Robust and adaptive non-parametric tests for detecting general distributional shifts in gene expression

Summary

Differential expression analysis is crucial in genomics, yet most methods focus only on mean shifts. Variance shifts in gene expression—especially in cellular signaling and aging—are increasingly recognized as being biologically important. We present QRscore (quantile rank score), a general non-parametric framework that extends the Mann-Whitney test to detect both mean and variance shifts through model-informed weights derived from negative binomial (NB) and zero-inflated NB (ZINB) distributions. QRscore offers high statistical power with false discovery rate (FDR) control, surpassing existing methods in detecting both types of distributional changes. When applied to bulk RNA sequencing (RNA-seq) data from Genotype-Tissue Expression (GTEx), QRscore uncovers numerous genes with dispersion shifts that are missed by mean-shift analysis. In pseudo-bulked single-cell RNA-seq data from the Asian Immune Diversity Atlas (AIDA), QRscore further reveals cell-type-specific variance shifts across age groups that remain undetectable in bulk analyses. QRscore augments the genome bioinformatics toolkit by offering a powerful and flexible approach for differential expression analysis.

Graphical abstract

Highlights

• •QRscore detects both variance and mean shifts in gene expression
• •It controls FDR and achieves high power under noise and zero inflation
• •Tissue-specific, age-related differentially dispersed genes are identified from GTEx data
• •Cell-type-specific variance shifts associated with aging are revealed from scRNA-seq data

Motivation

The increasing availability of large-scale RNA-seq datasets from both bulk and single-cell sources increases our power to detect biologically meaningful shifts in mean gene expression and expression variability. However, existing methods often prioritize mean shifts or rely on strong parametric assumptions that may not hold across heterogeneous populations. This has created a gap in methods that adapt to the alternative hypothesis of interest while also controlling type I and type II errors without requiring parametric assumptions to formulate the null hypothesis. We thus develop QRscore, an R/Bioconductor software that leverages non-parametric statistics and asymptotic theory to obtain powerful and false discovery rate (FDR)-controlled flexible tests of differential expression.

Zhou et al. present QRscore, a robust non-parametric framework for statistical hypothesis testing that detects shifts in both the mean and variance of gene expression. QRscore maintains high power and strict FDR control, even under noise and zero inflation. It reveals age-related variance changes in both bulk RNA-seq and pseudo-bulked single-cell RNA-seq data.

Introduction

In both bulk and single-cell genomics, differential expression analysis is a routine procedure for identifying genes whose expressions differ under varying conditions or treatments. To detect changes in average expression, widely used methods in bulk RNA sequencing (RNA-seq) differential analysis treat the dispersion of gene expression as a nuisance parameter, implementing techniques to shrink and stabilize the estimation of variance. The widespread use of these approaches can result in missed signals of gene expression variability that capture important biological mechanisms. In humans and C. elegans, variability indices are associated with gene regulation^1,2 and cell development,³ potentially explaining many biological phenomena, including adaptation to changing environments, disease susceptibility, and incomplete penetrance.^4,5 Recent studies also demonstrate that variability indices provide insights into aging,^6,7,8 therefore underscoring the crucial need to incorporate changes in variability into tests of differential gene expression.

To detect expression variability, previous studies on microarray datasets^9,10 applied traditional hypothesis testing methods, such as the F-test and Bartlett’s test. These tests are based on normal distribution assumptions of two samples and are designed to identify differences in standard deviation. To improve robustness, one approach is to analyze different measures for variability other than the standard deviation. Such measures include the median absolute deviation¹⁰ and the coefficient of variation.² Another approach to ensure the robustness of statistical tests is to employ methods that are more robust to deviations from normality for two samples, such as Levene’s test and permutation tests.¹⁰ The method diffVar¹¹ implements a modified version of Levene’s test that utilizes an empirical Bayes estimator instead of variance estimation for t statistics to obtain a more powerful test. However, this method maintains its parametric nature, making it susceptible to reduced performance due to deviations from the underlying assumptions. Additionally, non-parametric methods like the Kolmogorov-Smirnov (KS) test are not commonly used in the analysis of RNA-seq data because of their limited statistical power, especially when working with small sample sizes.

Because bulk RNA-seq data often show overdispersion, particularly in gene expression, where the variance for most genes exceeds the mean, they are commonly modeled using the negative binomial (NB) or zero-inflated NB (ZINB) distribution. One method for detecting differences in dispersion applies generalized additive models for location, scale, and shape (GAMLSS),¹² allowing the addition of regressions for dispersion parameters in the NB distribution into the main regression model and targeting changes in dispersion parameters through likelihood ratio tests.⁴ Another method, MDSeq,¹³ utilizes a reparameterization of the ZINB model to test the mean and dispersion separately from generalized linear models (GLMs). DiffDist¹⁴ uses a hierarchical Bayesian model based on the NB distribution, which can provide tests for differential dispersion and distribution for RNA-seq data.

Although these current methods are based on NB distributions and are widely applied in scenarios of small sample sizes, recent research has shown that parametric methods are prone to inflated false discovery rates (FDRs) in both studies with small sample sizes¹⁵ and population-level RNA-seq studies with large sample sizes.¹⁶ Additionally, they are highly sensitive to preprocessing procedures¹⁷ and are not robust to model misspecification.¹⁸

Here, we introduce QRscore (quantile rank score), a flexible rank-based method for conducting robust and powerful two-sample tests. QRscore generalizes the Mann-Whitney test and capitalizes on the adaptability of the chosen test statistic to yield substantial statistical power across various scenarios, including the detection of shifts in mean and variance. QRscore offers various approaches for designing test statistics that are effective in detecting shifts in mean and dispersion. These include a versatile naive approach applicable to various cases beyond count data, as well as methods designed specifically for RNA-seq count data with either a low occurrence of zeros or a higher proportion of inflated zeros. Also, we extend QRscore to K-sample tests, where $K\ge 3$. Through simulations that extensively compare QRscore with existing differential variability tests and differential centrality tests, we find that QRscore controls FDRs across diverse model specifications while simultaneously obtaining high statistical power. We use QRscore to detect both differentially dispersed genes (DDGs) with variance shifts and differentially expressed genes (DEGs) with mean shifts across multiple tissues in the Genotype-Tissue Expression (GTEx) project,¹⁹ showing that differential variability characterizes functionally distinct sets of genes from differential centrality. We also apply QRscore to pseudo-bulked single-cell RNA-seq (scRNA-seq) data, highlighting its potential to uncover cell-type-specific variance shifts that are obscured in bulk RNA-seq analyses. Our method contributes to the field of bioinformatics tools for genomics and can be adapted to other applications that require flexible non-parametric tests.

Results

Overview of QRscore

QRscore is a rank-based hypothesis testing method that achieves robust false positive control through its non-parametric foundation while gaining power from parametric model-informed weighting (Figure 1).

Figure 1.

Overview of QRscore

(1) Prefiltering and normalization: this initial data processing stage ensures quality and comparability of data. (2) Pooling and sorting: for gene g, expression levels $X_g=\left\{X_{1g},\dots ,X_{kg}\right\}$ and $Y_g=\left\{Y_{1g},\dots ,Y_{ng}\right\}$ are combined into a single ordered list. The rank of $X_{ig}$, denoted as $R_{X_{ig}}$, is determined after pooling and sorting the expression values of gene g. (3) Constructing the test statistic: a test statistic t is calculated using a weighted function based on modified score test functions, which enhances test power through parametric estimation and incorporates a generalized K-sample test. (4) Hypothesis and inference: this stage involves evaluating the null hypothesis for gene $g,H_{0g}$, $P_g=Q_g$, where $P_g$ and $Q_g$ represent the distributions of gene expression samples $X_g$ and $Y_g$, respectively. p values are computed and adjusted using the Benjamini-Hochberg procedure to control the false discovery rate. (5) Identification of DDGs and DEGs: genes with significant expression-level differences between the conditions are identified, as represented in the QRscore-Var and QRscore-Mean diagrams. Details of QRscore-Var and QRscore-Mean are provided in the STAR Methods and Methods S1.

Our null hypothesis posits that the two samples are drawn from the same distribution. Specifically, given a gene g, let $X_{ig}$ and $Y_{jg}$ denote the normalized gene expression for individuals i and j in the first and second conditions, respectively, where $i=1,\dots ,k$ and $j=1,\dots ,n$. With samples ${\left\{X_{ig}\right\}}_{i=1}^k\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}.}{\sim }{P}_g$ and ${\left\{Y_{jg}\right\}}_{j=1}^n\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}.}{\sim }{Q}_g$, the null and alternative hypotheses for gene g are

$$H_{0g}:P_g=Q_g\text{and}{H}_{1g}:P_g\ne Q_g\text{.}$$

Our rank-based test statistic, $T_g^w$, is computed as the normalized sum of weighted ranks of the $X_{ig}$ observations using the formula

$$T_g^w=\frac{1}{\sqrt{n+k}}\sum _{i=1}^k{w}_g\left(\frac{R_{X_{ig}}}{n+k}\right)\text{,}$$ (Equation 1)

where $R_{X_{ig}}$ denotes the rank of $X_{ig}$ after pooling and sorting observations from both groups. Here, $w_g$ is a weight function chosen to maximize the test power against user-specific alternatives. Its dependence on g indicates that it can be chosen differently across different genes.

