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Fold change

Fold change is a fundamental metric in biomedical and statistical research used to quantify the relative magnitude of differences between two groups or conditions, typically calculated as the ratio of the value in one group to that in another, such as treatment versus control.^[1] Originating in gene expression studies around the early 2000s, it has become widely applied across omics fields—including genomics, proteomics, and metabolomics—to identify differentially expressed genes, proteins, or other variables by highlighting proportional changes rather than absolute ones.^[1] For instance, a fold change of 2 indicates that the quantity in the experimental group is twice that of the reference group, while values below 1 signify decreases; to handle bidirectional changes and improve interpretability, it is often transformed into a logarithmic scale, such as log₂(fold change), where positive values denote upregulation and negative values downregulation.^[2] In practice, fold change is computed using various methods to aggregate data from multiple samples, including the arithmetic mean of ratios, geometric mean, or median, with the latter two preferred for their robustness against outliers, unequal variances, or skewed distributions common in biological data.^[1] It plays a central role in differential analysis pipelines, such as those for microarray or RNA-seq data, where it is combined with statistical tests (e.g., t-tests or moderated approaches) to rank features in volcano plots—graphical representations plotting log₂(fold change) against statistical significance (e.g., -log₁₀(p-value)).^[3] This visualization aids in selecting biologically relevant subsets for downstream applications like machine learning models or pathway enrichment analysis.^[1] Despite its intuitiveness and prevalence—evidenced by over 10,000 PubMed entries by 2024—fold change has limitations, including sensitivity to sample size, noise in high-dimensional data, and lack of inherent statistical inference, necessitating integration with p-values or false discovery rate corrections to avoid false positives.^[1] Researchers are encouraged to report the exact calculation method for reproducibility, as inconsistencies can lead to variable results across studies.^[1] Beyond omics, the concept extends to other domains like pharmacology (e.g., dose-response ratios) and ecology (e.g., population growth multiples), underscoring its versatility as a relative measure in comparative analyses.^[2]

Definition and Basic Concepts

Standard Definition

Fold change quantifies the ratio by which a quantity changes from an initial value, such as a baseline or control, to a final value, such as a treatment or experimental condition, typically expressed as the final divided by the initial.^[1] This multiplicative measure captures proportional alterations, making it suitable for comparing changes across varying scales in scientific contexts.^[4] For instance, if a biological quantity increases from 10 to 20, the fold change is 2, signifying it has doubled; if it decreases from 20 to 10, the fold change is 0.5, indicating it has been halved.^[1] In contrast to absolute change, which computes the arithmetic difference (e.g., +10 or -10 in the examples above), fold change emphasizes relative proportions, enabling scale-invariant assessments that are particularly valuable when initial values differ widely, as often occurs in experimental biology.^[4] Logarithmic transformations of fold change, such as log₂(fold change), are sometimes used to symmetrize increases and decreases around zero, though the standard form remains the direct ratio.^[1]

Alternative Formulations

While the standard fold change is typically expressed as the ratio of final to initial values, a related measure known as relative change is defined as (final - initial)/initial.^[5] This measure is equivalent to the standard ratio minus 1 and is useful for expressing the change as a proportion of the original value.^[5] A related variant is percentage change, calculated as $100 \times (final - initial)/initial, which scales the relative change by 100 for more intuitive reporting in everyday contexts.^[6] For instance, a doubling from an initial value corresponds to a 100% increase. To illustrate, consider values changing from 10 to 15: the standard fold change is 1.5, the relative change is 0.5, and the percentage change is 50%.^[5] In contrast to the multiplicative nature of the standard ratio, they provide a more direct assessment of proportional shifts relative to the baseline.^[5]

