MAD-FC: A fold change visualization with readability, proportionality, and symmetry

Bruce A Corliss; Yaotian Wang; Francis P Driscoll; Heman Shakeri; Philip E Bourne

doi:10.1371/journal.pone.0304632

. 2024 May 31;19(5):e0304632. doi: 10.1371/journal.pone.0304632

MAD-FC: A fold change visualization with readability, proportionality, and symmetry

Bruce A Corliss ^1,^2,^*, Yaotian Wang ³, Francis P Driscoll ¹, Heman Shakeri ¹, Philip E Bourne ^1,²

Editor: Stephen R Piccolo⁴

PMCID: PMC11142613 PMID: 38820396

Abstract

We propose a fold change transform that demonstrates a combination of visualization properties exhibited by log and linear plots of fold change. A fold change visualization should ideally exhibit: (1) readability, where fold change values are recoverable from datapoint position; (2) proportionality, where fold change values of the same direction are proportionally distant from the point of no change; (3) symmetry, where positive and negative fold changes of the same magnitude are equidistant to the point of no change; and (4) high dynamic range, where datapoint values are distinguishable across orders of magnitude within a fixed plot area and pixel resolution. A linear visualization has readability and partial proportionality but lacks high dynamic range and symmetry (because negative direction fold changes are bound between [0, 1] while positive are between (1, ∞)). Log plots of fold change have partial readability, high dynamic range, and symmetry, but lack proportionality because of the log transform. We outline a new transform, named mirrored axis distortion of fold change (MAD-FC), that extends a linear visualization of fold change data to exhibit readability, proportionality, and symmetry (but still has the limited dynamic range of linear plots). We illustrate the use of MAD-FC with biomedical data using various fold change plots. We argue that MAD plots may be a more useful visualization than log or linear plots for applications that do not require a high dynamic range (less than 8 units in log2 space).

Introduction

Bioinformatics research often requires analyzing datasets that are expressed in units of fold change [1]. This measurement type represents the ratio of the sample mean of an experiment group divided by a control group. Fold change data is visualized to summarize the spread of the dataset and identify interesting datapoints- often those with the largest magnitude in either direction of change. Typically, scientists visualize fold change using a log [2] or a linear transform, the latter of which simply presents raw fold change values [3]. Each transform has a unique set of properties that may be advantageous or disadvantageous depending on the particular purpose of the visualization. We propose a transform, mirrored axis-distortion of fold change (MAD-FC), and demonstrate its use with visualizing real research data. Our proposed fold change transform may enhance the understanding and interpretation of fold change data in scientific research because it combines some of the more useful properties of linear and log transforms.

To assist with our discussion of fold change visualization properties, we introduce the term point of no change, which is the value in a fold change transform space that denotes no change and separates negative fold changes from positive fold changes. For a log transform of fold change measurements, the point of no change is zero; for a linear transform, the point of no change is one. We also define a linear encoding for fold change, fold change units, that represents the number of fold changes from the point of no change. With this encoding, the raw fold changes of (2, ½, 3, 1/3) are mapped to (1, -1, 2, -2) fold change units, respectively. Fold change units, f_U, can be mathematically defined as:

f_{U} (x) = {\begin{matrix} x - 1 & x \geq 1 \\ 1 - \frac{1}{x} & x \in (0, 1) \\ u n d e f i n e d & o t h e r w i s e \end{matrix},

(1)

where x is the raw fold change measurement. With these terms defined, we can now propose the properties of a useful fold change visualization and then evaluate various types of plots (summary in Fig 1A):

Fig 1 — (A) Table summarizing visualization properties for each of the plot types. (B) Visualizing a dataset of positive fold changes with a linear (upper) and log2 (lower) scale to illustrate dynamic range (each datapoint is a staggered line, alternating between black and another color to aid with datapoint identification). (**C-E**) A dataset of fold changes ranging from 1/6 to 6 (equal to -5 to 5 in fold change units, see legend on right) is visualized with (C) linear, (D) log, and (E) MAD plots to illustrate their characteristics. Visualization properties of plot types are assessed with: readability based on the transform and the units of the axis tick labels, proportionality from linear fits between the point of no change and extreme datapoints (grey lines), symmetry from boxes that match points of same magnitude (grey shaded boxes to right of each plot), and dynamic range based on whether datapoints are mapped to linear space (medium) or log space (high). (D-E) Inner y-axis tick labels are raw transform units; outer y-axis tick labels are back-transformed to original fold change units.

Readability: a visualization exhibits readability if it allows the observer to easily recover the original fold change values from the datapoints within the visualization. To fulfill this condition, there must be a clear and direct mapping between the values of the visualized datapoints and their spatial locations. This property can be split into two steps: (1) extraction, where a datapoint value is extracted from the plot coordinate system based on its relative position to the nearest axis tick labels, and (2) conversion, where the extracted value is converted to the underlying fold change value (if necessary). For example, for a plot using raw logarithmic axis tick labels (e.g. inner y-axis labels of Fig 1D), the log transform must be reversed on each datapoint extracted from the plot’s coordinate system to obtain the original fold change measurement. A visualization that is readable facilitates rapid extraction and conversion of its datapoints with minimal effort from the observer.
Proportionality: a visualization exhibits proportionality if the fold change values are proportionally distant from the point of no change within the plot. This proportional relationship does not have to be identical for positive versus negative fold changes. Proportionality allows for direct comparisons between the magnitudes of positive fold changes (and separately for negative fold changes). This property holds true if there is an exact linear relationship between the transformed fold changes and corresponding fold change units for both positive fold changes and negative fold changes. Proportionality can be visually assessed by generating a test dataset of fold change datapoints and plotting the transformed value versus its fold change units. Within this plot, we connect a line between the largest positive fold change and the point of no change. If all positive fold change datapoints fall on the line, then positive fold changes are designated as proportional for the transform. The same procedure can be used to examine negative fold changes. If both fold change directions are proportional, then we designate the transform as having proportionality. Note that the slopes for these two lines are not required to be one (meaning they do not need a direct 1:1 correspondence to fold change units), and the slope does not have to be the same between directions (that is examined with the next property).
Symmetry: a fold change plot has symmetry if every datapoint would remain equidistant to the point of no change if its fold change direction were hypothetically reversed. This property helps scientists visually compare the magnitudes of positive and negative fold changes. Symmetry of a transform can be visually assessed by measuring the distance between synthetically generated pairs of fold changes of opposite direction with the same magnitude. For example, a fold change of 2 should be equidistant to the point of no change compared to a fold change of ½, a fold change of 3 should be equidistant compared to a fold change of 1/3, and so forth.
Dynamic Range: fold change measurements can span many orders of magnitude, and for some applications it is important to clearly distinguish the positions of points across this range. We illustrate this by visualizing a dataset of fold changes ranging from a value of 1 to 2^9 in magnitude on a linear and log scale (Fig 1B). It is easy to perceive the distinct lines when the datapoints are visualized with a log2 transform, but the smallest fold changes are difficult to perceive on a linear scale. When data spans multiple magnitudes on a linear scale, large outlying data values overwhelm the axis spatial encoding, often leaving insufficient space to distinguish differences between small values (e.g. the crowding between small fold changes on the linear axis in Fig 1B). While the number of orders of magnitude that a linear scale can capture depends on the pixel resolution of the plot, image compression artifacts, plot area, visual acuity, and viewing distance, we estimate a plot with a linear scale can typically span 2⁸ units (approximately 10^2.5 units in base 10). Based on this basic characterization, we designate a log transform as having high dynamic range, while a linear transform has medium dynamic range. It is important to note that plots with log transforms are not universally better at distinguishing closely positioned points when compared to linear- there are instances where two datapoints are located closer together in a log plot than a corresponding linear plot and therefore harder to distinguish.