While the space of all possible weight functions $w_g$ induces a family of test statistics of the form shown in Equation 1, particular choices of $w_g$ lead to well-known tests. For instance, $w_g\left(x\right)=x$ gives the Mann-Whitney test, while $w_g\left(x\right)={\left(x-0.5\right)}^2$ corresponds to Mood’s test.²⁰

Focusing on differential expression analysis applications, we design weight functions $w_g$ by composing the score function (the gradient of the log likelihood with respect to the parameters) with the inverse cumulative distribution function (see STAR Methods). We derive these score functions selectively from either an NB or a ZINB distribution, with QRscore-Var targeting DDGs and QRscore-Mean targeting DEGs. The rationale behind this design is to overcome the lower power of traditional rank-based models; our approach significantly boosts power under correct model specification and achieves efficiency comparable to the likelihood ratio test.

Despite the tailored weight design, our method remains distribution-free under the null hypothesis $H_{0g}$, which assumes identical distributions ($P_g=Q_g$). This is achieved because the rank-based inputs are uniformly distributed under the null, regardless of the underlying data distribution. This distribution-free property provides a significant advantage: it inherently reduces the impact of model misspecification and helps maintain strict control over false discoveries.

Moreover, our method also extends to support K-sample tests (see STAR Methods), broadening its applicability to multiple sample scenarios. Overall, QRscore is a versatile and powerful approach for detecting differential mean or dispersion in RNA-seq data, combining the robustness of non-parametric tests with the enhanced detection power of a special weight design. The details of the test statistic construction and its statistical inference are described in STAR Methods and Methods S1.

Simulation results

In this section, we evaluate existing methods for detecting genes with changes in dispersion or mean across groups using extensive simulations. To benchmark two-sample tests, we use mean and dispersion parameters estimated from GTEx whole-blood RNA-seq data as the baseline. We simulate fold changes in dispersion or mean across randomly selected DDGs, varying the magnitude across datasets. To test robustness, we introduce outliers and zero inflation in different proportions. We also vary sample sizes to reflect realistic experimental designs. Lastly, we compare QRscore’s performance in both two-sample and three-sample tests with varied fold changes and sample sizes. Simulation details are provided in the STAR Methods.

Key simulation results are qualitatively summarized in Table 1. The hypothesis tests can be categorized into two groups: those that rely on specific parametric distributions and those that are non-parametric. For DDG analysis, parametric methods include GAMLSS, MDSeq, diffVar, and Levene’s test, while Mood and QRscore-Var are distribution free. Similarly, for DEG analysis, methods like DESeq2, edgeR, and limma-voom rely on parametric models, whereas Mann-Whitney and QRscore-Mean are distribution free. To analyze the sensitivity and robustness of these methods, metrics such as the area under the precision-recall curve (AUPRC), power, and FDR under different simulation settings are computed. Non-parametric methods generally showed greater robustness to outliers and zeros. Notably, QRscore-Var and QRscore-Mean achieved the highest detection power for identifying DDGs and DEGs across various scenarios. In contrast, parametric methods were more susceptible to the negative effects of outliers and inflated zeros, often leading to an inflated FDR. Detailed results are provided in the following subsections.

Table 1.

Summary table for simulation results

Method	Assumption	Targeted direction	Efficient in NB model	Robust to outliers	Robust to zeros
QRscore	non-parametric	mean/variance	yes	yes	yes
GAMLSS	NB	variance	yes	no	no
MDSeq	ZINB	mean/variance	no	no	yes
Mood	non-parametric	variance	no	yes	yes
diffVar	Gaussian	variance	yes	yes	no
Levene	Gaussian	variance	yes	yes	no
DESeq2	NB	mean	yes	no	yes
edgeR	NB	mean	yes	no	yes
limma-voom	Gaussian	mean	yes	yes	no
Mann-Whitney	non-parametric	mean	yes	yes	yes

QRscore is a solid contender in differential dispersion and differential expression analysis under correctly specified NB models

In simulations based on NB distributions, QRscore-Var exhibits strong performance, slightly behind GAMLSS (setting 1 in Table 2) and achieving comparably high detection power with FDR controlled (Figure 2, “no zero inflation or outliers”). This close competition highlights QRscore-Var’s effectiveness in scenarios with correctly specified NB models. QRscore-Var also demonstrates higher power than naive methods like Mood’s test and Levene’s test (Figure S1A). In smaller sample size settings (settings 2–3 in Table 2), QRscore-Var consistently ranks as the second-best method after GAMLSS, reinforcing its reliability and showing notable improvements as sample sizes increase.

Table 2.

Average area under the AUPRC across all fold changes in all seeds for each simulation setting

Setting	$n_1$	$n_2$	α (outliers)	β (zeros)	DiffVar	GAMLSS	Levene	MDSeq	Mood	QRscore-Var
1	113	249	0	0	0.774	0.798	0.775	0.701	0.749	0.788
2	50	100	0	0	0.615	0.642	0.610	0.558	0.568	0.634
3	50	50	0	0	0.518	0.575	0.525	0.516	0.479	0.527
4	113	249	0.05	0	0.762	0.753	0.763	0.627	0.745	0.783
5	113	249	0.1	0	0.754	0.714	0.756	0.573	0.745	0.780
6	113	249	0.2	0	0.728	0.640	0.731	0.481	0.737	0.772
7	113	249	0	0.05	0.741	0.736	0.753	0.699	0.742	0.783
8	113	249	0	0.1	0.716	0.689	0.736	0.699	0.741	0.780
9	113	249	0	0.2	0.650	0.594	0.692	0.698	0.732	0.772
10	113	249	0.1	0.1	0.694	0.629	0.714	0.569	0.734	0.772
11	50	100	0.1	0.1	0.529	0.429	0.555	0.399	0.550	0.613
12	50	50	0.1	0.1	0.427	0.363	0.472	0.367	0.458	0.503

The simulation results in this table are from scenarios where the dispersions of group 1 are greater than those of group 2 for true DDGs. For each setting, the greatest value is shown in bold. Sample size for group 1: $n_1$; sample size for group 2: $n_2$; proportion of genes with random outliers: α; and proportion of genes with random zeros: β. Sample sizes $n_1=113$ and $n_2=249$ are chosen from GTEx whole-blood tissue data, corresponding to the 40–49 and 60–69 age groups, respectively. When perturbations are introduced, QRscore outperforms the others, and it ranks second when model assumptions are perfectly met. For a table that shows cases where the dispersions of group 2 are greater than those of group 1 for true DDGs, see Table S1. AUPRC, area under the precision-recall curve.

Figure 2.

Simulation results for QRscore-Var benchmarked against other methods for detecting DDGs

The true DDGs have greater dispersion in group 1 than in group 2.

(A) Line plots of power when true false discovery proportions are less than 0.05 across different dispersion fold changes.

(B) Line plots of FDR when adjusted p values are less than 0.05 across different dispersion fold changes.

Both plots use $n_1=113$ and $n_2=249$ (GTEx derived). We present four different settings in the plot: baseline NB model, 10% genes with outliers, 10% genes with zero inflation, and a combination of both. For additional methods benchmark, see Figure S1.

Likewise, as shown in Table S3, QRscore-Mean shows performance comparable to DESeq2 and edgeR in differential mean analysis, with a marginal difference in the AUPRC of about 1%. This slight difference in power and FDR is further illustrated in setting 1 of Figure S2.

QRscore-Var leads in robustness and effectiveness in gene expression variability analysis with outliers and excess zeros

Next, we simulate data with random outliers (settings 4–6 in Table 2), where QRscore-Var emerges as the best-performing method based on the average AUPRC. In contrast, methods like GAMLSS and MDSeq show much lower AUPRC scores, even with a small proportion of outliers, due to their reliance on the NB assumption, which makes them vulnerable to model misspecification. For instance, in setting 4 with 5% random outliers, the AUPRC for GAMLSS drops by 5.6% (from 0.798 to 0.753), while QRscore-Var experiences only a 0.6% drop (from 0.788 to 0.783). As the outlier proportion increases to 20%, GAMLSS’s score declines by 19.8% (to 0.640), whereas QRscore-Var remains stable, with only a 2.0% drop (to 0.772). MDSeq also shows considerable performance degradation, and both GAMLSS and MDSeq exhibit inflated FDRs in the presence of 10% outliers, as shown in Figure 2B (“outliers").

