Source-linked AI summary
Application of Volcano Plots in Analyses of mRNA Differential Expressions with Microarrays
Wentian Li
TL;DR
The review addresses whether analysts should use fold-change or t-statistic when variance estimates may be unreliable, especially with small samples. It advocates using both through volcano plots, examines regularized variance and filtering, and reports that SNP location on the plot is intrinsically related to minor allele frequency.
Problem
Analysts face a choice between fold-change and t-statistic, while variance may be estimated unreliably, especially with small samples.
Method
The review uses volcano plots to consider both fold-change and t-statistic, and examines regularized variance estimated as a weighted sum of prior and posterior information.
Results
SNP location on the volcano plot is intrinsically related to minor allele frequency.
Takeaways & Limitations
Volcano plots provide a visual framework for using fold-change and t-statistic together when evaluating gene filtering.
Takeaways & Limitations
The probability of the error described in the review does not seem to decrease with large sample sizes.
Abstract
from arXiv · showhide
Volcano plot displays unstandardized signal (e.g. log-fold-change) against noise-adjusted/standardized signal (e.g. t-statistic or -log10(p-value) from the t test). We review the basic and an interactive use of the volcano plot, and its crucial role in understanding the regularized t-statistic. The joint filtering gene selection criterion based on regularized statistics has a curved discriminant line in the volcano plot, as compared to the two perpendicular lines for the "double filtering" criterion. This review attempts to provide an unifying framework for discussions on alternative measures of differential expression, improved methods for estimating variance, and visual display of a microarray analysis result. We also discuss the possibility to apply volcano plots to other fields beyond microarray.
1 Introduction
Microarray differential-expression analysis must balance intuitive fold-change with noise-aware statistical evidence, while choices about averaging and transformation affect interpretation. The review presents the volcano plot as a way to display both fold-change and t-statistic together.
- Microarray analysis still faces issues involving cross-platform consistency, integration, batch effects, noise, and limited dynamic range.
- Differential-expression profiling identifies genes whose expression differs between two conditions and can inform disease biology, diagnosis, or prognosis.
- Arithmetic mean, geometric mean, and median provide different interpretations of average expression, and log transformation changes how these summaries relate.
- The central practical question is whether to use fold-change or t-statistic when selecting differentially expressed genes.Fold-change is intuitive but ignores noise; t-statistic incorporates noise, whose variance may be unreliable with small samples.
- The review answers the fold-change-versus-t-statistic question by recommending use of both through a volcano plot.It also covers relationships between the measures, regularized statistics, software, stratified plots, and broader applications.
2 Fold-change and t-statistic: signal and signal-to-noise ratio
Fold-change measures the observed expression difference, whereas the t-statistic standardizes that difference by estimated noise. Log transformation makes expression data more normal-like and links the two measures, but their relationship depends on variance and sample size.
- Log transformation makes fluorescence-intensity distributions more normal-like and stabilizes variance, supporting t-test assumptions for microarray expression data.
- The two fold-change definitions are mostly similar in the reviewed comparison, so the review does not distinguish them.
- Fold-change compares arithmetic mean expression between two groups, while t-statistic divides their mean difference by its estimated standard error.
- Log10(FC) approximately equals the difference between the group means of log10 expression, although averaging and logarithm operations generally do not commute.
- The t-statistic is log10(FC) standardized by pooled standard error, making it a noise-adjusted counterpart to the unstandardized signal.
- The relationship between t-statistic and standardized effect measures requires care because t-statistic changes with sample size whereas standardized effect size does not.
3 Volcano plot and its basic use
Volcano plots display fold-change against a noise-adjusted measure such as a t-statistic or −log10(p-value), helping researchers inspect genes selected by different differential-expression criteria. They support visual comparison, investigation of discordant candidates, and interactive gene selection.
- Basic volcano plot: Volcano plots commonly place log10FC on the x-axis and either the t-statistic or −log10(p-value) from a t-test on the y-axis.The t-statistic and −log10(p-value) are highly correlated, so the resulting plots are very similar.
- Filtering criteria: Double filtering selects genes with |log10 FC| > log10 FC0 and t > t0, equivalently using a p-value threshold, producing two rectangular regions away from the origin.Genes selected by only one filter occupy three disjoint regions outside the double-filtering areas.
- Interpreting discordant genes: Genes with large fold-change but weak significance or strong significance but low fold-change may reflect outliers, batch effects, low expression, or unusually low variance.These possibilities motivate examining genes in the single-filtering regions rather than relying on one criterion alone.
- Interpreting discordant genes: A gene with p-value = 7.7 × 10−17 and FC=1.379 illustrates strong t-test evidence despite fold-change below 2, whereas another with FC=2.66 has p-value = 3×10−3.The paper treats the first gene as significant if its low variance was correctly estimated from 17 control samples, while the second may reflect heterogeneity in the case group.
- Interactive use: Interactive volcano plots in R use identify to select the point nearest a mouse click and display information such as the gene name.The accompanying script transforms fold-changes and p-values before drawing the plot.
- Scope and interpretation: Volcano plots do not show average expression levels directly, although colors can add that information; they nevertheless encode relative standard-deviation magnitude through point geometry.Cross-hybridization and other technical variation may persist despite larger sample sizes, supporting simultaneous inspection of fold-change and t-statistic.
4 Robust variance estimation and regularization
The paper reviews variance-estimation strategies for small-sample microarray data, emphasizing regularization that stabilizes gene-specific variance and moderates extreme estimates. It also considers robust, resampling, Bayesian, and non-parametric alternatives.
- Motivation: Microarray studies often estimate variance from small samples, while the large number of genes can support reliable estimation of a common control-group variance.This small-n, large-p setting motivates methods that account for statistical noise when comparing genes.
