrays, the Meaning of the Values Being Z-scored Changes
In one-color microarrays such as Affymetrix GeneChip arrays, one sample is applied to one microarray, and the signal value of each gene is measured. Therefore, when one-color signal values are Z-score normalized, what is standardized is not the expression ratio between two samples but the distribution of log-transformed signal values across all genes within a single sample, forcing it to have a mean of 0 and a standard deviation of 1. Although the same term, sample-wise Z-score normalization, is used, the meaning of the values being standardized is fundamentally different between two-color and one-color arrays.
Why It Appeared Reasonable for Early GeneChips
Early Affymetrix GeneChip platforms such as HG-U95 improved on many of the technical problems that had been major issues in spotted two-color microarrays, including irregular spot formation and dye-related differences. In addition, compared with later comprehensive microarrays, they contained fewer genes, and the measured targets were biased toward relatively well-characterized genes for which sequence information was available at the time. As a result, the proportion of unexpressed or low-signal genes was relatively low.
In such datasets, log-transformed signal values often appeared to form a roughly unimodal distribution. Under those conditions, applying sample-wise Z-score normalization as a way to compare signal values from different microarrays on a common scale appeared to have some rationale. However, a distribution that looks unimodal does not mean that different samples inherently have the same expression distribution. Similar histogram shapes do not justify assuming that differences in the center or spread of the distributions are entirely technical.
There Is No Guarantee That Different Tissues Have the Same Expression Distribution
Consider comparing biologically distinct tissues such as brain, liver, muscle, and blood. These tissues differ both in which genes are expressed and in which genes are expressed strongly. One tissue may contain a small number of tissue-specific genes expressed at extremely high levels, whereas another may express a larger number of genes across a broader range. There is no biological basis for assuming that the distributions of all gene expression values in such samples should inherently be identical.
Nevertheless, if every sample is forced to have a mean of 0 and a standard deviation of 1, differences in expression distributions that arise from the biological characteristics of each tissue are also removed as if they were technical differences. Sample-wise Z-score normalization is not a method for confirming that samples have the same expression distribution. It is a method that forces their expression distributions to have the same center and spread.
As Coverage Increases, the Distribution Is No Longer a Single Population
In microarrays that measure larger numbers of transcripts, such as the GeneChip HG-U133 Plus 2.0, the measured targets include not only genes that are clearly expressed in the sample, but also many unexpressed genes and probe sets with signals close to the detection limit. As a result, the distribution of log-transformed signal values contains both a background or low-signal population and a clearly expressed population, producing a more complex shape in which the low-signal and expressed groups overlap rather than forming a single peak.
Figure: Example of how gene expression value distributions change as GeneChip platforms become more comprehensive and as analysis shifts toward RNA-Seq.
From left to right, GeneChip HG-U95, GeneChip HG-U133 Plus 2.0, and RNA-Seq are shown. Scatter plots between samples are shown in the upper row, and the distributions of values within each sample are shown in the lower row. HG-U95 shows a relatively compact, approximately unimodal distribution, whereas HG-U133 Plus 2.0 includes more low-signal probe sets, making the distinction between the low-signal and expressed populations clearer. RNA-Seq contains even more genes with zero or low Counts, producing discrete peaks toward the low-value end. The values from these measurement methods are not on the same scale; the purpose of this figure is to illustrate conceptually how the assumption that all genes can be treated as one homogeneous distribution becomes increasingly difficult to justify as measurement coverage expands.
Treating such a distribution as one homogeneous population and calculating Z-scores using a single mean and standard deviation is problematic. The mean calculated from the combined populations represents neither the center of the low-signal population nor the center of the expressed population. Likewise, the standard deviation is influenced not only by the variability within each population, but also by the distance between the centers of the two populations. Therefore, Z-scores calculated from that mean and standard deviation do not necessarily have a clear biological interpretation.
A Z-score can, of course, be calculated even when the original distribution is not normally distributed. However, it does not necessarily make sense to treat multiple populations with different characteristics as a single distribution and represent them using one mean and one standard deviation.
Quantile Normalization Was Also Proposed for One-Color Microarrays
For one-color microarrays, quantile normalization was proposed as a method that aligns sample distributions even more strongly than sample-wise Z-score normalization. Whereas sample-wise Z-score normalization adjusts each sample to a mean of 0 and a standard deviation of 1, quantile normalization transforms the entire distribution of values in each sample to the same shape. As a result, after quantile normalization, all samples have the same distribution when viewed as histograms. At first glance, this may appear to be an even stronger and more complete form of normalization than Z-score normalization.
However, forcing distributions to align more strongly does not necessarily make gene expression data more appropriate for comparison.