It is important to note that while these properties are explained in the context of spatial encodings, they could potentially be extended to color encodings. However, applying these encodings to color is complex because of the nonlinear relationship between color and the human eye’s spectral sensitivity [4]. We illustrate the potential of our transform for heatmaps in Fig 8, but a more detailed investigation of these properties applied to color encodings is beyond the scope of this study. Additionally, each of these visualization properties could be formally defined by mathematical relationships between fold changes and transformed outputs, but a more rigorous exploration of these properties is reserved for future research.

Fig 8 — Comparison of heatmaps with (A) linear, (B) log2, and (C) MAD fold change color mapping of differential gene expression of papaya leaves tissue after drought stress compared to control. Datapoints are annotated as significantly upregulated (Up, blue), significant downregulated (Down, red), or not statistically significant (NS, black) based on an FDR < 0.05 and a fold change greater than ± 1.

We evaluate these properties for linear plots, log plots, and MAD plots (summary found in Fig 1A). We refer to raw log plots as those with the raw log-transformed axis tick marks and labeled log plots as those with the tick marks back-transformed to the original fold change values (e.g. inner y-axis labels of Fig 1D compared to outer labels, respectively). The same notation is used for MAD plots. MAD plots denote any type of plot that uses the MAD-FC transformed fold changes for one of its visual encodings. Raw MAD plots display the raw MAD-FC transformed fold changes, while labeled MAD plots have the axis tick marks back-transformed to the original fold change units. In theory, these plot types are not mutually exclusive if both axes are included, but we discourage this approach because multiple axes can make interpretation more difficult.

A linear plot of fold change displays raw fold change values with a linear transform (Fig 1C). A linear plot of fold change is readable because extracting coordinates from a linear axis tick mapping can be done based on the relative position between tick marks, and no conversion is required since the axis tick marks in a linear plot are the original fold change values. Positive fold changes in a linear plot are proportional since they have a linear relationship to fold change units (Fig 1C, upper right quadrant), while negative fold changes are not proportional (lower left quadrant). This type of plot does not have symmetry since negative fold changes are compressed between [0, 1] while positive fold changes span between (1, ∞). Linear plots of fold change have medium dynamic range because they use a linear axis tick label mapping (see illustrative example in Fig 1B, and Fig 8 for an example using real data).

Raw log plots of fold change use axis tick labels in log units (Fig 1D, inner y-axis labeled Log₂(FC)). Extracting the value of a datapoint with this plot is simple because it can be linearly interpolated between axis tick marks (log units are linearly mapped to the axis scale). However, an observer must reverse the log transform to convert the datapoint to the original fold change value, which is a nontrivial process, and even more so for a collection of points. Therefore, a raw log plot is only partially readable. In contrast, a labeled log plot (Fig 1D, outer y-axis labeled FC) has a nontrivial process for extraction because the datapoint value cannot be estimated based on linear interpolation between axis tick marks (back-transformed tick marks are not linearly mapped to the axis scale since the underlying scale is not linear). A labeled log plot has simple conversion because the original fold change values are extracted from the plot directly. Log plots of both types do not have proportionality because of the nonlinearity of the log transform. Log plots are symmetrical, and the log scale gives them a high dynamic range.

A raw MAD plot (that displays the raw output of the MAD-FC transform) has the negative fold changes stretched out to match the scale of the positive fold changes (Fig 1E, inner y-axis labeled MAD-FC). Extracting the value of a datapoint with this plot is simple because the fold change value can be directly interpolated from the axis tick marks. However, the conversion process is nontrivial to obtain fold changes from datapoints since the MAD-FC transform must be reversed. Raw MAD plots are therefore partially readable. In contrast, a labeled MAD plot (Fig 1E, outer y-axis labeled FC) is fully readable because linear interpolation is used for datapoint extraction and axis tick labels use raw fold change units, with no conversion required. MAD plots are proportional because negative fold changes are distorted to match the proportionality of positive fold changes. MAD plots are symmetrical by design and have a medium dynamic range.

While raw log plots and labeled log plots have a trade-off in readability between extraction and conversion (Fig 1A), we note that there is no such trade-off between raw MAD plots and labeled MAD plots. We therefore strongly recommend to never use raw MAD plots to visualize fold change data, because the raw MAD-FC transform units will reduce the readability of the plot and potentially confuse the observer.

Methods

MAD-FC transform

We start with a fold change dataset with n measurements that are positive real numbers. To produce a visualization fold change with readability, proportionality, and symmetry, we will perform two transforms on the raw data and then reverse the transforms on the axis tick labels. For this explanation, we use a test dataset comprised of pairs of positive and negative fold changes of the same magnitude (Fig 2A). An actual dataset does not need matching pairs of datapoints; we use these to illustrate how symmetry is maintained with the transform. Notice that reversing the direction of a fold change is simply the reciprocal of its value (i.e., f(x) = 1/x). In order for a plot of fold changes to be symmetric, these pairs of points must become equidistant to the point of no change.

Based on the table in Fig 2A, we can define a transform that stretches the negative fold change values to match the spacing of the corresponding positive fold changes. We use the same transform that is used to reverse the direction of a fold change and then multiply by negative one. Since we only want to transform negative fold changes, we define a case equation that only transforms negative fold change measurements. We denote this mirror transform as f_M, where

f_{M} (x) = {\begin{matrix} x & x \geq 1 \\ - \frac{1}{x} & 0 < x < 1 \\ u n d e f i n e d & o t h e r w i s e \end{matrix} .

(2)

This transform shifts all negative fold changes to be symmetrically spaced from zero to their corresponding positive fold changes (MFC column in Fig 2A, visualized in Fig 2C). But this transform leaves a discontinuous region between [–1,1] where no fold change values can exist. This would give a misleadingly large spatial distance between negative and positive fold changes close to one (grey region in Fig 2C) and for interval estimates that cross this region. We correct for this by translating all datapoints closer to zero by one unit, defined as a contraction transform f_C:

f_{C} (x) = {\begin{matrix} x - 1 & x \geq 1 \\ x + 1 & x < - 1 \\ u n d e f i n e d & o t h e r w i s e \end{matrix} .