In simulations with random zeros (settings 7–9 in Table 2), QRscore-Var again demonstrates robust performance. GAMLSS and diffVar, by contrast, show inflated FDRs and reduced power under zero-inflated conditions (Figure 2, “zero inflation"). Despite MDSeq being designed to accommodate technical zeros, QRscore-Var consistently outperforms it across all fold changes. Among non-parametric methods, QRscore-Var also achieves the highest power in zero-inflated settings (Figure S1, “zero inflation”). Furthermore, QRscore maintains reliable power and FDR control under both up- and down-regulation of variance, underscoring its robustness across different shift directions (see Tables 2 and S1).

To assess QRscore-Var’s robustness in settings involving small samples, we perform additional simulations with group sizes ranging from 5 to 15, incorporating noise through randomly inserted outliers and excess zeros. We observe that parametric methods such as MDSeq and GAMLSS exhibit severe FDR inflation as sample sizes decrease and fail to provide reliable discoveries with controlled FDRs. In contrast, QRscore-Var consistently maintains strict FDR control across all scenarios and achieves the highest power when group sizes are 8 or higher (Figure S3A). We find that all methods have low power below group sizes of 8. Therefore, we recommend using at least 8 samples per group to ensure true discoveries while maintaining FDR control.

Finally, to assess the robustness of QRscore-Var under additional model misspecification, we conduct simulations where gene expression follows bimodal distributions—an important scenario given that real RNA-seq data can exhibit multimodal patterns due to heterogeneous cell populations or mixed regulatory states. As shown in Table S2, QRscore-Var consistently achieves the highest average AUPRC across all scenarios with outliers and excess zeros. These results demonstrate that QRscore-Var maintains strong FDR control and detection power even when the data distribution departs from standard unimodal assumptions.

QRscore-Mean leads in robustness and effectiveness in differential gene expression analysis with outliers and excess zeros

QRscore-Mean, similar to QRscore-Var, also shows the strongest performance in differential gene expression analysis under high proportions of random outliers (setting 6 in Table S3). While edgeR and DESeq2 show marginally higher AUPRC scores with fewer outliers, they suffer from inflated FDRs, making their detection results less reliable in practice (Figure S2B, “outliers"). Among the methods that control the FDR, QRscore-Mean performs the best in terms of the average AUPRC and detection power.

QRscore-Mean outperforms other methods in differential gene expression analysis when there are excess zeros (settings 7–9 in Table S3). This improved performance is due to its robustness in handling data distribution irregularities, particularly those characterized by excess zeros. While random zeros weaken the signal without inflating FDR, this loss of power impacts methods like edgeR and DESeq2, which lack explicit handling of zero inflation.

We also conduct additional “small sample" simulations with group sizes ranging from 5 to 15, incorporating realistic noise through randomly inserted outliers and excess zeros. QRscore-Mean and the Mann-Whitney test preserve FDR control across all scenarios. Although DESeq2 and edgeR show slightly higher AUPRC values at very small sample sizes, their inflated FDR limits practical utility, as shown in Figure S3B. In contrast, QRscore-Mean consistently offers more accurate results. Similar to variance shift testing, we recommend a minimum sample size of 8 per group to ensure sufficient detection power.

Three-sample tests show superiority compared to pairwise two-sample tests in diverse simulations for QRscore-Var

We investigated the performance of the three-sample QRscore test (see STAR Methods) compared to the commonly used approach of sequential pairwise testing followed by p value aggregation. In practice, the user may prefer to test the heterogeneity of gene expression across multiple groups or conditions and obtain a single, well-calibrated p value for each gene. Results from Figure S4 demonstrate that the three-sample QRscore test outperforms the pairwise two-sample tests in terms of the AUPRC, particularly when the sample sizes for different groups are not equal. Additional simulations explore scenarios where the three samples share the same sample size and cases with varying degrees of dispersion among all three groups. These simulations, which also include the presence of random outliers and zeros, show that the three-sample tests consistently provide superior AUPRC results, demonstrating better adaptability and precision even under model perturbations. Based on the simulation results, we will apply three-sample tests to GTEx data in application to GTEx.

Application to GTEx

Having demonstrated QRscore’s error control capabilities and enhanced power, we now apply it to the GTEx dataset. The GTEx project provides RNA-seq data from bulk tissue samples, gathered from nearly 1,000 deceased donors across various age groups and both sexes. This dataset covers a wide array of tissue types, enabling tissue-specific gene expression analyses.

The flexibility of QRscore allows for the detection of tissue-specific DDGs, as well as distinct gene sets that differ in mean expression across tissues. We also show that these gene sets exhibit unique functional effects.

QRscore-Var detects tissue-specific genes with changes in dispersion

We apply QRscore-Var across 33 tissues to identify DDGs across three age groups: 20–39 (younger), 40–59 (middle), and 60–79 (older). We detected distinct trends in variance shifts from younger to older age groups (Figure 3A). In tibial artery and skeletal muscle tissues, most DDGs show an increase in variance from the middle to the older age group. In contrast, in whole-blood tissue, we observe more DDGs showing an increase in variance from the younger to the middle age group, followed by a decrease from the middle to the older group. Our latter finding agrees with a recent study on gene expression in blood samples from twins, which reported a general decline in expression variability with age.⁶ In total, 10,362 unique tissue-specific DDGs were detected.

Figure 3.

QRscore-Var results on GTEx data

(A) A stacked bar plot illustrates the quantity of genes identified by QRscore-Var, categorized by variance trends. For instance, the “decrease-decrease" label indicates a progressive reduction in gene expression variance from the youngest age group (20–39) through the middle age group (40–59) and into the oldest age group (60–79).

(B) Histogram of the number of tissues in which each DDG detected only by QRscore-Var is found. The red bars indicate DDGs found from more than 3 tissues.

(C) UpSet plot showing overlaps of DDGs from the three-sample test and corresponding pairwise comparisons.

(D) Bar plots display the log counts of DDGs identified in various tissues from female and subsampled male individuals, respectively.

To better interpret these patterns, we perform pairwise two-sample tests between age groups (Figure 3C). In the tibial artery, variance shifts are primarily driven by differences between the youngest group and the other two. In the transverse colon, the oldest group contributes most to the detected differences. These patterns reflect how age range and tissue context shape dispersion trends.

The majority of DDGs are tissue specific, where each gene is detected in only one tissue (Figure 3B). However, a few genes show differential dispersion across multiple tissues (Table S5). For example, EDA2R, a gene previously known to be associated with aging in the lung²¹ and skeletal muscle,²² is detected as a DDG in 16 tissues. Additionally, PTCHD4, previously associated with aging in the thyroid gland,²³ is found to have differential dispersion in 16 tissues.

Sex can significantly influence the detection of variance shifts, and the GTEx dataset includes more male than female samples. To address this potential confounding factor, we stratify the data by sex and subsample male samples to match the number of females, aiming for a fair comparison (Figure 3D). In tissues such as the tibial artery, transverse colon, and whole blood, the numbers of DDGs identified in male and female samples are similar, and there are sex-specific DDGs discovered in each tissue. Interestingly, within adipose visceral tissue, even after subsampling to equalize sample sizes, DDGs are identified in male samples but not in female samples. These findings provide insights into sex-specific DDGs during aging.

QRscore-Var identifies different sets of genes from QRscore-Mean

In parallel with QRscore-Mean, designed to identify genes with mean expression changes using tailored weights, we used QRscore-Var for its capacity to detect variance shifts. Although QRscore-Var detects fewer genes overall compared to QRscore-Mean, it identifies 2,652 unique DDGs across at least one tissue that QRscore-Mean does not capture (that is, DDGs that are not also DEGs; see Figure 4A), indicating its ability to uncover different aspects of gene expression changes.

Figure 4.

Analysis of age-related differential gene expression across human tissues using QRscore-Var and QRscore-Mean

(A) Pie charts visualizes the proportion of gene biotypes across various tissues, analyzed through the QRscore-Var and QRscore-Mean methods. The slices represent protein-coding genes, long noncoding RNAs (lncRNAs), and small noncoding RNAs (sncRNAs), with immunoglobulin (Ig) genes separated into their own category due to their substantial proportion in some tissues. Pseudogenes are excluded. A white circle at the center indicates the total gene count. Tissues are ranked by the number of DDGs unique to QRscore-Var.

(B) Bar plots for selected tissues, highlighting the top 20 significant genes only detected by QRscore-Var. These plots juxtapose the significance levels ($-{\log }_{10}\left(p\text{value}\right)$) of identified genes between the QRscore-Var and QRscore-Mean analyses, with a focus on protein-coding genes.