Mathematical Properties

Calculation Methods

The fold change is calculated as the ratio of the final value to the initial value, expressed as \text{fold change} = \frac{\text{final}}{\text{initial}}.^[7] This direct division provides a measure of relative change, where values greater than 1 indicate an increase and those less than 1 indicate a decrease.^[8] When the initial value is zero, direct division results in undefined or infinite fold changes, which is common in count-based data such as RNA sequencing reads. To handle this, a small pseudocount \epsilon is added to both the numerator and denominator, yielding \text{fold change} = \frac{\text{final} + \epsilon}{\text{initial} + \epsilon}, where \epsilon is typically set to 1 or a value specific to the data scale, such as the minimum detectable count.^[9] This approach stabilizes computations while approximating the true ratio without introducing substantial bias for non-zero values.^[10] For experiments with replicates, normalization precedes fold change computation to account for variability. Replicate expression values are first averaged—often using the geometric mean for ratio-based data, as it better preserves multiplicative relationships—before calculating the fold change between group means.^[1] This step reduces noise and ensures the ratio reflects central tendencies rather than outliers.^[11] Computationally, fold changes are efficiently calculated in batch using statistical software that implements linear models or Bayesian shrinkage. In R, the limma package fits moderated linear models to normalized data, deriving fold changes as contrasts between conditions while incorporating empirical Bayes adjustments for stability across multiple features.^[12] Similar algorithmic steps in Python libraries, such as those in scikit-learn or scanpy, involve data normalization, model fitting, and ratio extraction, emphasizing vectorized operations for large datasets. For edge cases where the initial value is near zero even after pseudocount addition, results are sometimes reported qualitatively as "greater than N-fold" (e.g., >10-fold), where N is derived from the detection threshold or pseudocount limit, to convey magnitude without implying infinity.^[9]

Logarithmic Transformations

Logarithmic transformation of fold changes addresses the multiplicative nature of ratios by converting them to an additive scale, facilitating statistical analysis and interpretation. The log fold change is computed as \log \left( \frac{\text{final}}{\text{initial}} \right) = \log (\text{final}) - \log (\text{initial}), where the logarithm is typically base 2 (\log_2) in biological contexts for intuitive interpretation of doublings or halvings— for instance, \log_2(2) = 1 indicates a twofold up-regulation, while \log_2(0.5) = -1 indicates a twofold down-regulation.^[8] This transformation yields several key mathematical properties that enhance its utility. On the log scale, fold changes exhibit symmetry around zero: positive values denote increases (up-regulation), and negative values denote equivalent proportional decreases (down-regulation), avoiding the asymmetry of raw ratios where a twofold increase (2) and twofold decrease (0.5) do not balance. Additionally, the log scale converts multiplicative effects to additive ones, enabling straightforward averaging, variance stabilization, and application of linear models for differential analysis across multiple conditions or replicates.^[8] A fundamental derivation underscores the appropriateness of averaging on the log scale: the arithmetic mean of log fold changes equals the logarithm of the geometric mean of the corresponding raw fold changes, given by \frac{1}{n} \sum_{i=1}^n \log (FC_i) = \log \left( \prod_{i=1}^n FC_i^{1/n} \right), where FC_i is the i-th fold change. This property is particularly valuable in multi-condition experiments, as it provides a multiplicatively interpretable summary of changes without bias toward extreme ratios. For example, consider a 4-fold increase (\log_2 = 2) and a 2-fold decrease (\log_2 = -1); their average on the log scale is \frac{2 + (-1)}{2} = 0.5, corresponding to \log_2(1.414) \approx 0.5 or a net 1.4-fold increase via the geometric mean. This contrasts with averaging raw fold changes (4 and 0.5), which yields 2.25—an arithmetically misleading result that overstates the net effect.

Applications

In Genomics and Molecular Biology

In genomics and molecular biology, fold change serves as a fundamental metric for quantifying differences in gene expression levels between biological conditions, such as treated versus control samples or diseased versus healthy tissues.^[8] This measure is particularly prominent in the analysis of data from DNA microarrays and RNA sequencing (RNA-seq), where it assesses the magnitude of differential expression for thousands of genes simultaneously.^[13] The concept gained widespread adoption in the 1990s with the advent of DNA microarray technology, which enabled high-throughput monitoring of gene expression patterns, and played a key role in functional genomics studies following the Human Genome Project around 2000. In microarray experiments, fold change is calculated as the ratio of normalized expression intensities between two conditions, often using complementary DNA (cDNA) probes spotted on arrays to detect relative mRNA abundances.^[13] Similarly, in RNA-seq, it derives from normalized read counts, capturing changes in transcript levels across conditions like drug treatment in cell lines.^[8] A common threshold for identifying biologically relevant differential expression is a fold change greater than 2 (upregulation) or less than 0.5 (downregulation), as these cutoffs highlight genes with substantial magnitude shifts while filtering noise.^[14] For instance, log-transformed fold changes are frequently employed in these analyses to symmetrize up- and downregulation and facilitate statistical modeling.^[8] A representative application occurs in cancer genomics, where a gene exhibiting a 3-fold upregulation in tumor tissue compared to normal tissue may signal a potential oncogene driving proliferation.^[15] Such findings, derived from microarray or RNA-seq profiles, guide hypothesis generation for oncogenic pathways, as seen in studies of breast cancer where upregulated genes like MTOR show consistent fold changes across cohorts.^[16] Fold change integrates with statistical metrics like p-values in visualization tools such as volcano plots, where it assesses expression magnitude on the x-axis (often as log2 fold change) while p-values evaluate significance on the y-axis, prioritizing genes with both large effect sizes and low error probabilities.^[17] This combination underscores fold change's role in emphasizing biologically impactful differences beyond mere statistical detection.^[18]