(3)

This transform removes the discontinuous region, but the datapoint values no longer reflect the actual fold change values because they are shifted one unit (MAD-FC column in Fig 2A, visualized in Fig 2D). Eqs (2) and (3) represent the MAD-FC transform, and visualizing this output would represent a raw MAD plot (Fig 3D). If we wish to produce a labeled MAD plot, we will now reverse both transforms on the axis tick labels so that the datapoint’s value can be read from the labels. We will perform this label reassignment by reversing the two transforms we performed on the data (i.e., f_C(x) and f_M(x)) on the axis tick labels. We first reverse the transform f_C with the function f_C^-1:

Fig 3 — Volcano plots of p-value versus (A) linear, (B) log, and (C) MAD fold change. MA plots using (E) log2, (F) linear, and (G) MAD fold change versus normalized mean count. Datapoints are annotated as significantly upregulated (Up, red), significant downregulated (Down, blue), or not statistically significant (NS, black) based on a Wald test adjusted p-value < 0.1 and a fold change greater than ± 1.

f_{C}^{- 1} (x) = {\begin{matrix} x + 1 & x \geq 0 \\ x - 1 & x < 0 \end{matrix} .

(4)

This transform changes the point of no change of the plot from 0 back to 1 (since the first case in f_C^-1 includes zero). We now reverse the mirrored fold change transform f_M with f_M ^-1_, where

f_{M}^{- 1} (x) = {\begin{matrix} x & x \geq 1 \\ - \frac{1}{x} & x < - 1 \\ u n d e f i n e d & o t h e r w i s e \end{matrix} .

(5)

After these transforms, the original fold change values can be read from the plot. We then can display the resulting negative fold change axis tick labels as fractions (Fig 2E), decimals, or with an exponent. We are left with a visualization that not only retains the readability of a linear plot of fold change, but also exhibits proportionality and symmetry around a fold change of 1.

Data analysis

The dataset used in Fig 3 is from the airway package [5] available through Bioconductor package in R [6]. Differential gene expression analysis is performed with DSeq2 using the Wald test without a beta prior specified, with log fold change shrinkage used to refine the dispersion estimates. The fold change datapoints used for Fig 4D–4F were estimated directly from the left panel in Fig 6 from the reference publication [7] using WebPlotDigitizer (https://automeris.io/WebPlotDigitizer/). Figs 5 and 6 used gene expression data from the cancer genome atlas using the RTCGA and RTCGA.mRNA packages [8] for five genes of interest. Specifically, RNA expression was extracted from the Breast invasive carcinoma (BRCA), Ovarian serous cystadenocarcinoma (OV) and Lung squamous cell carcinoma (LUSC) datasets. Gene expression measurements were used without any further processing. Fig 7 contains protein expression from a study with quantification for ubiquitin interactors [9]. This data was loaded from the UbiLength dataset contained within the DEP R package [10] and processed with the standard LFQ workflow with default settings, as done in the Introduction to DEP vignette page (please see source code for more information). Fig 8 uses a dataset of RNA expression from leaf tissue of the papaya plant under drought stress [11]. We used preprocessed data made available by the first author (found at https://github.com/sdgamboa/misc_datasets) of the source publication [11]. Fold change measurements from the preprocessed dataset were used without any further modifications.

Fig 4 — Fold change interval estimates with the same interval width across groups visualized with a (A) linear, (B) log2, and (C) MAD plot (from a simulated dataset with identical dispersion in fold change units across all groups with interval estimates spanning from -2 to +2-fold change units from the point estimate, error bars are confidence intervals, color gradient used to visually separate groups). Comparison of (D) linear, (F) log2, and (F) MAD fold change plot of protein expression and phosphorylation in HCC1954-P cells treated with either 300nM refametinib (MEKi) or 15nM copanlisib (PI3Ki) alone or in combination (MEKi– 300nM: PI3Ki– 15nM) (error bars are standard deviation).

Fig 6 — Fold change violin plots with the same dispersion of measurements across groups visualized with a (A) linear, (B) log2, and (C) MAD plot (from a simulated dataset with identical dispersion in fold change units across all groups, color gradient used to visually separate groups). Comparison of violin plots with (D) log, (F) linear, and (F) MAD fold change of mRNA expression of various genes measured from patients with breast invasive carcinoma (BRCA), ovarian serous cystadenocarcinoma (OV), and lung squamous cell carcinoma (LUSC) (data from the Cancer Genome Atlas).

Fig 5 — Fold change boxplots of simulated data visualized with a (A) linear, (B) log2, and (C) MAD plot (from a simulated dataset with identical dispersion in fold change units across all groups, 2-fold change unit differences between each quartile boundary, color gradient used to visually separate groups). Comparison of (D) linear, © log2, and (F) MAD plots of mRNA expression of various genes measured from patients with breast invasive carcinoma (BRCA), ovarian serous cystadenocarcinoma (OV) and lung squamous cell carcinoma (LUSC) (datasets from the Cancer Genome Atlas).

Fig 7 — Comparison of heatmaps with a (A) linear, (B) log2, and (C) MAD-FC transformed color mapping of differential expression of proteins that interact with ubiquitin, a regulatory protein found in all eukaryotic organisms. The rows are proteins that are identified as Ubiquitin-protein interactors while the columns are experiment groups that represent ubiquitins of specific chain lengths (linear mono (Ubi1), tetra (Ubi4), and hexa-ubiquitin (Ubi6)). Experiments were performed on HeLa cells in vitro and expression for each group are averaged across three replicates. Note: Color scale mapped to fold changes after each of the transforms.

R packages and their version numbers are listed in Table 1.

Table 1. Packages used in code repository.

Package	Version	Reference
DESeq2	1.40.2	[2]
EnhancedVolcano	1.18.0	[12]
tidyverse	2.0.0	[13]
BiocManager	1.30.22	[14]
magrittr	2.0.3	[15]
Bioconductor	1.30.22	[14]
airway	1.20.0	[5]
RTCGA	1.30.0	[8]
RTCGA.mRNA	1.28.0	[8]
DEP	1.23.0	[10]

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Results

Different data visualizations can leave very different impressions of the same dataset and influence the conclusions drawn from the data. We demonstrate the importance of plot types for fold change measurements by comparing log plots, linear plots, and MAD plots of fold change with various biomedical datasets. We highlight the differences in their portrayal of the data, along with the unique advantages of using MAD plots with various plot types, including scatterplots, volcano plots, box plots, violin plots, and heatmaps.

We start with an RNA-Seq dataset that probes differential gene expression from airway smooth muscle cells to investigate the therapeutic mechanisms of glucocorticoid treatment for asthma [5]. Four human airway smooth muscle cell lines were treated with dexamethasone and a control, and gene expression was compared between the two study groups using the DESeq2 package. When performing differential gene expression analysis, it is common to visualize the data with a volcano plot and MA plot. Volcano plots are scatterplots that visualize statistical significance versus effect size and enable the quick identification of genes that exhibit both high statistical significance and large fold change [16].