A notable number of DDGs are identified only by QRscore-Var, but not QRscore-Mean, in several tissues, in line with studies emphasizing variance shifts in the transverse colon and thyroid.²⁴ While most DDGs are protein-coding genes, immunoglobulin (Ig) genes are also prominently detected, especially in tissues like esophagus mucosa, transverse colon, and lung (Figure 4A). Their contribution suggests a link between age-related variability and immune-response alterations. These results indicate that variance shifts can reveal biologically meaningful changes that complement mean-based analyses in understanding aging.

We summarize the top 20 protein-coding DDGs with age-associated variance shifts that are not classified as DEGs across six tissues (see Figure 4B). Variance shifts in genes like C-C motif chemokine ligand 2 (CCL2) in whole blood and SAA1 in the transverse colon underscore their roles in shaping the inflammatory landscape during aging. CCL2 is pivotal in chronic immune responses during aging,²⁵ while SAA1 contributes to an inflammatory microenvironment that varies across individuals, leading to diverse impacts on epithelial cell function via MMP9 up-regulation in aging tissues.²⁶ In the tibial artery, ADRA2B regulates vascular tone, with its variance shifts suggesting its importance in maintaining vascular health with age. BNIP3 in skeletal muscle indicates diverse mitochondrial balance and autophagy responses, signaling varied susceptibility to age-related muscle degradation.²⁷ Meanwhile, SERPINB1 in the lung safeguards against protease-induced tissue damage, with its variance shifts revealing intricate age-related changes that impact lung maintenance and contribute to chronic lung conditions.²⁸ These gene expression shifts reveal the intricate, tissue-specific regulatory changes that contribute to aging and its associated dysfunctions.

We compile lists of genes related to human aging from three databases, GenAge,²⁹ LongevityMap,³⁰ and CellAge³¹ to annotate age-related genes. We rank the genes based on p values for each tissue and age group, using QRscore-Var or QRscore-Mean tests, and apply one-sided Wilcoxon rank-sum tests to evaluate the methods’ effectiveness at prioritizing aging-related genes. In Table S4, QRscore-Var shows statistical significance ($p<0.05$) in ranking aging-related genes in twelve out of fourteen tissues. However, the Wilcoxon test p values are less significant than QRscore-Mean in most tissues and are not significant in lung and lower leg skin (sun exposed). This suggests that, as the existing databases are mostly annotated by mean expression shifts, they may miss age-related DDGs that QRscore-Var can identify, and QRscore-Var provides a complementary perspective on the genetic basis of aging.

QRscore-Var and QRscore-Mean identify genes that are functionally distinct

To investigate the biological functions of genes identified by QRscore, we performed Gene Ontology (GO) enrichment analysis for the top 500 DDGs and DEGs ranked by QRscore-Var and QRscore-Mean, respectively. In whole blood, while QRscore-Mean highlights fundamental processes such as mRNA processing and RNA splicing, QRscore-Var reveals enrichment in pathways related to chemotaxis, angiogenesis, cell motility, and regulation of the ERK1/2 cascade—reflecting broader regulatory changes beyond mean expression shifts (Figures 5A and 5B). Specifically, genes with large variance shifts, such as HRG, ANGPTL3, and SERPINE1, are linked to angiogenesis, while FGB and FGG are involved in immune signaling and blood coagulation (Figure 5C). Additionally, the analysis reveals notable variance shift among age groups in the CCL gene family, such as CCL14, CCL18, and CCL20, which are enriched in the chemokine-mediated signaling pathway. This further highlights the intricate gene expression dynamics involved in immune responses and inflammation.

Figure 5.

Comparative analysis of GO enrichment in whole-blood tissue based on significant genes identified by QRscore-Var and QRscore-Mean

(A) A dot plot illustrates the enriched GO terms within the biological process ontology for the top 500 genes significantly detected by QRscore-Var.

(B) Similar enrichment analysis results for genes identified by QRscore-Mean.

(C) A heat plot, generated using ClusterProfiler, shows the fold change in expression variance between the 20–39 and 40–59 age groups for significant genes associated with the top 5 enriched GO terms identified by the QRscore-Var analysis.

(D and E) A dot plot (D) and a density plot (E) further investigate the expression patterns of two highly coexpressed genes identified by QRscore-Var as being involved in angiogenesis.

We further identify the top QRscore-Var genes associated with significantly enriched GO terms across 20 tissues and compare their QRscore-Mean rankings (see Table S6). In whole blood, CCL2 is uniquely identified as significant by QRscore-Var and exhibits non-monotonic changes in dispersion with age. Among the top 25 DDGs contributing to enriched GO terms in this tissue, 23 are ranked within the top 200 by QRscore-Var but fall outside the top 1,000 under QRscore-Mean, underscoring the distinct insights gained from analyzing expression variability. We also identify coexpressed DDG pairs linked to key biological processes—for example, SERPINE1 and ANGPTL4 are strongly coexpressed (Spearman’s $\rho =0.808$) and are implicated in angiogenesis and blood coagulation (Figures 5D and 5E).

Across other tissues—including the tibial artery, transverse colon, and skeletal muscle—QRscore-Var reveals biological processes and genes with significant expression variability during aging. In the tibial artery, variance shifts in DDGs highlight tissue-specific processes such as sprouting angiogenesis and the positive regulation of cell migration. Beyond these processes, SEMA6A, identified as a DDG but not a DEG, is associated with the semaphorin-plexin signaling pathway and the negative regulation of axon extension involved in axon guidance, pointing to additional molecular mechanisms at play in the tibial artery. In the transverse colon, DDGs are enriched for processes such as digestion, steroid and hormone metabolism, and xenobiotic response, highlighting their roles in maintaining gastrointestinal function, regulating metabolic activity, and responding to environmental stimuli. In skeletal muscle, coexpressed DDGs like COMP and FMOD underscore their importance in collagen fibril organization and maintaining the structural integrity of muscle tissue. The DDGs, enriched pathways, and coexpression patterns described here are shown in Tables S6 and S7.

Application to pseudo-bulked scRNA-seq data

Here, we apply QRscore to data from the Asian Immune Diversity Atlas (AIDA),³² a large-scale scRNA-seq resource profiling peripheral blood mononuclear cells (PBMCs) from healthy donors across Asia. Our analysis focuses on individuals of East Asian identity to minimize confounding due to population heterogeneity (see STAR Methods). The single-cell resolution of AIDA enables detailed exploration of age-related regulatory changes in specific immune cell types, which are often obscured in bulk tissue analyses. To increase detection power, stabilize variance estimates, and enable robust population-level statistical testing, we aggregate single-cell profiles into pseudo-bulk samples by donor, age group, and cell type. This approach preserves cell-type resolution while capturing inter-individual variability, aligning with our goal of characterizing gene expression variance across the population. Applying QRscore-Var across 25 immune cell types and three age groups (20–34, 35–49, and 50+), we detect both cell-type-specific and shared patterns of variance shifts, highlighting distinct regulatory programs across the immune system.

We identify widespread shifts in gene expression variability with aging. Among the analyzed cell types, naive thymus-derived CD4-positive, alpha-beta T cells; central memory CD4-positive, alpha-beta T cells; and CD16-positive, CD56-dim natural killer cells exhibit the highest DDG counts, suggesting greater transcriptional reprogramming in these cell types during aging (Figure S5A). Most DDGs exhibit monotonic variance trajectories across age groups, with the majority showing either progressive increases (increase-increase) or decreases (decrease-decrease) in dispersion from younger to older individuals. A smaller subset of DDGs display non-monotonic patterns, with predominantly increase-decrease trajectories and a minority showing decrease-increase variance shifts. Although the majority of DDGs are specific to individual cell types (Figure S5B), several genes, such as RBM38, RPL27A, and BRD4, are recurrently identified across multiple immune cell-type populations. Notably, RBM38, a p53-regulated RNA-binding protein involved in mRNA stability and cellular senescence, is linked to aging and tumor suppression.³³ Furthermore, pairwise comparisons reveal that variance shifts are often strongest between the youngest and oldest groups (Figure S5C). In the top three cell types with the highest DDG counts, two-sample tests show that more DDGs exhibit significant variance shifts in the 20–34 vs. 50+ and 35–49 vs. 50+ comparisons, indicating that the oldest age group displays the most pronounced and distinct variance patterns.

Next, we identify genes with significant variance shifts detected in both GTEx whole-blood and pseudo-bulked scRNA-seq analyses, with overlaps of 11%–32% across immune cell types (Figure S6A). Among the shared genes, key regulators such as RBM38, RGS1, and the NR4A family (NR4A1/2/3) emerge as recurrent across multiple cell types, suggesting their involvement in a core aging-related regulatory program. These genes are well-established mediators of immune activation, stress responses, and inflammatory regulation—processes tightly linked to immunosenescence.

