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CARhy: Comprehensive Analyses of Circadian Rhythms in Transcriptomic Experiments with Multiple Conditions

This paper introduces CARhy, a unified statistical framework implemented as an R package that enables comprehensive analysis of circadian rhythms across multiple experimental conditions by testing for rhythmicity, amplitude, phase, and baseline differences while robustly handling heteroscedastic noise and unbalanced designs.

Original authors: Weiyi Huang, Jerome S. Menet, Samiran Sinha

Published 2026-04-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Weiyi Huang, Jerome S. Menet, Samiran Sinha

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine your body is a bustling city that never sleeps, but it operates on a strict 24-hour schedule. Just like the city has rush hour, quiet nights, and specific times for garbage collection, your cells have "circadian rhythms"—internal clocks that tell genes when to wake up, work, and rest. When these clocks get messed up, it can lead to health problems like diabetes, heart disease, or cancer.

Scientists want to study these clocks by looking at how genes "talk" (express themselves) over a full day. But here is the problem: most existing tools for studying these clocks are like a pair of scissors that can only cut two pieces of paper at a time. They can compare Condition A (like eating at night) vs. Condition B (eating freely), but they struggle when you have three or more different scenarios to compare at once. They also often assume that the "noise" or messiness in the data is the same for everyone, which isn't true in real life.

Enter CARhy (Comprehensive Analysis of Rhythmicity). Think of CARhy as a high-tech, multi-lens camera that can take a single, sharp photo of a complex scene with many moving parts.

Here is how the paper explains what CARhy does, using simple analogies:

1. The Problem with Old Tools

Imagine you are trying to figure out why three different groups of runners (Group A, B, and C) have different running styles.

Old tools usually only let you compare Group A vs. Group B, then Group B vs. Group C, one pair at a time. They also often assume all runners are running on the same smooth track with the same wind conditions. If the track is muddy for one group and dry for another, the old tools get confused and might say a runner is faster just because the mud slowed them down, not because they are actually faster.
The Gap: There wasn't a tool that could look at all three groups at once, handle the different "muddy tracks" (uneven data noise), and tell you exactly what changed: Did they run faster? Did they start at a different time? Did they run a different distance?

2. What CARhy Does

CARhy is a new statistical "camera" designed to handle multiple conditions (more than two) all at once. It uses a mathematical model based on waves (like the tides) to track how gene activity rises and falls over 24 hours.

It breaks down the "running style" of a gene into three specific parts:

The Mesor (The Average Level): Imagine the average height of the tide. Is the water generally higher or lower in one group compared to another?
The Amplitude (The Swing): How high does the wave go? Is the gene swinging wildly from high to low, or is it just a gentle ripple?
The Phase (The Timing): When does the peak happen? Does the gene hit its high point at 6:00 AM in one group, but at 2:00 PM in another?

3. How It Handles "Messy" Data

In real biology, data is messy. Some groups might have fewer samples (like having only 2 runners instead of 10), and the "noise" (random errors) might be louder in one group than another.

The Innovation: CARhy doesn't force the data to be "clean" or equal. It uses a clever mathematical trick (called the Satterthwaite-Welch approximation) that acts like a smart filter. It adjusts its calculations on the fly to account for the fact that Group A might be noisier than Group B. This prevents it from making false alarms (saying a gene changed when it didn't) or missing real changes.

4. The Results: A Better Detective

The authors tested CARhy against two other popular tools (DODR and dryR) using computer simulations and real data from mouse livers.

The Simulation: They created fake gene data with known "cheats" (genes that were supposed to change). CARhy caught the cheats more often than the others, especially when the data was messy or the groups were uneven. It also rarely made false accusations.
The Real Data: When they applied CARhy to real mouse liver data (comparing mice fed at night, freely, or on a weird schedule), it found hundreds of genes that changed their rhythms.
- It confirmed that genes in the "weird schedule" group had very weak rhythms (the wave was flat).
- It pinpointed exactly which genes changed their timing (phase) and which changed their intensity (amplitude).
- It was also much faster than the other tools, processing thousands of genes in seconds.

5. The Bottom Line

The paper claims that CARhy is a unified, reliable tool for scientists who need to compare circadian rhythms across three or more conditions. It doesn't just say "something changed"; it tells you how it changed (timing, intensity, or baseline) and does so even when the experimental data is imperfect or unbalanced.

The authors have made this tool available as a free software package (an R package) so other scientists can use it to study their own biological clocks without needing to be math experts.

In short: CARhy is the new, all-seeing lens that lets scientists compare complex, messy biological schedules all at once, telling them exactly how the internal clocks of genes are shifting, speeding up, or slowing down.

1. Problem Statement

Circadian rhythms are endogenous 24-hour oscillations regulating physiological processes. Disruptions are linked to diseases like cancer and metabolic disorders. While transcriptomic studies increasingly involve multiple experimental conditions (e.g., different genotypes, treatments, or feeding regimens), existing statistical tools for analyzing rhythmicity suffer from critical limitations:

Pairwise Limitations: Most methods (e.g., DODR, CircaCompare) are restricted to comparing only two conditions.
Lack of Decomposition: Existing tools often test for "differential rhythmicity" as a monolithic concept, failing to disentangle specific changes in amplitude (magnitude), phase (timing), or mesor (baseline level).
Rigid Assumptions: Many methods assume homoscedasticity (equal variance across conditions) and balanced designs, which are frequently violated in real biological data where sample sizes are small and noise levels vary.
Computational/Statistical Trade-offs: Bayesian approaches (e.g., dryR) can handle multiple conditions but are computationally expensive and rely on subjective thresholds (BIC weights) rather than formal p-values.