Fig 3A–3C show volcano plots with linear, log, and MAD transforms of fold change for this dataset. The linear plot facilitates comparing the magnitude of positive fold change datapoints, but the negative fold changes are compressed between 0 and 1 (Fig 3A). While the log plot (Fig 3B) has symmetry and facilitates comparisons between positive and negative fold changes, the values of fold change datapoints are not proportional to their distance to the point of no change. The MAD plot of fold change allows for comparison between positive and negative fold changes symmetrically, along with comparing the magnitudes proportionally between fold changes of the same direction (Fig 3C). This type of plot is often used to prioritize candidate genes for further investigation. While the value of each individual datapoint can be read from a log plot, the spatial distribution of the point cloud gives a distorted summary when compared to their actual fold change values. The log plot gives a potentially misleading impression that there are many genes that are reasonably close to the max datapoint value for fold change, but the MAD plot highlights most of these candidates are less than 50% of the max fold change observed in the dataset. This is an important consideration when deciding which gene candidates should be prioritized for follow up studies.

MA plots are used to compare fold change of differential gene expression versus the average count of the same gene between both groups (essentially a Bland-Altman analysis for fold change data with two study groups [17]). MA plots are used to highlight systematic bias or highly differentially expressed genes. We produce MA plots with linear, log, and MAD transforms (Fig 3D–3F) of the same dataset used in Fig 3A–3C. The advantages of the MAD MA plot are the same as with volcano plots- the values of the datapoints are presented without the spatial distortion found with the log and linear plots.

The MAD-FC transform is especially useful for visualizing the sample distribution and uncertainty associated with fold change measurements (whether the standard deviation, standard error, quartile, distribution, confidence interval, credible interval, or support interval). To illustrate this, we produce a simulated dataset of fold change measurements with a 95% confidence interval that extends 2-fold change units above and below the point estimate for each study group. These study groups all have identical interval widths in fold change units, yet the MAD plot is the only visualization where this consistency is apparent (Fig 4A–4C). Both the log and linear fold change plots heavily distort the interval width, making it impossible to perceive that all the study groups have the same confidence interval widths. The MAD plot not only preserves the proportional distance of each datapoint to the point of no change, but also preserves the interval estimate width regardless of the value of the fold change measurement. This key behavior is apparent with a dataset measuring protein expression and phosphorylation in HER2-positive breast cancer cell lines treated with the MEK inhibitor refametinib [7] (Fig 4D–4F). The linear visualization of fold change (Fig 4D) used in the cited publication makes it easy to mistakenly conclude that the standard deviation appears approximately consistent across measurements and the negative fold change measurements are smaller in magnitude than the largest positive fold change measurement. However, when the same data is viewed by log plot (Fig 4E) or MAD plot (Fig 4F), the negative fold changes are revealed to have much larger standard deviations than the other measurements and they are larger in magnitude than the largest positive fold change datapoint. The advantage with the MAD plot over the log plot is that the width of the bounds for standard deviation can be compared directly and proportionally between groups regardless of their distance from the point of no change.

The distortions exhibited by log and linear plots are even more pronounced in boxplots. A simulated dataset visualized with boxplots with the sample median swept from 1/9 to 9-fold changes, with quartiles evenly spaced 2-fold changes between themselves, shows that not only the visualized width of the boxplots changes dramatically, but also the relative widths of quartiles within a single boxplot can become heavily distorted depending on the plot used (Fig 5A–5C). This characteristic gives a false impression that the 4^th and 6^th study group have skewed distributions in fold change within the log plot (Fig 5A) when they are in fact symmetrical (Fig 5C). With boxplots of mRNA expression of several genes measured from tumor tissue of several cancer types, the log, linear, and MAD plots appear as if they represent entirely different datasets (Fig 5D–5F). Log plots of fold change exaggerate graphical features that are close to zero and compress those that are further away. Linear plots of fold change heavily distort any features that extend into the region of negative fold change.

The same trends are observed when comparing violin plots between each of the three visualizations. With a simulated dataset of fold change distributions uniformly translated to different fold change values while maintaining the same degree of dispersion in fold change units (Fig 6A–6C), only the MAD plot reveals that each of these study groups have the same distribution shape, while the log and linear plots heavily distort the distribution depending on the fold change values. When displaying the same dataset as Fig 5D–5F with violin plots, the overall appearance and trends of the datasets are again dramatically different for each of the visualizations (Fig 6D–6F).

The MAD-FC transform can also be useful for mapping fold change data to color gradients. A comparison of fold change heatmaps for differential protein expression (Fig 7A–7C) reveals that MAD-FC emphasizes measurements with the largest fold change values in the dataset. Mapping logarithmically transformed fold changes to a color gradient makes it more difficult to clearly discern the largest fold change measurements (dataset from a study measuring protein expression of Ubiquitin-protein interactors [9]).

As mentioned previously, using the MAD-FC transform to visualize fold changes may not be useful with datasets with a dynamic range of more than 8 units in a log2 scale. Although most bioinformatics datasets do not satisfy this condition, there are exceptions. One such example is a dataset from a study investigating gene expression in response to drought stress in the papaya plant [11] (Fig 8A–8C). This dataset spans ±15 units on the log2 scale. While the log plot spreads out the points for effective visual inspection, the linear and MAD plots compress nearly the entire dataset onto the point of no change, limiting the usefulness of such visualizations. For this specific data set, a log-transform would be more useful for most applications.

Discussion

Here we propose a transform that enhances the usefulness of a linear visualization of raw fold changes. A major shortcoming of linear plots of fold change is that negative fold change values are compressed from [0, 1] and cannot be compared to positive fold change values by their spatial position. This limitation hinders acquiring a holistic summary of fold change values.

Log plots allow the comparison of fold changes from both directions because positive and negative fold changes are positioned symmetrically about the origin. Yet the proportional relationship between fold change value and spatial position is lost for log plots. While individual points can be read from a log plot with those familiar with the log scale spacing of values between axis tick labels, it is difficult to identify linear trends from point clouds in a log scale. In contrast, MAD plots are designed to maintain readability, symmetry, and proportionality. This combination of characteristics allows spatial position to be used as a proportional encoding for fold change value regardless of direction or magnitude of the measurement. As a consequence, MAD plots are especially useful for visualizing the distribution, quartiles, and interval estimates of fold change measurements since these visual features are not distorted in a spatially dependent manner as in log and linear plots of fold change.

We enhance linear visualization of fold change with a transform strategy that is borrowed from labeled log plots, where the axis tick labels undergo a reverse transform to display the original fold change values. While this visualization lacks the high dynamic range found in log plots, in many applications comparing fold change across 2.5 orders of magnitude (8 units on the log 2 axis) is sufficient to identify interesting datapoints. This new visualization may be a useful tool for more intuitively summarizing fold change values. Such visualizations could perhaps be used for other applications than what is shown here, such as meta-analysis techniques [18] and comparing effect size across broadly related experiments as we have done in previous studies [19–21].

Acknowledgments

We thank the reviewers for their insightful feedback that greatly improved the quality of this manuscript.

Data Availability

Code used to generate all data and figures is written in R and available at: https://github.com/bacorliss/mirrored_axis_distortion.

Funding Statement

BAC This work was funded by PEB’s endowment for the School of Data Science, University of Virginia. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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PLoS One. doi: 10.1371/journal.pone.0304632.r001

Decision Letter 0

Stephen R Piccolo

5 May 2023

PONE-D-23-12801MAD-FC: a fold change visualization with readability, proportionality, and symmetryPLOS ONE

Dear Dr. Corliss,

Thank you for submitting your manuscript to PLOS ONE. We invite you to submit a revised version of the manuscript that addresses the point raised below.