To explore the functional relevance of these shared DDGs, we perform ontology enrichment analysis on the overlap between pseudo-bulk and bulk results for CD14-positive monocytes. The analysis reveals enrichment for inflammatory cytokine responses, including pathways related to interleukin-1, tumor necrosis factor (TNF), and chemokine signaling—key processes in age-associated inflammation (Figure S6B). Genes such as ZC3H12A, CXCL8, CCL4, and CCL20 exemplify how transcriptional variability in monocytes is linked to enhanced responsiveness to pro-inflammatory cytokines and immune aging (Figure S6C).

While some genes are shared between the GTEx whole-blood and pseudo-bulked scRNA-seq analyses, many DDGs are uniquely identified in the pseudo-bulk analysis, highlighting the added resolution provided by cell-type-specific variance analysis. Figure S5D shows bar plots of the top 20 DDGs for the three cell types with the highest DDG counts, including cell-type-specific DDGs shown in black. In naive thymus-derived CD4-positive, alpha-beta T cells, FURIN and EXT1 suggest modulation of cytokine signaling and extracellular immune interactions. In CD16-positive, CD56-dim natural killer cells, variance shifts in NFIL3 and IL1B reflect dynamic regulation of development and inflammatory responses. In central memory CD4-positive, alpha-beta T cells, CSNK1E and KLHL11 exhibit variability changes linked to immune rhythm and tolerance. These genes highlight the unique contributions of cell-type-specific analysis in uncovering regulatory mechanisms with age-related variability that were missed in our bulk analysis.

Discussion

We introduced QRscore, a suite of rank-based non-parametric tests for identifying DDGs and DEGs from bulk RNA-seq data. We showed, through extensive simulations, that QRscore is robust at detecting both mean and dispersion shifts, particularly under challenging and realistic conditions where datasets have inflated zeros and outliers. Notably, QRscore controls false discovery well and achieves high power even in such settings. Applying QRscore to GTEx data, we find different DDGs and DEGs in each tissue that are also functionally distinct, implicating novel candidates for further biological investigation.

Model misspecification is a critical issue in RNA-seq data analysis that potentially explains the inflation of false discoveries observed in practice. For instance, a study³⁴ showed that methods relying on specific distributional assumptions, like NB or ZINB, identify different sets of DEGs across mouse strains and human tissues, with elevated FDRs in the presence of outliers. As a rank-based non-parametric method, QRscore is less sensitive to data distribution and effectively handles technical zeros and outliers. By being less prone to model misspecification, QRscore yields more reliable and replicable results, improving the likelihood of downstream experimental validation.

As pointed out in the introduction, gene expression variability plays an important role in aging, adaptation to changing environments, and disease susceptibility. Focusing on aging in our present work, senescence-related changes in gene expression variability are often driven by shifting regulatory efficiency and the accumulation of epigenetic modifications,³⁵ which may go unnoticed when focusing solely on mean expression levels. QRscore fills this gap by detecting variance shifts, revealing biologically relevant DDGs that highlight aging-related processes like angiogenesis in whole blood. These findings reinforce the value of analyzing gene expression variability to uncover insights beyond mean-level changes.

We further demonstrated that QRscore can be applied to pseudo-bulked scRNA-seq data, enabling analysis of cell-type-specific regulatory variability. This is important because bulk tissue data, like those of GTEx, aggregate signals from mixed cell populations, potentially obscuring variability specific to a particular lineage. In our analysis, many DDGs with cell-type-specific roles were uniquely identified in pseudo-bulk data. Additionally, QRscore identified shared signals across GTEx and pseudo-bulked scRNA-seq analyses, despite differences in sequencing technology, cell population composition, cell numbers per sample, and population ancestry. Together, these results highlight QRscore’s ability to uncover meaningful, cell-type-specific insights and robust signals of gene expression variability across diverse contexts.

The future of differential expression analysis lies in detecting diverse expression changes while accounting for confounders and technical noise. Flexible methods like QRscore support this goal by capturing distributional shifts. As tools improve, biologists will gain deeper insight into how gene expression varies across myriad contexts.

Limitations of the study

One limitation of QRscore is that it tends to be conservative when the sample size is small. This is an inherent constraint of rank-based tests, which generally require larger sample sizes to ensure reliable decisions. As large-scale datasets like GTEx become more common, future work will extend QRscore to even larger and more diverse populations.

Resource availability

Lead contact

Requests for further information and resources should be directed to and will be fulfilled by the lead contact, Yun S. Song (yss@berkeley.edu).

Materials availability

This study did not generate new unique reagents.

Data and code availability

• •This paper analyzes existing, publicly available data. The accession numbers for the datasets are listed in the key resources table. The data used to reproduce the figures in this paper are available at https://doi.org/10.5281/zenodo.16416084.
• •The R package for QRscore is available at https://github.com/songlab-cal/QRscore and Bioconductor. The code for reproducing the figures in this paper is available at https://doi.org/10.5281/zenodo.16416084.
• •Any additional information required to reanalyze the data reported in this work paper is available from the lead contact upon request.

Acknowledgments

This research is supported in part by NIH grant R35-GM134922 and grant number CZF2019-002449 from the Chan Zuckerberg Initiative Foundation.

Author contributions

All authors contributed to the conception of the work. F.Z., A.J.A., D.D.E.-P., and Y.S.S. contributed to the study design and method development. F.Z. and A.J.A. conducted the simulation studies and data analyses. Y.S.S. supervised the project. F.Z. and A.J.A. wrote the initial draft. All authors edited and approved the final manuscript.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Deposited data
GTEx bulk RNA-seq data (v8)	GTEx Portal	dbGaP: phs000424.v8.p2; RRID: SCR_013042
AIDA single-cell RNA-seq data	Kock et al.32	AIDA Data: https://doi.org/10.1016/j.cell.2025.02.017
GenAge database	de Magalhães and Toussaint29	GenAge: https://doi.org/10.1016/j.febslet.2004.07.006
LongevityMap database	Budovsky et al.30	LongevityMap:
CellAge database	Avelar et al.31	CellAge: https://doi.org/10.1186/s13059-020-01990-9
Data for reproducing figures	This paper	Zenodo: https://doi.org/10.5281/zenodo.16416084
Software and algorithms
R statistical programming environment	https://www.r-project.org/	v4.4.0
ggplot2	https://ggplot2.tidyverse.org/	R package version 3.5.2
gamlss	https://www.gamlss.com/	R package version 5.4–22
MDSeq	https://github.com/zjdaye/MDSeq	R package version 1.0.5
missMethyl	https://bioconductor.org/packages/missMethyl/	R package version 1.40.0
DESeq2	https://bioconductor.org/packages/DESeq2/	R package version 1.48.1
edgeR	https://bioconductor.org/packages/edgeR/	R package version 4.6.3
limma	http://bioinf.wehi.edu.au/limma	R package version 3.5.2
clusterProfiler	https://bioconductor.org/packages/clusterProfiler/	R package version 4.16.0
pscl	https://github.com/atahk/pscl	R package version 1.5.9
QRscore	This paper. https://github.com/songlab-cal/QRscore	R package version 1.1.1
Custom code for reproducing figures	This paper	https://doi.org/10.5281/zenodo.16416084

Method details

We present QRscore, a rank-based test that capitalizes on the benefits of parametric modeling while remaining non-parametric in nature. This approach inherits the strengths of non-parametric tests, such as false positive control, while also providing increased detection power if the data roughly follow a parametric model.

Weight construction approach for QRscore

To obtain the weight functions $w_g$ appearing in Equation 1, we adopt the approach of Erdmann-Pham (2022),³⁶ which combines likelihood gradients (i.e., scores) with rank statistics to construct $w_g$. Erdmann-Pham showed that for many asymptotically efficient non-parametric tests (e.g., Mann-Whitney, van der Waerden, and Klotz tests) — specifically, tests that surprisingly achieve similar power to their parametric counterparts in large samples when the parametric assumptions are met — their weight functions satisfy a first-order Taylor approximation to the difference between the log likelihood under the null hypothesis $P_g=Q_g$ and the log likelihood under a “slightly perturbed” alternative hypothesis. (This type of perturbation lies at the heart of Le Cam’s local asymptotic normality theory,³⁷ which allows asymptotic distributions to be derived.) Here, $P_g$ and $Q_g$ are parametric distributions for which the non-parametric test is compared against its parametric counterpart, and “slightly perturbed” refers to a very small perturbation of the associated parameter. For example, $P_g$ and $Q_g$ could be normal distributions parameterized by different means ${\theta }_P$ and ${\theta }_Q$, with the Van der Waerden test being asymptotically as powerful as the t-test when samples themselves are drawn from normal distributions with different means (and the Mann-Whitney test also being comparable to the t-test), and the perturbation is ${\theta }_Q={\theta }_P+1/\sqrt{N}$ where $N=n+k$ is the pooled sample size.