2. Methodology

The authors propose CARhy (Comprehensive Analysis of Rhythmicity), a unified statistical framework based on first-harmonic Fourier regression.

A. Statistical Model

For a gene $g$ under condition $k$ , the expression profile $Y(t)$ is modeled as:
$Y_{j,k}(t) = \delta_k + \alpha_k \cos\left(\frac{2\pi t}{24}\right) + \beta_k \sin\left(\frac{2\pi t}{24}\right) + \epsilon_{j,k}$

Parameters: $\delta_k$ (mesor/baseline), $\alpha_k, \beta_k$ (rhythmic coefficients).
Derived Metrics: Amplitude $A_k = \sqrt{\alpha_k^2 + \beta_k^2}$ and Phase $\phi_k = \text{atan2}(\beta_k, \alpha_k)$ .
Noise Assumption: The model explicitly allows for heteroscedasticity, where error variances $\sigma^2_k$ can differ across conditions. It accommodates unbalanced designs (varying numbers of replicates).

B. Hypothesis Testing Framework

CARhy performs five distinct types of inference using contrast matrices and novel test statistics:

Rhythmicity Test (CARhy-TR): Tests $H_0: \alpha_k^2 + \beta_k^2 = 0$ $H_{0} : α_{k}^{2} + β_{k}^{2} = 0$ (is the gene rhythmic in a specific condition?).
- Innovation: Uses a Monte Carlo method to approximate the reference distribution of the test statistic, as the exact distribution is a weighted sum of non-central $\chi^2$ variables.
Differential Rhythmicity (CARhy-TDR): Tests $H_0: (\alpha_1, \beta_1) = \dots = (\alpha_K, \beta_K)$ (are rhythms identical across all $K$ conditions?).
Differential Mesor (CARhy-TDM): Tests equality of baseline levels ( $\delta_k$ ) across conditions.
Differential Amplitude (CARhy-TDA): Tests equality of amplitudes. Uses a cube-root transformation ( $\theta_k = (\alpha_k^2 + \beta_k^2)^{1/3}$ ) to improve normality of the estimator.
Differential Phase (CARhy-TDP): Tests equality of phases.

C. Distributional Approximation (Key Innovation)

To avoid unreliable large-sample asymptotic approximations (which fail in typical small-sample circadian experiments, e.g., $n=18$ ), the authors derive finite-sample null distributions for the test statistics using the Satterthwaite–Welch approximation.

The test statistics (quadratic forms) are scaled to follow an approximate F-distribution.
The degrees of freedom ($df$) and scaling constants ( $c$ ) are calculated using moment-matching techniques that account for condition-specific variances and sample sizes.
This approach eliminates the need for the homoscedasticity assumption.

3. Key Contributions

Unified Multi-Condition Framework: CARhy is the first method to provide a coherent statistical framework for analyzing circadian data across more than two conditions simultaneously.
Decomposition of Rhythmicity: It uniquely separates and tests for differences in amplitude, phase, and mesor independently, rather than treating differential rhythmicity as a single black box.
Robustness to Heteroscedasticity: By deriving exact finite-sample distributions via Satterthwaite approximation, CARhy remains valid even when variances differ significantly between groups and sample sizes are unbalanced.
Computational Efficiency: Unlike Bayesian model selection methods (dryR) that require fitting numerous candidate models per gene, CARhy fits a single linear model per condition, making it significantly faster and scalable to high-dimensional transcriptomic data.
Open-Source Implementation: Implemented as an R package, making the methodology accessible to the biological community.

4. Results

The authors validated CARhy through extensive simulations and real-world application.

A. Simulation Studies

Type I Error Control: CARhy maintained Type I error rates close to the nominal 0.05 level across all scenarios, including heteroscedastic noise, heavy-tailed distributions (Student's t), and unbalanced designs.
- Comparison: Competitors like DODR and dryR showed inflated Type I errors under heteroscedasticity and unbalanced sampling.
Statistical Power: CARhy consistently achieved higher power than DODR and dryR, particularly in challenging settings (unequal sample sizes, heavy-tailed noise).
FDR Control: In simulations with 2,000 genes, CARhy achieved the highest F1 scores while maintaining FDR near 0.05, outperforming dryR across various BIC thresholds.

B. Real Data Application (Mouse Liver)

Dataset: Analyzed 12,365 genes from a mouse liver study with three feeding regimens (Nighttime, Ad Libitum, Arrhythmic).
Rhythmicity Detection: CARhy-TR identified more rhythmic genes overlapping with an external database (CGDB) compared to the non-parametric JTK CYCLE, demonstrating higher sensitivity.
Differential Analysis:
- Identified 469 genes with differential rhythmicity (DR) and 724 non-differential (NDR).
- Validation: DR genes showed significantly lower temporal profile similarity, higher amplitude divergence, and greater phase dispersion compared to NDR genes, confirming the biological validity of the statistical findings.
- Specificity: CARhy-TDA and CARhy-TDP identified specific subsets of genes driven by amplitude changes (222 genes) or phase shifts (506 genes), providing mechanistic insights.
Performance: CARhy was substantially faster than dryR (seconds vs. minutes) and provided raw p-values for flexible thresholding, unlike dryR's reliance on arbitrary BIC cutoffs.

5. Significance

CARhy addresses a critical gap in circadian biology by enabling rigorous, comprehensive statistical inference for complex, multi-condition experimental designs. Its ability to handle heteroscedasticity and unbalanced designs without sacrificing power makes it highly suitable for realistic biological studies where sample sizes are limited. By decomposing rhythmic changes into amplitude, phase, and baseline components, CARhy offers deeper biological insights into how external perturbations (e.g., diet, drugs) reshape circadian gene expression. The method sets a new standard for reproducibility and statistical rigor in transcriptomic rhythm analysis.