The Fundamental Contradiction in Quantile Normalization
For quantile normalization to be appropriate, the differences in expression distributions observed among samples must be assumed to be primarily technical, while the underlying biological expression distributions are assumed to be broadly similar. However, this creates a fundamental contradiction. If the underlying expression distributions are already similar across samples and the measurements are stable, applying quantile normalization will not substantially change the data. In such cases, there is little need to force the entire distributions to match.
On the other hand, when sample distributions differ substantially and quantile normalization appears necessary, those differences may reflect not only technical factors but also biological differences such as tissue type, cell state, differentiation, disease, or drug response. In that situation, the assumption that the underlying expression distributions should be the same no longer holds. In other words, when the assumptions of quantile normalization are satisfied, strong distribution correction is often unnecessary; when strong distribution correction appears necessary, those assumptions may already be violated.
The same issue applies to sample-wise Z-score normalization, which forces the mean and standard deviation to match. Neither method determines whether differences among sample distributions are technical or biological. They simply transform the center, spread, or entire shape of the distributions without first determining why those differences exist.
Reshaping Distributions Does Not Remove Bias within the Data
Even if quantile normalization forces the histograms of all samples to have the same shape, batch effects within the data are not necessarily removed. Batch effects do not simply raise or lower all genes uniformly. They commonly affect particular gene groups, specific signal ranges, or probe sets with particular sequence characteristics. Therefore, even after the overall distributions of samples have been made identical, PCA or clustering may still separate samples according to experimental date, reagent lot, measurement facility, instrument, or other technical factors rather than biological condition.
The same is true for sample-wise Z-score normalization. Even after the mean and standard deviation have been standardized, the internal structure of the data—such as which genes are biased and in which direction—remains. In the figure above, even if the overall sample distributions shown in the lower panels were transformed to have the same shape, there is no guarantee that the gene-specific biases and internal structures visible in the scatter plots in the upper panels would be corrected appropriately. Simply aligning the distributions of samples does not remove technical bias within the data.
The goal of normalization is to reduce technical differences while preserving biological differences. However, a procedure that only aligns distributions cannot determine which differences are technical and which are biological. As a result, technical biases that should be removed may remain, while biological differences that should be preserved may be lost. When evaluating a normalization method, the important question is not whether the normalized distributions look neatly aligned, but whether the differences that should disappear have actually disappeared and the differences that should remain have been preserved.
To make that judgment, it is also important to understand which parts of gene expression data can be treated as reliable signal in the first place, where the noise lies, and how reproducibility and dynamic range should be interpreted. These fundamental aspects of gene expression data quality are discussed in more detail in What Is Good Gene Expression Data? | Understanding Noise, Reproducibility, and Dynamic Range.
These Are No Longer Standard Normalization Methods for RNA-Seq
Since the widespread adoption of RNA-Seq, sample-wise Z-score normalization and quantile normalization have generally no longer been used as standard normalization methods. RNA-Seq analysis typically uses normalization approaches that account for factors such as library size and RNA composition. However, if sample-wise Z-score normalization is deliberately applied to RNA-Seq data, essentially the same concerns arise as when it is applied to comprehensive microarrays such as the GeneChip HG-U133 Plus 2.0.
The Problem of Forcing Distributions to Match Still Remains
This issue can also be observed in actual RNA-Seq expression data. In the analysis example using GSE159751, quantile normalization was applied to FPKM distributions that differed substantially among samples. After normalization, the histograms appeared to show that the sample distributions had been aligned to the same shape. However, clustering revealed that simply matching the distributions did not produce data that were necessarily suitable for biological comparison.
Again, making distributions look the same and making data genuinely comparable are not the same thing.
Summary: Aligning Distributions Is Not the Goal of Normalization
Sample-wise Z-score normalization had some practical value in early microarray analysis, when technical variability was large and there was a need to place different datasets onto a common scale. However, as microarray technology moved from two-color to one-color systems, the meaning of the values being Z-score normalized changed substantially. As measurement became more comprehensive, unexpressed and low-signal genes increasingly coexisted with clearly expressed genes, making the assumption that all genes could be treated as one distribution increasingly difficult to justify.
Quantile normalization was later used to force entire distributions into the same shape, but aligning distributions does not necessarily remove batch effects or gene-specific technical biases within the data. The purpose of normalization is not to make sample distributions look neat and identical. It is to reduce technical differences while preserving the biological differences that are relevant to the analysis.
When sample distributions differ so strongly that a distribution-aligning procedure appears necessary, the assumption that those differences can safely be treated as technical should itself be questioned. Many methods have been proposed in the pursuit of better normalization, and new methods will continue to appear and become widely used.
However, the fact that post-normalization histograms look neatly aligned is not enough to conclude that the procedure was appropriate. PCA, clustering, individual gene expression patterns, sample metadata, and other information should be considered together to determine whether the differences that should be removed have disappeared and whether the differences that should be preserved remain.