The asymptotic efficiency for above non-parametric tests, however, is limited to normal distributions. Our method extends this approach by designing weights informed by any parametric model while maintaining a non-parametric framework. In this study, we construct $w_g$ tailored to the (zero inflated) negative binomial family, which better models gene expression data. As we will show, these weights are derived by computing the score of a transformed rank statistic, which handles over-dispersion and zero inflation in RNA-seq data (see STAR Methods: Extension to zero-inflation model).

Concretely, for each gene g, we compute ranks based on normalized expression values to mitigate technical artifacts like differences in library size. Meanwhile, to construct the weights used in our rank-based test, we model the raw count data. Let $Y_{jg}^{\text{raw}}$ denote the raw count for sample j in condition 2. We model it using a negative binomial distribution $\text{NB}\left({\mu }_g{s}_j,r_g\right)$, where ${\mu }_g$ and $r_g$ are the gene-specific mean and dispersion parameters, and $s_j$ is the library size factor. For sample i in condition 1, we assume $X_{ig}^{\text{raw}}\sim \text{NB}\left({\theta }_1{\mu }_g{s}_i,{\theta }_2{r}_g\right)$, where ${\theta }_1$ and ${\theta }_2$ represent fold changes in mean and dispersion, and $s_i$ is the size factor. Importantly, this modeling step is not used to define the null hypothesis—it is only employed to construct weight functions.

To target dispersion shifts, we fix ${\theta }_1=1$ and define the score function as $z_{{\theta }_2}^{g,\text{disp}}=\frac{\partial }{\partial {\theta }_2}\log P_{1,{\theta }_2}^g\text{,}$ where $P_{{\theta }_1,{\theta }_2}^g$ is the probability mass function for $\text{NB}\left({\theta }_1{\mu }_g,{\theta }_2{r}_g\right)$. Let $F_{{\theta }_1,{\theta }_2,{\mu }_g,r_g}$ denote the corresponding cumulative distribution function (CDF). The weight function is then defined by

$$w_g^{\text{disp}}\left(x\right)=z_1^{g,\text{disp}}\circ F_{1,1,{\hat{\mu }}_g,{\hat{r}}_g}^{-1}\left(x\right),x\in \left(0,1\right)\text{,}$$

where ${\hat{\mu }}_g$ and ${\hat{r}}_g$ are maximum likelihood estimates obtained from raw counts, using a model that includes log-transformed estimated size factors as offsets. This modeling approach allows the score and probability mass functions used for weight construction to omit explicit size factors, as ${\hat{\mu }}_g$ and ${\hat{r}}_g$ characterize the distribution on the normalized scale. The symbol $\circ$ denotes function composition: the inverse CDF is applied first to the rank-based input x, and the result is passed through the score function.

This weight construction employs a modified Score test, where the original samples are replaced by functions that take in rank-based statistics. Under the null hypothesis that gene expression distributions are identical under two conditions, the ranks $R_{X_{ig}}$ are uniformly distributed in $\left\{1,2,\dots ,n+k\right\}$. In this context, the inverse CDF becomes pivotal, mapping this uniform distribution back into the scale of negative binomial distribution. This step adapts rank-based data for use in the Score test, while maintaining its non-parametric nature.

Crucially, the use of rank-based statistics as inputs guarantees that the test is distribution-free. This ensures robustness against model misspecification and provides stringent control over false discoveries. Equally important is the method’s efficiency: when parameters and the score function are tailored within a negative binomial model, the test’s ability to detect changes in dispersion is remarkably enhanced. Most notably, with an accurately specified model, this test is asymptotically as efficient as that of the likelihood-ratio test. This elevates it above other non-parametric tests in scenarios where the model precisely captures the underlying distribution.

Similarly, we can set ${\theta }_2=1$ and derive the corresponding score function and test statistic $w_g^{\text{mean}}\left(x\right)$ that targets on shifts in mean parameters. A detailed derivation of the functions used in constructing the weights, along with numerical considerations and adjustments, is provided in Method S1.

Inference

The asymptotic normality and efficiency of the general test statistic are proved in Erdmann-Pham.³⁶ In the context of this work, suppose $k,n\to \infty$, with $k/\left(n+k\right)\to \alpha >0$. The statistic $T_g^{w,\text{disp}}=\frac{1}{\sqrt{n+k}}{\sum }_{i=1}^k{w}_g^{\text{disp}}\left(\frac{R_{X_{ig}}}{n+k}\right)$ is asymptotically normal under null hypothesis $H_{0g}$ with

$$T_g^{w,\text{disp}}\stackrel{H_{0g}}{\to }N\left(0,\alpha \left(1-\alpha \right)H_g^{\text{disp}}\right)\text{,}$$

where $H_g^{\text{disp}}$ is the Fisher information specific to the dispersion parameter in the NB distribution. An analogous result holds for the mean-based test statistic used in differential expression analysis.

For each gene, the p values are obtained using the asymptotic distribution of the above statistics and adjusted for multiple testing using the Benjamini-Hochberg (BH) procedure. We refer to our method as QRscore-Var when assessing dispersion changes and QRscore-Mean when assessing mean changes.

For smaller sample sizes ($\le 50$), we also apply a permutation-based approach to compute p values. Specifically, if both groups consist of samples smaller than 10, we compute exact p values by enumerating all possible permutations. Otherwise, we use a Monte Carlo resampling procedure. Due to the rank-based nature of the test and limitations of permutation methods under extreme sparsity, we do not recommend applying QRscore when any group sample size is $\le 5$, especially in the context of multiple testing.

Extension to zero-inflation model

Large-scale RNA-seq data often contain a substantial number of genes with technical excess zeros,¹³ which can pose a challenge for accurate analysis of gene expression means and variability. To properly account for such zeros to ensure reliable and interpretable results, it is customary to model the counts for each gene in bulk RNA-seq data by a zero-inflated negative binomial distribution $\text{ZINB}\left({\mu }_g{s}_j,r_g,{\pi }_g\right)$:

$$Y_{jg}^{\text{raw}}\sim \{\begin{array}{cc}0& \text{with}\text{probability}{\pi }_g,\\ \text{NB}\left({\mu }_g{s}_j,r_g\right)& \text{with}\text{probability}1-{\pi }_g,\end{array}$$

where $0\le {\pi }_g<1$ is the probability of technical excess zeros. This richer model allows one to distinguish between zero counts resulting from technical variations at the excess zero state and those arising from biological variation at the negative binomial state.

We model $X_{ig}^{\text{raw}}$ by $\text{ZINB}\left({\theta }_1{\mu }_g,{\theta }_2{r}_g,{\pi }_g\right)$. The score function, CDF, and Hessian values for this model are derived from the ZINB probability mass function (see Method S1). We estimate the parameters ${\mu }_g,r_g$, and ${\pi }_g$ using numerical maximum likelihood estimates, with log-transformed estimated size factors as offsets, obtained from the pscl package.³⁸

Since technical zeros are not present in all genes, we fit the negative binomial model and designed weights for genes with expression levels all greater than zero in each sample. For genes containing zeros, we applied the zero-inflated model in further analyses. It is important to note that if all zeros in a gene’s expression profile originate from the negative binomial model (${\pi }_g=0$), the numerical maximum likelihood estimate of ${\pi }_g$ would be very close to zero. Consequently, this scenario results in a test that behaves similarly to the negative binomial weights.

$\mathit{K}$-sample test

We next generalize our procedure to a K-sample scenario, where we have K distinct samples. Suppose there are K samples for each gene g, and our null hypothesis is

$$H_{0g}:P_{1g}=P_{2g}=\cdots =P_{Kg}\text{.}$$

The alternative hypothesis $H_{1g}$: at least one distribution is different from the others.

Our approach efficiently integrates multiple group comparisons into a single unified analysis. For each gene, we compare each group separately against the remaining pooled groups, yielding a vector of two-sample test statistics. Rather than performing separate tests and obtaining multiple p values, we project this vector into an appropriate subspace to eliminate correlations among group-specific statistics. After this projection, we derive the covariance structure of the transformed statistics and apply a chi-square test, resulting in a single well-calibrated p value per gene. Below, we detail this construction and inference procedure explicitly for the case $K=3$, as used in our real data analysis.

For gene g, consider three samples $X_g$, $Y_g$, and $Z_g$ with respective sizes $n_1$, $n_2$, and $n_3$, and define the total sample size as $N=n_1+n_2+n_3$. The corresponding sample proportions are ${\alpha }_X=\frac{n_1}{N}$, ${\alpha }_Y=\frac{n_2}{N}$, and ${\alpha }_Z=\frac{n_3}{N}$.