In Other Scientific Fields

In pharmacology, fold change is commonly used to quantify differences in drug potency, particularly through metrics like the half maximal inhibitory concentration (IC₅₀), where a 10-fold decrease in IC₅₀ value between compounds indicates a 10-fold increase in potency.^[19] For instance, in studies of antiretroviral drug resistance, fold changes in IC₅₀ relative to wild-type viruses help assess how mutations reduce efficacy, with values often exceeding 10-fold signaling significant clinical implications.^[19] In economics, fold change describes relative growth in indicators such as gross domestic product (GDP), highlighting expansion over time periods. Emerging markets have shown substantial increases; for example, Brazil's nominal GDP rose from approximately $237 billion in 1980 to $1.44 trillion in 2020, representing about a 6-fold growth.^[20] This metric aids analysts in comparing economic trajectories across regions, such as the collective rise in emerging and developing economies' share of global GDP from 25% in 2000 to 45% by the mid-2020s, driven by compounded annual growth rates.^[21] In physics and engineering, particularly signal processing, fold change quantifies amplitude variations, such as attenuation in filters to suppress unwanted frequencies. For example, anti-aliasing filters in analog-to-digital converters are often designed for 60 dB attenuation, equivalent to a 1,000-fold reduction in signal amplitude to prevent noise folding.^[22] In wireless communications, materials like brick can cause 40 dB attenuation in millimeter-wave signals, corresponding to a 100-fold amplitude decrease, which informs system design for propagation losses.^[23] In climate science, fold change measures relative shifts in atmospheric concentrations, such as carbon dioxide (CO₂). From pre-industrial levels of about 280 parts per million (ppm) to approximately 425 ppm as of 2025, this represents a 1.5-fold increase, underscoring human-induced enhancements beyond natural variability over millennia.^[24]

Interpretation and Limitations

Statistical Significance

Assessing the statistical significance of an observed fold change requires evaluating whether the difference in expression or measurement levels between conditions is likely due to chance, rather than a true biological effect. Fold change alone is insufficient for this determination, as it does not account for variability in the data; instead, statistical tests such as the t-test or analysis of variance (ANOVA) are applied to log-transformed data to compute p-values that indicate the probability of observing the difference under the null hypothesis of no change.^[25]^[26] This approach uses the log fold change, typically log2-transformed, as the variable for testing, which normalizes the data and stabilizes variance across scales.^[27] Fold change serves as a measure of biological effect size, quantifying the magnitude of change, but its interpretation is strengthened by combining it with confidence intervals, such as the 95% confidence interval (CI) around the log2 fold change estimate. These intervals provide a range within which the true effect size likely lies, helping to assess precision and reliability; for instance, a narrow CI around a log2 fold change of 1 (corresponding to a 2-fold increase) supports a more robust finding than a wide one.^[28] In practice, a 2-fold change accompanied by a p-value less than 0.05 from a t-test on replicates may indicate statistical significance, but small sample sizes can inflate false positive rates by increasing the likelihood of spurious detections due to noise.^[14] In high-throughput experiments, such as those in genomics, multiple testing across thousands of features necessitates correction to control the overall error rate. The false discovery rate (FDR) correction, particularly the Benjamini-Hochberg procedure, is widely recommended to adjust p-values and limit the proportion of false positives among significant results, ensuring that only a controlled fraction (e.g., 5%) of declared changes are expected to be erroneous.^[29] This method is particularly valuable when fold changes are modest, as it balances discovery power with reliability in large-scale analyses.^[30]