Please submit your revised manuscript by Jun 19 2023 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

We look forward to receiving your revised manuscript.

Kind regards,

Stephen R. Piccolo

Academic Editor

PLOS ONE

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Additional Editor Comments (if provided):

I'm sending this back to you (before sending it out for review) to ask that you submit a revised version with higher resolution figures. I'm not asking for the figures to be at extremely high resolution. However, the resolution is so low in the current submission that the figures look quite grainy, so it is difficult to decipher some of the patterns and text. Because visualization is the whole point of this paper, it will be important to make sure the reviewers will be able to see the figures well.

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Reviewers' comments:

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PLoS One. 2024 May 31;19(5):e0304632. doi: 10.1371/journal.pone.0304632.r002

Author response to Decision Letter 0

11 May 2023

Figures were reproduced at higher resolution (600 DPI) and processed through Plos One's PACE tool. Unfortunately, the figures embedded within the PDF are still compressed and at low resolution, but there is a download link at the top of each page where the original high-resolution files can be downloaded. I talked to the editorial office, and they said there is no way I can boost the resolution of the embedded images within the PDF.

PLoS One. doi: 10.1371/journal.pone.0304632.r003

Decision Letter 1

Stephen R Piccolo

31 Jul 2023

PONE-D-23-12801R1MAD-FC: a fold change visualization with readability, proportionality, and symmetryPLOS ONE

Dear Dr. Corliss,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

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Stephen R. Piccolo

Academic Editor

PLOS ONE

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Additional Editor Comments:

Two reviewers provided feedback on your manuscript. Both had positive things to say. One of them provided a detailed list of suggestions for improving the manuscript. I am inclined to trust that these recommendations will indeed improve the manuscript. But if you can provide convincing justification that some are unnecessary, feel free to make that case.

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: (No Response)

Reviewer #2: (No Response)

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Partly

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: No

**********

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Reviewer #1: Yes

Reviewer #2: No

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The authors present a new data transformation method that aims to improve data visualization and interpretation. Large datasets with fold-change data are often presented on a log scale for its large dynamic range. This scale, however, has the disadvantage that it is not proportional. The new approach, MAD-FC, yield a proportional scale which improves readability and, as a consequence, interpretation.

In my opinion, the work is original and relevant. I have no comments on the content of the article and I only want to encourage the authors to make the process as easy and transparent for potential future users. It would be nice to see people try and adapt this method.

The code to produce all figures is available and that’s great start. Also, the procedure is detailed in the manuscript, step-by-step, but instructions on how to do this on ones own data can be improved.

So, for a true novice that wants to implement this, an excel sheet with formula’s (if possible), a video tutorial or a more detailed walk-through in R (or even better R markdown) may be helpful.

These are suggestions only and should not be regarded as requirements that need to be met before publication.

Reviewed by Joachim Goedhart (University of Amsterdam, the Netherlands).

Reviewer #2: Please see attachment. (The website wants me to include at least 100 characters of text here and isn't letting me submit the form until I finish this sentence...? Okay, now it's good.)

**********

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Reviewer #1: Yes: Joachim Goedhart (University of Amsterdam, the Netherlands)

Reviewer #2: Yes: Marcus W. Fedarko

**********

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

Attachment

Submitted filename: CommentsToAuthor.pdf

pone.0304632.s001.pdf^{(126.7KB, pdf)}

PLoS One. 2024 May 31;19(5):e0304632. doi: 10.1371/journal.pone.0304632.r004

Author response to Decision Letter 1

18 Oct 2023

>> Note: Author responses to each query being with “>>”.

PONE-D-23-12801R1

MAD-FC: a fold change visualization with readability, proportionality, and symmetry

PLOS ONE

Dear Dr. Corliss,

Please include the following items when submitting your revised manuscript:

• A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.

• A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

• An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

We look forward to receiving your revised manuscript.

Kind regards,

Stephen R. Piccolo

Academic Editor

PLOS ONE

Journal Requirements:

Additional Editor Comments:

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

>> We want to thank both reviewers for their constructive feedback that enhanced the quality of the manuscript.

Reviewer's Responses to Questions

Comments to the Author

Reviewer #1: (No Response)

Reviewer #2: (No Response)

________________________________________

2. Is the manuscript technically sound, and do the data support the conclusions?

Reviewer #1: Yes

Reviewer #2: Partly

________________________________________

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: No

________________________________________

4. Have the authors made all data underlying the findings in their manuscript fully available?

Reviewer #1: Yes

Reviewer #2: No

________________________________________

5. Is the manuscript presented in an intelligible fashion and written in standard English?

Reviewer #1: Yes

Reviewer #2: Yes

________________________________________

6. Review Comments to the Author

So, for a true novice that wants to implement this, an excel sheet with formula’s (if possible), a video tutorial or a more detailed walk-through in R (or even better R markdown) may be helpful.

These are suggestions only and should not be regarded as requirements that need to be met before publication.

>> You make a great point that this work can only be useful if it is accessible. Although optional, we went ahead and started the process by making a website with vignettes in RMarkdown for each of the examples in the manuscript (link added to methods section). In the future, we hope to make an R package, and then expand to other programming languages.

Reviewed by Joachim Goedhart (University of Amsterdam, the Netherlands).

Reviewer #2: Please see attachment. (The website wants me to include at least 100 characters of text here and isn't letting me submit the form until I finish this sentence...? Okay, now it's good.)

________________________________________

7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: Yes: Joachim Goedhart (University of Amsterdam, the Netherlands)

Reviewer #2: Yes: Marcus W. Fedarko

________________________________________

Review of Corliss et al., 2023 (“MAD-FC: a fold change visualization with readability, proportionality, and symmetry”)

Corliss et al. describe MAD-FC, a transform for fold change values designed to produce easier-tointerpret visualizations (compared to visualizations of raw fold changes and visualizations of log fold changes). The authors outline four desirable properties of fold change visualizations (readability, proportionality, symmetry, and high dynamic range), argue that plots using MAD-FC-transformed fold changes exhibit readability, proportionality, and symmetry, and present various examples of these plots compared with plots of raw and log fold changes.

Visualizations of raw fold changes do not represent negative fold changes well: although positive fold changes fill the interval of (1, ∞), negative fold changes are constrained to the interval [0, 1). The scale of fold changes in a MAD-FC plot is adjusted to account for this, allowing for negative and positive fold changes of equal magnitude to be symmetrically placed on both sides of 1 (the point marking “no change”). Log plots of fold changes also exhibit symmetry, since log(A/B) = -log(B/A) for all positive A and B; however, unlike log plots, MAD-FC plots preserve fold changes’ linearity.

I believe that the proposed methods have merit and could be widely useful. However, the paper has some serious issues that should be addressed before publication. Among other major and minor criteria, some important issues I see include:

• I am not convinced that the four described properties of fold change visualizations are well defined;

• The methods detailing the transform are not clearly written;

• The paper lacks details describing how Figures 3–7 were created, and does not provide information about the locations of the corresponding data;

• The paper uses inconsistent terminology and has a nontrivial amount of typographic errors; and

• I believe that the paper places too much of an emphasis on axis ticks and labels rather than on the MAD-FC transform itself.