Step 1: Vector formation

Form a vector of length 3, with its first entry being the sum of the weight function $w_g$ over the ranks of $X_g$, normalized by $\sqrt{N}$. The second and third entries are computed similarly for $Y_g$ and $Z_g$, respectively. The weight function is the same defined in two-sample test, which can be either weights for detecting shifts in mean or weights for detecting shifts in variance. We define

$$S_{X,g}=\frac{1}{N}\sum _{i=1}^{n_1}{w}_g\left(\frac{R_{X_{ig}}}{N}\right)\text{,}$$

$$S_{Y,g}=\frac{1}{N}\sum _{i=1}^{n_2}{w}_g\left(\frac{R_{Y_{ig}}}{N}\right)\text{,}$$

$$S_{Z,g}=\frac{1}{N}\sum _{i=1}^{n_3}{w}_g\left(\frac{R_{Z_{ig}}}{N}\right)\text{,}$$

where $R_{X_{ig}}$ denotes the rank of $X_{ig}$ after pooling and sorting the values from $X_g$, $Y_g$, and $Z_g$. The ranks $R_{Y_{ig}}$ and $R_{Z_{ig}}$ are similarly defined for the values in $Y_g$ and $Z_g$, respectively. Let

$$v_g={\left[S_{X,g},S_{Y,g},S_{Z,g}\right]}^{\top }\text{.}$$

This vector resides in a co-dimension 1 subspace as its entries sum to zero due to the centered score function $w_g$.

Step 2: Projection matrix formation

For $K=3$, a matrix B that serves as an orthonormal basis for the subspace can be chosen as

$$B=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ -1& 1\\ 0& -2\end{array}\right]\text{.}$$

Each column vector in B is an orthonormal basis vector of the subspace where the vector from Step 1 resides.

Matrix A is a basis for the orthogonal complement of $\left(\sqrt{{\alpha }_X},\sqrt{{\alpha }_Y},\sqrt{{\alpha }_Z}\right)$. One example for $A{A}^T$ such that A satisfy such condition is

$$A{A}^{\top }=\left[\begin{array}{ccc}1-{\alpha }_X& -\sqrt{{\alpha }_X{\alpha }_Y}& -\sqrt{{\alpha }_X{\alpha }_Z}\\ -\sqrt{{\alpha }_Y{\alpha }_X}& 1-{\alpha }_Y& -\sqrt{{\alpha }_Y{\alpha }_Z}\\ -\sqrt{{\alpha }_Z{\alpha }_X}& -\sqrt{{\alpha }_Z{\alpha }_Y}& 1-{\alpha }_Z\end{array}\right]\text{.}$$

Finally, the diagonal matrix $\mathrm{\Lambda }$ is

$$\mathrm{\Lambda }=\left[\begin{array}{ccc}\frac{1}{\sqrt{{\alpha }_X}}& 0& 0\\ 0& \frac{1}{\sqrt{{\alpha }_Y}}& 0\\ 0& 0& \frac{1}{\sqrt{{\alpha }_Z}}\end{array}\right]$$

With these matrices, we proceed with the computation of the test statistic and perform the necessary tests.

Step 3: Test statistic construction and inference

We multiply the vector $v_g$ by matrix B and $\mathrm{\Lambda }$. Let $T_g=B^T\mathrm{\Lambda }{v}_g$ and $H_{\text{new}}=H_g{B}^{T}A{A}^{T}B$. Note that $H_g$ is the same as the Hessian value estimated in the two sample test. Under the null hypothesis that the 3 samples are sampled from the same distribution, we know that the two dimensional random variable $T_g$ asymptotically follows $N\left(0,H_{\text{new}}\right)$³⁶. Therefore, when $n_1,n_2,n_3\to \infty$,

$$T_g^T{H}_{\text{new}}^{-1}{T}_g\stackrel{d}{\to }{\chi }_2^2$$

and a chi-square test is applied to the resulting scalar for inference.

Simulation details

We conduct simulations to evaluate the performance of methods for detecting differential expression driven by changes in either dispersion or mean across groups. Simulations are based on parameters estimated from bulk RNA-seq count matrices from whole blood tissue of individuals aged 40–49 ($n_1=113$) and 60–69 ($n_2=249$) from the GTEx project.

We begin by filtering out genes with more than 20% zero counts—this step is applied only during the simulation setup to avoid unintended inflation of zeros unless they are explicitly introduced. We then estimate size factors (${\hat{s}}_i$) for each individual using the median-of-ratios method implemented in DESeq2. After normalizing raw counts with these size factors, we estimate gene-specific mean (${\hat{\mu }}_g$) and dispersion (${\hat{r}}_g$) parameters for use in simulation. We provide the simulation settings in detail below.

Simulation for dispersion shifts in large samples

In the following settings, we set the fold change of dispersion parameter $\left({\theta }_2\right)$ to be $1.1,1.3,\dots ,2.9$ and for each fold change we set 20 random seeds to repeat the experiments. We simulate $N\times G$ count matrices, where $N=$ $n_1+n_2$ and $G=3000$.

Setting 1 (Basic dispersion shifts)

Counts are generated using the negative binomial model with $n_1=133$ and $n_2=249$. We randomly sampled 10% of genes as true DDGs ($G_D$). For $g\in G_D$, we simulate Group 2 counts as $\text{NB}\left({\hat{\mu }}_g·{\hat{s}}_i,{\hat{r}}_g·{\theta }_2\right)$ or $\text{NB}\left({\hat{\mu }}_g·{\hat{s}}_i,{\hat{r}}_g/{\theta }_2\right)$ depending on whether Group 1 or Group 2 has larger variance. Other counts are simulated from $\text{NB}\left({\hat{\mu }}_g·{\hat{s}}_i,{\hat{r}}_g\right)$ where the dispersions remain unmodified.

Settings 2 and 3 (reduced sample size)

Same as Setting 1, but decrease the number of individuals in groups to $n_1=50$, $n_2=100$ (Setting 2) and $n_1=n_2=50$ (Setting 3).

Settings 4–6 (random outliers)

Following Setting 1, we randomly select $5\%$, $10\%$, $20\%$ of the genes to have outliers ($G_{ol}$). For $g\in G_{ol}$, we select a proportion ($p_g$) of individuals to be outliers, where $p_g\sim \text{Unif}\left(0.25,0.1\right)$. The outliers are multiplied by ${\delta }_{i,g}$ where ${\delta }_{i,g}\sim \text{Unif}\left(4,8\right)$.

Settings 7–9 (inflated zeros)

Following Setting 1, we randomly select $5\%$, $10\%$, $20\%$ of the genes to have random zeros ($G_{zi}$). For $g\in G_{zi}$, we select a proportion ($p_g$) of individuals to be zeros, where $p_g\sim \text{Unif}\left(0.25,0.1\right)$.

Settings 10–12 (simulation with both outliers and zeros)

Following Setting 1, we randomly select $10\%$ of the genes to have outliers ($G_{ol}$) and $10\%$ of the genes to have inflated zeros ($G_{zi}$). These selections are made separately, with the possibility that some genes may be included in both groups. The way to generate outliers and zeros are the same as Setting 5 and Setting 7 if genes are in $G_{ol}$ or $G_{zi}$. We then decrease the number of individuals in groups to $n_1=50$, $n_2=100$ (Setting 11) and $n_1=n_2=50$ (Setting 12).

Simulation for dispersion shifts in small samples

To evaluate the performance of methods under challenging conditions, we conducted additional simulations using small sample sizes. We set $\left(n_1,n_2\right)=\left(n,n\right)$ with $n=5,6,\dots ,15$ and repeat 20 times. For each replicate, we sample ${\theta }_2\sim \text{Unif}\left(4,16\right)$, outlier and zero-inflation rates from $\text{Unif}\left(0,0.2\right)$. DDGs comprise 10% of genes. Other details follow Setting 10. This simulation framework allows us to rigorously evaluate method performance under small sample conditions with realistic biological variability and noise.

Simulation for dispersion shifts in bimodal distribution

Since we observe bimodal distribution in real data scenarios, in the following experiments, we introduce bimodal expression distributions to test model robustness. For each gene g, draw

$${\mu }_{g,1}\sim \text{Unif}\left(0.6{\hat{\mu }}_g,0.95{\hat{\mu }}_g\right),{\mu }_{g,2}\sim \text{Unif}\left(1.05{\hat{\mu }}_g,1.4{\hat{\mu }}_g\right)\text{.}$$

Mixing weights $w_1,w_2$ satisfy

$$w_1{\mu }_{g,1}+w_2{\mu }_{g,2}={\hat{\mu }}_g,w_1+w_2=1\text{.}$$

For each gene g, we generated bimodal expression within each group by randomly assigning each individual to one of two modes with probabilities $w_1$ and $w_2$. This procedure creates a bimodal expression distribution centered around the global mean ${\hat{\mu }}_g$ for each gene. As in previous settings, we randomly selected $10\%$ of the genes to be true DDGs. For DDGs, the dispersion parameter ${\hat{r}}_g$ was perturbed according to the fold change ${\theta }_2$ for Group 2, following the same logic as in Setting 1 stated above. To assess robustness under technical artifacts, we also introduced synthetic outliers and inflated zeros using the same procedures as in Settings 4–10.