- Robust estimation: Robust variance estimation can iteratively remove outliers before recalculating the mean and variance, but this further reduces the number of samples used.The approach addresses inaccurate mean estimates but introduces a direct sample-size trade-off.
- Alternative approaches: Bootstrap resampling can artificially increase the sample size, whereas Mann–Whitney–Wilcoxon tests avoid variance estimation by replacing the t-test.These alternatives address variance-estimation concerns through different assumptions and procedures.
- Regularization: Adding a constant penalty term s0 to the sample-estimated standard deviation prevents estimated variance from becoming too low, an approach known as regularization.The penalty is also associated with SAM and can stabilize variance estimation even when s0 is small.
- Bayesian variance estimation: Bayesian moderation represents posterior variance as a weighted combination of prior variance and sample-estimated variance, with the weight tending toward 1 as sample size increases.This draws gene-specific variance toward a middle value while allowing larger samples to have greater influence.
5 Regularized t-statistic as a joint filtering criterion
The regularized t-statistic combines log-fold-change and the t-statistic, but its filtering boundary is curved rather than a straight line. The curvature makes selection depend jointly on effect size and standard error, with larger regularization favoring larger fold-changes.
- Definition: The regularized t-statistic is a weighted sum of log10(FC′) and the t-statistic, with weights determined by s and s0.tsam = aδ + b(δ/s), where a = 0.5/(s + s0) and b = 0.5s/(s + s0).
- Geometric interpretation: The apparent straight-line interpretation of the regularized-statistic criterion is incorrect because its coefficients vary with the variables.The decomposition can misleadingly suggest a linear discriminant, but the coefficients are not constants.
- Geometric interpretation: The standard error s has a geometric meaning as tan(θ), where θ is the angle between the y-axis and the line from the point to the origin.The y-intercept and x-intercept arise under different limiting behaviors of the volcano-plot point.
- Filtering boundary: The joint-filtering discriminant is a curve that moves upward for smaller standard errors, angles, and fold-changes.Consequently, a large t-statistic with a small fold-change is harder to retain under the regularized criterion.
- Filtering boundary: With larger s0, more genes with lower significance but larger fold-changes are selected when the threshold is adjusted to select similar numbers of genes.The review contrasts this curved boundary with a linear interpretation that could allow large fold-change and small t-statistic combinations.
- Empirical comparison: SAM selects genes not chosen by fold-change or t-test p-value alone and balances their variance-selection tendencies.Fold-change favors high-variance genes, t-test selection favors relatively low-variance genes, and SAM selects intermediate variance values.
6 Relevant Bioconductor programs
The review organizes Bioconductor programs around volcano plotting, differential-expression measures, and error or variance estimation. It emphasizes that volcano plots support multiple expression-data types and that alternative statistical measures are available.
- Program categories: Bioconductor programs are grouped into volcano plots, alternative differential-expression measures, and improvements in error or variance estimation.The review limits its software summary to Bioconductor packages.
- Volcano plots: Volcano-plot implementations usually place log-fold-change on the x-axis and another differential-expression measure on the y-axis.The listed applications include fluorescence-light-intensity mRNA data and other expression data.
- Data types: The reviewed programs cover microarray, RNA-seq count, and protein-expression analyses.The review mentions Affymetrix and Illumina data alongside count-based mRNA expression and protein levels.
- Alternative measures: Alternative differential-expression measures include Bayesian posterior probabilities, F-statistics, B-statistics, moderated statistics, and shrinkage statistics.Examples include EBAM, DEDS, and GeneSelector.
- Variance estimation: Variance-estimation approaches include using replicated-sample variation, variance between similar probe sets, and pooled errors.The review presents robust and reliable variance estimation as central to choosing between t-statistic and fold-change criteria.
7 Stratified volcano plots by external information
External information can be overlaid on volcano plots to create stratified plots that highlight genes by chromosome, pathway, function, or process. In the example, coloring cytokine-related genes reveals a coherent down-regulation pattern.
- Scope: Volcano plots can incorporate external information relevant to differential expression beyond the plotted expression statistics.The review also notes applications beyond microarray expression data, including RNA-seq and protein-expression levels.
- Stratification: A volcano plot can become a stratified volcano plot by coloring or marking genes using external annotation information.Examples include chromosome location and gene-ontology categories for pathways, cellular components, functions, or processes.
- Interpretation: Stratified volcano plots help identify candidate genes associated with the selected external categories.The review states that the plot makes such candidates easy to identify visually.
- Interpretation: The cytokine-linked top-ranking genes in the example are all down-regulated rather than up-regulated.Examples include CLCF1, SOCS2, and SOCS3, with the cited fold-changes and p-values.
8 Discussion and conclusions
The discussion presents volcano plots as visual tools that jointly display standardized and unstandardized differential-expression signals, clarify regularized statistics, and support gene-selection interpretation. It also extends the framework to genetic association studies, where plot location relates to minor allele frequency and can inform the balance between chi-square results and odds ratios.
- Log-fold-change and t-statistic form the two principal quantities used for gene filtering, which volcano plots naturally visualize.
- Volcano plots jointly display noise-level-standardized and unstandardized signals for mRNA differential expression.
- In genetic association studies, volcano plots can display association signals using odds ratios or allele-frequency differences alongside noise-adjusted measures.
- The location of a SNP on the volcano plot is intrinsically related to its minor allele frequency, informing how chi-square results and odds ratios should be balanced when selecting associated genes.
- Regularized test statistics and joint filtering have an intuitive geometric interpretation in volcano plots and an advantage over double filtering.
- As scatter plots, volcano plots can incorporate external information such as gene annotation to support hypothesis generation.