>> We think that all of these are great points; we discuss specific corrections for each of them below. We rewrote much of the introduction and rethought our definitions used for the visualization properties. We thank the reviewer for providing extensive feedback that we believe significantly increased the quality of the manuscript.

Below are my comments. Please note that I will refer to page numbers in the draft PDF (spanning 43 pages): specifically, I will refer to pages in the first version of the manuscript within this PDF (within the range of pages 7–28).

Major points

1. The words “graph,” “plot,” and “chart” are all used (apparently interchangeably) throughout the manuscript (for example, the abstract says “plots of fold change” and “fold change charts” in multiple places).

The third reference that the authors cited (Midway 2020) agrees that these terms are often used interchangeably (see Box 1 of that paper). For the sake of clarity, I recommend sticking to only one of these terms throughout the manuscript, since the definition of “chart” given in the Background section (page 9) makes it seem as if charts are distinct, somehow, from plots and graphs. In particular, I suggest replacing “graph” and “chart” with “plot” (since by my count “graph” is used 4 times in the paper, “chart” is used 5 times, and “plot” is used around a hundred times), although I leave the decision up to the authors.

>>> Great point, we changed all mentions of “graph” and “chart” to plot.

2. Many parts of the paper refer to “log plots” and “linear plots” of fold change, and a recurring implicit claim in the paper (including at least the title, abstract, and introduction) is that MADFC represents a new visualization method, in addition to a new mathematical transform.

I contend that referring to MAD-FC in this way overcomplicates the paper, and that it would be clearer to just refer to MAD-FC as a transform (albeit a transform which has been defined primarily for the purposes of visualization). This is analogous to how we can define a logarithmic transform, but the logarithm (by itself) is not a visualization method: it is a way we can adjust the scale(s) being used in a visualization. We might sometimes refer to “log plots” in shorthand, but these are more precisely described as “plots of log-transformed data.”

Similarly, the authors use the MAD-FC transform in multiple types of “MAD-FC plots”—in ordinary scatter plots (Figure 1F, 2E–G), volcano plots (Figure 3C), box plots (Figure 5C), heatmaps (Figure 7C), etc. It’s subtle (and I understand if the authors have reservations about making this change throughout the entire paper), but I strongly recommend adjusting the paper to move away from the term “MAD-FC plot” in favor of describing the “MAD-FC transform,” the results of which can then be visualized in many different types of plots. (Another fix would be explicitly defining “MAD-FC plot” to mean “any plot that uses MAD-FC-transformed fold changes,” etc.).

When I first began reading through this paper, I assumed that the authors were defining a new type of plot—but the main contribution here (and it is a worthwhile contribution!) is the transform that the authors present, rather than the way this transform is plotted. This is what is worth emphasizing.

>> We followed your recommendation and restructured the paper to place more emphasis on the MAD-FC transform and defined MAD plot to mean any plot that uses MAD-FC transformed fold changes for one of its linear encodings.

3. I propose replacing the term “reference point” with “point of no change,” or something similar. This will make it consistent with the term “fold change from no change” (also, the terms “reference point of no change” and “point of no change” pop up multiple times in the paper anyway).

>> Solid point, changed all wording of “reference point” to “point of no change”.

4. The paper introduces the term “fold change from no change”, but also makes use of the term

“fold change units” throughout. It seems to me like these are different ways of describing the

same concept, so I recommend sticking to one of them. (If I’m reading this wrong and they are two separate things, that should be made clearer—I don’t see much reference information online when I search for “fold change units”, so I expect other readers will be similarly confused.)

>> Great point, and your assumption is correct. We removed all mentions of “fold change from no change” and only used “fold change units” (it seemed less awkward to use that form in sentences and axis labels).

5. I think the definitions of “readability” (Background, page 10: “a visualization with readability has a clear and direct mapping between the value of the datapoints and their spatial location”) and “proportionality” (Background, page 10: “a visualization exhibits proportionality if the fold change datapoints going the same direction are proportionally distant from the value denoting no change within the plot”) need some work. I suspect that it might be best to merge “readability” and “proportionality” together, since these terms’ definitions seem to overlap.

1. Arguably, a visualization using a logarithmic scale also has such a clear and direct mapping —the only difference is that it is a logarithmic mapping, and thus can be slightly harder to interpret. You may want to consider renaming “readability” to “linearity,” or saying “a visualization with readability has a

Attachment

Submitted filename: ResponseReviewers_MAD-FC.docx

pone.0304632.s002.docx^{(78.5KB, docx)}

PLoS One. doi: 10.1371/journal.pone.0304632.r005

Decision Letter 2

Stephen R Piccolo

14 Nov 2023

PONE-D-23-12801R2MAD-FC: a fold change visualization with readability, proportionality, and symmetryPLOS ONE

Dear Dr. Corliss,

Please submit your revised manuscript by Dec 29 2023 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.
A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.
An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

We look forward to receiving your revised manuscript.

Kind regards,

Stephen R. Piccolo

Academic Editor

PLOS ONE

Journal Requirements:

Additional Editor Comments:

Thank you for addressing the reviewers' concerns. The reviewers have looked at the revised version. One reviewer provided a detail list of additional suggestions. I do think that many of these will improve the paper, so I ask you go through them and consider whether you agree. If so, please make those changes. If you do not agree any of the changes, please provide a short response on those items, and I will evaluate them (likely without sending it out again for review).

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

Reviewer #1: All comments have been addressed

Reviewer #2: (No Response)

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

Reviewer #1: Yes

Reviewer #2: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

Reviewer #1: Yes

Reviewer #2: Yes

**********

6. Review Comments to the Author

Reviewer #1: (No Response)

Reviewer #2: Please see the attached PDF of comments. I feel that the paper has strongly improved since the last version, although I believe a few issues remain that should be addressed or at least acknowledged before publication.

**********

7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: Yes: Joachim Goedhart

Reviewer #2: Yes: Marcus W. Fedarko

**********

Attachment

Submitted filename: 202311_CommentsToAuthors.pdf

pone.0304632.s003.pdf^{(214KB, pdf)}

PLoS One. 2024 May 31;19(5):e0304632. doi: 10.1371/journal.pone.0304632.r006

Author response to Decision Letter 2

13 May 2024

Response to Reviewer Comments

Review 1

We thank reviewer 1 for their feedback with the first revision and are happy that they feel we have addressed their suggestions.

Review 2 of Corliss et al., 2023 (“MAD-FC: a fold change visualization with readability, proportionality, and symmetry”)

The authors have done an excellent job improving the paper. The additional rigor and detail introduced in this revision, including the additions to Figure 1 and the new Figure 8, makes the paper much stronger—thank you!

In particular, the revisions made to the four desirable criteria of a fold change visualization (readability, proportionality, symmetry, and high dynamic range) make the foundation of the paper much clearer. I still have some lingering questions about the readability criterion, however, and I have a few ideas that I think would improve the presentation of the other three criteria.

The majority of the remaining suggestions I have now are minor points; many of these are debatable issues with phrasing or inessential suggestions, but there are some remaining clear errors that should be addressed.