Common Pitfalls and Considerations

One common pitfall in interpreting fold changes is ignoring the baseline expression scale, where large relative changes from low baseline levels often reflect technical noise rather than biological significance. For instance, a 10-fold increase from 0.1 to 1 unit may arise from stochastic variation in low-count data, whereas a 1.1-fold change from 100 to 110 units represents a more substantial absolute shift despite the modest ratio.^[31]^[8] Fold changes are inherently context-dependent and directional, with positive values indicating upregulation and negative values (in log scale) signaling downregulation, but their biological relevance varies by experimental system and absolute expression levels. To enhance clarity, fold changes should always be reported alongside absolute values or normalized counts, avoiding isolated reliance on ratios that can mislead without this context.^[8] A best practice for reliable interpretation, particularly in small-sample studies, involves using moderated fold changes within Bayesian frameworks, such as limma's empirical Bayes method, which shrinks estimates toward a prior to stabilize variability across genes and reduce false positives from noisy data.^[32] Early genomics analyses frequently overlooked batch effects, leading to inflated or spurious fold changes; post-2010 standards have emphasized normalization techniques to mitigate these technical artifacts, improving reproducibility in high-throughput data.^[33]

References

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Fold change provides an intuitive measure of the magnitude of the difference between groups. Using l o g 2 allows direct interpretation of how many times ...
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• Thus, we often take the log of fold change values for data science purposes = log fold change (LFC). ➢ A negative LFC indicates lower overall values in ...Missing: definition | Show results with:definition
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### Summary of Percentage Change Calculation and Use in Economics
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How to Calculate Fold Change in Excel - Bricks
Feb 12, 2025 · Fold change is calculated in Excel using the formula: Final Value / Initial Value. For example, =B2/A2.
[8]
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Dec 5, 2014 · We present DESeq2, a method for differential analysis of count data, using shrinkage estimation for dispersions and fold changes to improve stability and ...Missing: formulations percentage
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We build on a statistical framework for fold changes, where pseudocounts correspond to the parameters of the prior distribution used for Bayesian inference of ...
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Last, we designed a Pseudo-Count (PC) model that does not model any ZGPs, but rather avoids numerical issues with zeros by adding a fixed positive pseudo-count ...
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No information is available for this page. · Learn why
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limma is an R/Bioconductor software package that provides an integrated solution for analysing data from gene expression experiments.
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How to choose log2FC thresholds for DGE analysis - biostatsquid.com
Common values are: |log₂FC| ≥ 1 (2-fold change) – most common, good balance; |log₂FC| ≥ 0.5 (1.4-fold change) – more sensitive, catches subtle changes; |log₂FC| ...
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A pan-cancer transcriptomic study showing tumor specific alterations ...
Jul 1, 2021 · The color bar on the right represents the log-fold changes; red shows genes upregulated in tumors and blue shows genes downregulated in tumors ...
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Gene expression trend changes in breast cancer populations over ...
We determined that three oncogenes, PD-L2, ETV5, and MTOR and 113 long intergenic non-coding RNAs (lincRNAs) were constantly up-regulated, whereas two oncogenes ...Missing: upregulation | Show results with:upregulation
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an interactive tool for pathway guided visualization of differential ...
A common tool for visualizing this type of data is the volcano plot where each feature is plotted as the log2 transformed fold change along the x-axis and the ...
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Within the fold: assessing differential expression measures and ...
'Fold-change' cutoffs have been widely used in microarray assays to identify genes that are differentially expressed between query and reference samples.<|separator|>
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Dose–response curve slope is a missing dimension in the analysis ...
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A first approach to identify differentially expressed genes is known as the Fold-Change estimation (FC). It evaluates the average log-ratio between two ...<|control11|><|separator|>
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State the statistical hypotheses in terms of the fold change (ratio) of the means. 2. Transform this into hypotheses about a difference by taking logarithms.
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Statistical tests for differential expression in cDNA microarray ...
Genes with large fold-change values will lie outside a pair of vertical threshold lines.
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Topconfects: a package for confident effect sizes in differential ...
Mar 28, 2019 · We propose a method of ranking by confidence bounds on the log fold change, based on the previously developed TREAT test.
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Testing significance relative to a fold-change threshold is a TREAT
The biological significance of a given fold-change is likely to depend on the gene and on the experimental context. On the other hand, it is reasonable to ...
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When Two-Fold Is Not Enough: Quantifying Uncertainty in Low-Copy ...
Our empirical data highlight that at low template concentrations, stochastic sampling dominates technical noise, impairing the reliability of even large fold ...<|separator|>
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limma powers differential expression analyses for RNA-sequencing ...
For each coefficient in the linear model or contrast, empirical Bayes moderated t-statistics and their associated P-values are generally used to assess the ...
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Tackling the widespread and critical impact of batch effects in high ...
Sep 14, 2010 · After normalization, these global differences are greatly reduced (FIG. 1b), and the sTCC study properly normalized the data. However, it is ...Missing: post- | Show results with:post-