Simulation for 3-sample tests

To compare the performance of three-sample QRscore-Var tests with sequential pairwise two-sample tests, we simulate gene expression data for both equal sized ($n_1=n_2=n_3=100$) and unequal sized samples ($n_1=50,n_2=100$, and $n_3=150$). We randomly select 10% of the genes to be true DDGs. We assess heterogeneity across multiple groups, evaluating two setups: one with identical distributions for Groups 1 and 2 and fold changes in Group 3, where the dispersion fold change between Group 3 and Group 1 is set to 1.5, 2, and 2.5; and another with differences in all three groups, specifically, dispersion fold change from Group 2 to Group 1 is 1.5, and from Group 3 to Group 2 is 2, 2.5, and 3. We simulate the data using pure negative binomial distributions or randomly selected 10% genes with outliers and 10% genes with random zeros, incorporating fold changes for selected true DDGs.

Simulation for mean shifts

We simulated $N\times G$ count matrices, where $N=n_1+n_2$ and $G=3000$. We set the fold change of mean parameter $\left({\theta }_1\right)$ to be $1.1,1.2,\dots ,2$ and for each fold change we set 20 random seeds to repeat the experiments.

In Setting 1, we simulate data from a purely negative binomial model with $n_1=133$ individuals in Group 1 and $n_2=249$ individuals in Group 2. We randomly select $10\%$ of the genes to be true DEGs, denoted by the set $G_E$. For each gene $g\in G_E$, we apply a fold change in dispersion, denoted by ${\theta }_2$. Specifically, for individuals in Group 1 $\left(X_{ig}\right)$, the simulated counts are generated from $\text{NB}\left({\hat{\mu }}_g·{\hat{s}}_i·{\theta }_1^z,{\hat{r}}_g\right)$, where $z_g$ follows a Rademacher distribution. For genes $g\notin G_E$ or for individuals in Group 2 $\left(Y_{ig}\right)$, the simulated counts are generated from $\text{NB}\left({\hat{\mu }}_g·{\hat{s}}_j,{\hat{r}}_g\right)$. For other settings, we use the same way to simulate random outliers, zeros or reduced sample sizes.

Evaluation metrics

The definition of FDR and Power are as follows.

$$\text{FDR}=\mathbb{E}\left[\frac{\text{Number}\text{of}\text{False}\text{Positives}}{\text{Total}\text{Number}\text{of}\text{Discoveries}}\right]\text{,}$$

where discoveries are genes with an adjusted p value $\le 0.05$.

$$\text{Power}=\max \left(\frac{\text{Number}\text{of}\text{True}\text{Positives}\text{Detected}}{\text{Total}\text{Number}\text{of}\text{True}\text{Positives}}|\text{FDP}\le 0.05\right)\text{,}$$

where FDP is the false discovery proportion. In other words, the p value threshold is adaptively chosen for each dataset with the false discovery proportion truly controlled.

For calculating the Area Under the Precision-Recall Curve (AUPRC), Precision and Recall are defined as follows, where the p value threshold is varied continuously from 0 to 1, rather than a fixed threshold:

$$\text{Precision}=\frac{\text{Number}\text{of}\text{True}\text{Positives}\text{Detected}}{\text{Total}\text{Number}\text{of}\text{Discoveries}}$$

and

$$\text{Recall}=\frac{\text{Number}\text{of}\text{True}\text{Positives}\text{Detected}}{\text{Total}\text{Number}\text{of}\text{True}\text{Positives}}\text{.}$$

Quantification and statistical analysis

Analysis of GTEx data

In real data applications, we use publicly available bulk RNA-seq data across multiple human tissues provided by the Genotype-Tissue Expression (GTEx) Consortium.¹⁹

We divide bulk RNA-seq count data by three age groups, namely 20–39, 40–59 and 60–79. We selected 33 tissues, each containing more than 25 donors, for differential analysis. We also filtered out lowly expressed genes, defined as those with an average count of less than 2 and more than 80% zeros. Before differential analysis, we apply a normalization procedure provided by DESeq2, which divides counts by sample-specific size factors determined by median ratio of gene counts relative to geometric mean per gene.

We apply QRscore on GTEx data after prefiltering and normalizing the data. Using QRscore-Var and QRscore-Mean, we conduct three-sample tests on three age groups for each tissue, adjusting the p values with BH adjustments to identify sets of DDGs and DEGs (threshold 0.05). For further insights, we calculate the variance ratio for pairwise two-samples and performed pairwise two-sample tests on selected DDGs to determine which groups drive the signal. Additionally, we separate individuals by sex and perform the same tests for each sex group respectively. We also conduct the tests on subsampled male data to match the sample size with females for a fair comparison.

For futher annotation, we compile lists of genes related to human aging from three databases: GenAge,²⁹ LongevityMap,³⁰ and CellAge³¹ to annotate age-related genes. We rank the genes based on p values for each tissue and age group, using QRscore-Var or QRscore-Mean tests, and applied one-sided Wilcoxon rank-sum tests for the age-related genes against other genes. We conduct ontology enrichment analysis using the ClusterProfiler package. For visualization of ontology terms in whole blood tissue, we use the simplify() function from the clusterProfiler package to remove highly similar terms, ensuring greater diversity and interpretability among the displayed results. For coexpressed gene analysis, we extracted pairs of DDGs enriched in each GO term and calculated their Spearman correlation. Gene pairs with an absolute correlation greater than 0.6 were retained.

Analysis of pseudo-bulked single-cell data

To achieve cell type-specific resolution while enhancing statistical power, we aggregate single-cell RNA-seq data into pseudo-bulk profiles, enabling downstream analyses equivalent to those applied in bulk RNA-seq datasets. Below, we describe the dataset and the preprocessing steps used to construct these pseudo-bulked profiles.

Dataset description

The Asian Immune Diversity Atlas (AIDA)³² is a large-scale single-cell RNA-sequencing resource profiling peripheral blood mononuclear cells (PBMCs) from healthy donors across Asia. Our analysis combines two versions of the experiments and focuses exclusively on individuals of East Asian identity — self-reported as Korean, Japanese, or Singaporean Chinese — to control for population heterogeneity. We use cell type annotation and donor metadata to perform age-related analyses of immune cell type-specific gene expression.

Preprocessing and pseudo-bulk aggregation

We exclude cells with low sequencing depth (total UMI count $\le$ 1,000) and genes expressed in fewer than 100 cells. Donor age is extracted from metadata and stratified into three groups: 20–34, 35–49, and 50+ years, to ensure sufficient sample sizes within each age group, particularly given the higher number of individuals in the younger cohorts.

To avoid confounding effects due to unequal cell numbers across age groups—particularly in variance-based analyses—we apply the a quantile-based subsampling procedure. For each cell type, we count the number of cells contributed by each donor within each age group and exclude donors with fewer than 20 cells. Within each age group, we rank the remaining donors by cell count and assign them to 50 quantile bins. We then align corresponding bins across age groups by quantile position. For each aligned bin, we compute the minimum number of cells contributed by any donor across age groups and randomly subsample cells without replacement from each eligible donor within that bin to match this target. This ensures that donor-level cell count distributions are approximately matched across age groups within each cell type, reducing bias in downstream variance-based analyses.

This procedure is important because unequal cell numbers across donors and age groups can introduce spurious variability in variance-based analyses. Standard normalization methods like library size scaling correct for mean expression but do not fully account for the increased noise in samples with fewer cells, which can inflate variance estimates. Samples with low cell counts contribute less stable measurements, biasing comparisons across groups. By aligning the distribution of donor-level cell counts across age groups within each cell type, the quantile-based subsampling procedure minimizes these artifacts, enabling more accurate and interpretable inference of true biological variability.

We then generate pseudo-bulk profiles by summing raw gene expression counts across all retained cells within each combination of cell type and individual. We further filter genes with an average count of less than 2 and more than 80% zeros after aggregation. These aggregated count matrices serve as input for downstream differential mean and variance analysis using bulk RNA-seq methodologies.

DDG discovery and downstream analysis

Using QRscore-Var and QRscore-Mean on preprocessed data, we conduct three-sample tests on the three age groups for each tissue, adjusting the p values with Benjamini-Hochberg correction to identify sets of DDGs and DEGs (threshold 0.05). For further insights, we calculate the variance ratio for pairwise two-sample comparisons and perform pairwise two-sample tests on selected DDGs to determine which groups drive the signal. We conduct Reactome pathway enrichment analysis using the ClusterProfiler package.

Published: September 2, 2025