In the comments below, I will refer to line numbers in the “clean” copy of the revised manuscript.

>> We thank this reviewer for all of the fantastic feedback!

Major points

1. Definitions used in “Readability”: The new definitions used in the context of readability help illustrate your intent much better—however, I still have a few remaining issues with these terms.

1. The definition of readability given in the introduction (lines 60–69) seems to only describe this in terms of plots where the fold change is used as a spatial encoding, i.e. not including Figure 7 (which shows heatmaps of fold changes).

1. You could probably define some sort of relationship between spatial and color-based encodings that allows us to apply “readability” to the latter, but the introduction section does not seem to do this as of writing.

2. It’s fine if “readability” only applies to spatial plots (I think going very in-depth about how MAD-FC can be used for heatmaps is probably out of scope of this paper). However, I would at least recommend adding a small disclaimer (likely to the introduction) that the descriptions of these four properties are mainly given in the context of spatial plots.

3. This also impacts the other three properties (proportionality, symmetry, and dynamic range) somewhat—however, these three properties seem a bit easier to describe in the context of a color-based encoding like that used in Figure 7. The possible more formal definitions of proportionality and symmetry I mentioned below (see “Introduction, lines 75–83 and 86–88”) may help with such a description? That being said, as stated above and below I do not think these particular issues are essential to resolve before publication.

>> Great points, we added this disclaimer on line 107: “It is important to note that while these properties are explained in the context of spatial encodings, they could potentially be extended to color encodings. However, applying these encodings to color is complex because of the nonlinear relationship between color and the human eye’s spectral sensitivity (4). We illustrate the potential of our transform for heatmaps in Figure 8, but a more detailed investigation of these properties applied to color encodings is beyond the scope of this study.”

2. Distinguishing raw and labeled plots has made the explanation of readability much clearer, I think! The main issue I have with the current presentation is that the paper implies that these are two mutually exclusive options—but, as Figures 1D and 1E show, it is possible to have both types of tick labels in the same plot.

1. Of course, in practice most plots (like Figures 3–8) will only show one type of label, so I think your distinction still makes sense. I will leave the exact way of handling it up to the authors, but I recommend at least adding a small disclaimer somewhere that there doesn’t have to be a “dichotomy” between raw and labeled plots. (Unless I am missing something—or, if you feel that combining labels in this way makes the plot hard to read, then it would be worth discussing that in the text.)

2. I think taking some more time in the introduction section that explains these terms (lines 107–109) to go into more detail would help clarify things.

>> Valid points, we added text the line 116: “In theory, these plot types are not mutually exclusive if both axes are included, but we discourage this approach because multiple axes can make interpretation more difficult.” Also added text to line 150: “While raw log plots and labeled log plots have a trade-off in readability between extraction and conversion (Figure 1A), we note that there is no such trade-off between raw MAD plots and labeled MAD plots. We therefore strongly recommend to never use raw MAD plots to visualize fold change data, because the raw MAD-FC transform units will potentially confuse the read and degrade the readability of the plot.”

2. Evaluating proportionality and symmetry: Introduction, lines 75–83 and 86–88: The descriptions of how to visually evaluate proportionality and symmetry have made this section much clearer—thank you! This is not an essential suggestion, so feel free to skip it if you’d like, but I propose more formally defining the ways in which you evaluate these properties.

1. For example, rather than saying that “Symmetry of a transform can be visually assessed by measuring the distance between synthetically generated pairs of fold changes of opposite direction with the same magnitude”, it would be more correct to provide a mathematical definition of symmetry (e.g. given a transform function f(x), this transform exhibits symmetry if f(x) = -f(1/x)).

1. This accounts for the unlikely case where a transform is only partially symmetric for some reason (e.g. only up to within a certain distance from the point of no change)— such a transform should not be called “symmetric,” and yet the visual assessment method currently described in the paper would result in this transform being called “symmetric” unless the synthetic dataset generated for testing extends far enough.

2. Similarly, instead of describing the process of drawing a line between the largest positive fold change and the point of no change and then checking to see if all positive points fall on this line, you could say that a transform function f(x) is proportional if, for all fold changes x > 1, f(x) = mfU(x) – mfU(1) + f(1) given some real m > 0.

1. At least, I think this correctly describes this relationship—this is based on the idea that m, which is the slope of the line passing through the point of no change in Figures 1C– 1E (i.e. (fU(1), f(1))) and an arbitrary positive fold change (i.e. (fU(x), f(x)), should be equal to the slope of the analogous line for all other positive fold changes. There might be a more elegant way to formulate this, though

3. As mentioned, I don’t think you need to make these changes—the current descriptions (using visual evaluation methods) are sufficient. However, being more exact here would help future researchers evaluate their transformations on the same criteria with less work,

and it would open the possibility for objectively quantifying how well a transform adheres to these criteria.

1. (It might also make it easier to adapt proportionality, symmetry, and dynamic range to color-based encodings like Figure 7—if you’re only dealing with numbers rather than plots, it should be easier to apply these properties to color gradients.)

4. If you choose to not make these changes (which is fine), I propose just adding a short note somewhere in the introduction (maybe alongside the proportionality and symmetry definitions) mentioning that the visual evaluation methods are slightly informal ways of testing these properties.

>> These are good points, but we wish to save this for future research. We added text to 114: “Additionally, each of these visualization properties could be formally defined by mathematical relationships between fold changes and transform outputs, but a more rigorous exploration of these properties is reserved for future research.”

3. Explaining “Dynamic Range”: Similarly, the new information about dynamic range (including Figure 1B) helps explain things a lot. Thank you!

1. It took me a while to understand the new changes, but I think I understand the point being made in the “Dynamic Range” section of the introduction now. To help other readers see your intent clearly, I have two suggestions:

1. Maybe explain more directly (in the “Dynamic Range” section of the introduction?) that, when visualizing data spanning many orders of magnitude, linear scales make it harder to distinguish small values—i.e. that these scales are dwarfed by “outliers.” This point is already made in the “Dynamic Range” section of the introduction, but (if this seems reasonable) I recommend going into more detail in the text about why exactly we see the effect in Figure 1B.

2. The sentences in the introduction explaining why linear and MAD plots have only medium dynamic range should be expanded in order to clarify this. Maybe refer to Figure 1B and/or Figure 8 from lines 117–118 and line 136? When I first read line 136, I was confused as to why MAD plots had only medium dynamic range; seeing Figure 8 helped everything click.

>> Added this sentence to line 98: “When data spans multiple magnitudes on a linear scale, large outlying data values overwhelm the axis spatial encoding, often leaving insufficient space to distinguish differences between small values (e.g. the crowding between small fold changes on the linear axis in Figure 1B).” And on line 129: “Linear plots of fold change have medium dynamic range because they use a linear axis tick label mapping (see illustrative example in Figure 1B, and Figure 8 for an example using real data).”

Minor points

1. Abstract, line 35: This is an extremely minor point, but you use the term “MAD-FC plot” here (despite only defining the term “MAD plot” later in the paper). There is thus a very slight inconsistency.

>> Changed to “MAD plot”.

2. You could probably leave this as is (I do not think it is essential to fix), but maybe this sentence could be rewritten to be consistent with the rest of the paper—perhaps something like “We argue that MAD-FC transformed fold changes may yield more useful visualizations than log or linear transformed fold changes […]”? But the decision is the author’s call—whatever you think would be best.

>> That is fair point, we would prefer to leave as is because we added text to differentiate the raw MAD-FC transformed values versus the MAD plot. We want to de-emphasize using just the raw MAD-FC transformed values/ raw MAD plot.

3. Introduction, line 44–45: “Typically, scientists visualize fold change with a plot using a log or a linear transform, the latter of which presents raw fold change values (2).”

1. “with a plot” could probably be removed from this sentence, since I believe it goes without saying (very minor, though, so feel free to disregard).

>> Removed as recommended.

2. The term “linear transform” might confuse some readers—if it wouldn’t be too much trouble, I would recommend just removing the four uses of “linear transform” throughout the paper (since when you visualize the raw fold changes you’re not really transforming the data at all).

1. As a less dramatic alternative, maybe you could add a “simply” to the end of this sentence, i.e. “simply presents raw fold change values”, or otherwise rephrase it a bit to be extra clear that your use of “linear transform” just amounts to visualizing the raw fold changes.

>> Great point, we went with the alternative option.

3. A slightly more important issue: to my understanding, the reference to (2) here (Love et al. 2014, the DESeq2 paper) supports the first claim in this sentence (that people typically visualize log-transformed fold changes), but that paper doesn’t provide any examples of visualizing raw / linear-transformed fold changes. Unless I am missing something, I recommend moving the reference to (2) earlier in this sentence to be after you mention log fold changes, just to be clearer that (2) is only a reference regarding log fold changes.

1. If you know of any examples of papers that visualize raw fold changes, you could cite those here after citing Love et al. 2014—I think reference (5) (O’Shea et al. 2017, the source of the data in Figure 4D–F) should be sufficient. (Before I realized that (5) would be a good example of a paper that visualizes raw fold-changes, I dug up another example—Figure 4 of https://onlinelibrary.wiley.com/doi/full/10.1111/eva.13142.)

>> Moved reference (2) to earlier in sentence, added your suggested reference.

3. Introduction, line 45: minor phrasing issue—I suggest adjusting “Both transforms have a unique set of properties” to something that indicates that these transforms (or scales, see above point) do not have the exact same set of properties (which seems to be what the current sentence implies, although it’s clear to me what you mean).

1. I think the easiest way to fix this would just be adjusting the sentence’s start to say that “Each transform has” rather than “Both transforms have,” if this seems reasonable.

>> Great point, changed as recommended.

4. Introduction, line 56: The changes in this section are great; thank you! The only tiny comment I have is that you may want to adjust this line to say “With this encoding, [the raw] fold changes of […]”, rather than just “With this encoding, fold changes of […]”. This will make it unambiguously clear to the reader that 2, 1/2, 3, and 1/3 are raw / linear fold changes, rather than log- or MAD-FC-transformed fold changes.

1. This could also apply to line 58, I believe.

>> Added “raw” to both instances to clarify as recommended.

5. Introduction, line 62: minor grammar issue—I think you mean to say “values” and “locations” to match the plural tense of “visualized datapoints”. I bring this up because the use of the singular-tense words “value” / “location” could confuse readers.

>> Changed as recommended.

6. Introduction, line 66: the term “raw logarithmic axis tick labels” has not yet been defined, and the meaning is not immediately obvious without reading other parts of the paper (e.g. Figure 1D). I suggest either (1) briefly defining this term, (2) rewriting this sentence in a clearer way, or (3) referring to one of your figures to explain this term.

1. Option (3) might be the best; maybe you could say something like “[…] raw logarithmic axis tick labels (e.g. rightmost y-axis labels of Figure 1D) […]”?

2. This also applies to line 107 of the introduction.

>> Added reference to figure for both instances.

7. Introduction, line 73: Minor grammatical error (“magnitude” → “magnitudes”), I think.

>> Changed as recommended.

8. Introduction, line 74: I assume “transformed units” here refers to the linear / log / MAD-FC transformed fold changes; this is fine as is, but if feasible you may want to make it a bit clearer (since the term “transformed units” has not been defined yet and is a bit vague).

1. Maybe something like “[…] between the transformed fold changes and corresponding fold change units […]”? This could still be made clearer, but I think it would help.

>> Changed wording as recommended.

9. Introduction, line 78: Minor, but I’m not sure “trend line” is the correct term here—I think you could just say “line”, unless I’m misunderstanding something.

>> Removed “trend”.

10. Introduction, lines 98–99: “[…] a linear scale can typically capture 8 orders of magnitude on a log2 scale, or about 2.5 orders of magnitude on a log10 scale.”

1. The notion of “a linear scale … on a log scale” seems confusing to me. I think your intent is to say that a typically-sized plot can span about 2^8 or 10^2.5 units? I think saying “units” or (like in the abstract) “log2 space” would help clarify this.

2. Also: it’s a small point, but these numbers (8 units in log2 space, 2.5 units in log10 space) are repeated a couple of times throughout the paper (in the abstract, here in the introduction, in the results near figure 8, and in the discussion).

1. This is fine as is, but I propose just giving these estimates once (probably h

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PLoS One. doi: 10.1371/journal.pone.0304632.r007

Decision Letter 3

Stephen R Piccolo

16 May 2024

MAD-FC: a fold change visualization with readability, proportionality, and symmetry

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The comments from the reviewers have been addressed, and I congratulate you on a fine paper!

PLoS One. doi: 10.1371/journal.pone.0304632.r008

Acceptance letter

Stephen R Piccolo

21 May 2024

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Associated Data

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Supplementary Materials

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Data Availability Statement

Code used to generate all data and figures is written in R and available at: https://github.com/bacorliss/mirrored_axis_distortion.

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MAD-FC: A fold change visualization with readability, proportionality, and symmetry

Bruce A Corliss

Yaotian Wang

Francis P Driscoll

Heman Shakeri

Philip E Bourne

Roles

Abstract

Introduction

Fig 1. Evaluating the visualization properties of linear, log, and MAD plots.

Fig 8. Comparison of volcano plots with a dataset of fold change values with high dynamic range.

Methods

MAD-FC transform

Fig 2. Illustration of MAD-FC transform and plot.

Fig 3. Comparison of log, linear, and MAD fold change plots for RNA-Seq data.

Data analysis

Fig 4. Comparison between fold change plots with interval estimates.

Fig 6. Comparison between fold change violin plots.

Fig 5. Comparison between fold change box plots.

Fig 7. Comparison of heatmaps with different encodings between fold change and color.

Table 1. Packages used in code repository.

Results

Discussion

Acknowledgments

Data Availability

Funding Statement

References

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Stephen R Piccolo

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Decision Letter 2

Stephen R Piccolo

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Author response to Decision Letter 2

Decision Letter 3

Stephen R Piccolo

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Acceptance letter

Stephen R Piccolo

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Associated Data

Supplementary Materials

Data Availability Statement

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