Distribution-free complex hypothesis testing for single-cell RNA-seq differential expression analysis

State-of-the-art methods for single-cell RNA sequencing (scRNA-seq) Differential Expression Analysis (DEA) often rely on strong distributional assumptions that are difficult to verify in practice. Furthermore, while the increasing complexity of clinical and biological single-cell studies calls for greater tool versatility, the majority of existing methods only tackle the comparison between two conditions. We propose a novel, distribution-free, and flexible approach to DEA for single-cell RNA-seq data. This new method, called ccdf, tests the association of each gene expression with one or many variables of interest (that can be either continuous or discrete), while potentially adjusting for additional covariates. To test such complex hypotheses, ccdf uses a conditional independence test relying on the conditional cumulative distribution function, estimated through multiple regressions. We provide the asymptotic distribution of the ccdf test statistic as well as a permutation test (when the number of observed cells is not sufficiently large). ccdf substantially expands the possibilities for scRNA-seq DEA studies: it obtains good statistical performance in various simulation scenarios considering complex experimental designs (i.e. beyond the two condition comparison), while retaining competitive performance with state-of-the-art methods in a two-condition benchmark. We apply ccdf to a large publicly available scRNA-seq dataset of 84,140 SARS-CoV-2 reactive CD8+ T cells, in order to identify the diffentially expressed genes across 3 groups of COVID-19 severity (mild, hospitalized, and ICU) while accounting for seven different cellular subpopulations.


Introduction
Single-cell RNA sequencing (scRNA-seq) makes it possible to simultaneously measure gene expression levels at the resolution of single cells, allowing a refined definition of cell types and states across hundreds or even thousands of cells at once. Single-cell technology significantly improves on bulk RNA-sequencing, which measures the average expression of a set of cells, mixing the information in the composition of cell types with different expression profiles. New biological questions such as detection of different cell types or cellular response heterogeneity can be explored thanks to scRNA-seq, broadening our comprehension of the features of a cell within its microenvironment (Eberwine and others, 2014).
Several challenges arise from the sequencing of the genetic material of individual cells like in transcriptomics (see Lähnemann and others (2020) for a thorough and detailed review). Differen-tial expression analysis (DEA) is a major field of exploration to better understand the mechanisms of action involved in cellular behavior. A gene is called differentially expressed (DE) if its expression is significantly associated with the variations of a factor of interest. Single-cell data have different features from bulk RNA-seq data that require special consideration for developing DEA tools. Indeed, scRNA-seq data display large proportions of observed zeros (i.e. "dropouts"), due either to biological processes or technical limitations (Lähnemann and others, 2020).
The large amount of single-cell measurements provides an opportunity to estimate and characterize the distribution of each gene expression and to compare it under different conditions in order to identify DE genes. In fact, scRNA-seq distributions usually show complex patterns. Therefore, the scDD method (Korthauer and others, 2016) condenses the difference in distribution between two conditions into four categories: the usual difference in mean, the difference in modality, the difference in proportions and the difference in both mean and modality. Since scRNA-seq data analysis lay unique challenges, new statistical methodologies are needed.
Several strategies making strong distributional assumptions on the data have been proposed to perform single-cell DEA. MAST (Finak and others, 2015) and SCDE (Kharchenko and others, 2014) are two well-know differential methods, the former using a two-part generalized linear model to take into account both the dropouts and the non-zero values by making a Gaussian assumption of each gene and the latter relying on a Bayesian framework combined with a mixture of Poisson and negative binomial distributions. scDD (Korthauer and others, 2016) makes use of a Bayesian modeling framework to detect differential distributions and then to classify the gene into four differential patterns using Gaussian mixtures. DEsingle (Miao and others, 2018) proposes a zero-inflated negative binomial (ZINB) regression followed by likelihood-ratio test to compare two samples. D 3 E (Delmans and Hemberg, 2016) also applies a likelihood-ratio test after fitting a Poisson-Beta distribution.
Yet, Risso and others (2018) have advanced that scRNA-seq data are zero-inflated and have proposed to use zero-inflated negative binomial models, while Svensson (2020) have argued that the number of zero values is consistent with usual count distributions. Then, Choi and others (2020) illustrated that scRNA-seq data are zero-inflated for some genes but "this does not necessarily imply the existence of an independent zero-generating process such as technical dropout".
In fact, biological information (e.g. cell type and sex) may explain it. The authors also discourage imputation as zeros can contain relevant information about the genes. In addition, Townes and others (2019) argue that single-cell zero inflation actually comes from normalization and logtransformation. The distribution and the sparsity of scRNA-seq data remains difficult to model, and -as there is no consensus on which model is the best one -it is of utmost importance to develop general and flexible methods for analyzing scRNA-seq data which do not require strong parametric assumptions.
Fewer distribution-free tools have been developed to model single-cell complex distributions without making any parametric assumption. EMDomics (Nabavi and others, 2016) and more recently SigEMD (Wang and Nabavi, 2018) are two non-parametric methods based on the Wassertein distance between two histograms, the latter including data imputation to handle the problem of the great number of zero counts. D 3 E offers in addition the possibility to perform either the Cramer-von Mises test or the Kolmogorov-Smirnov test to compare the expression values' distributions of each gene, thus delivering a distribution-free option. In a comparative review, Wang and others (2019) illustrated that non-parametric methods, i.e. distribution-free, perform better in distinguishing the four differential distributions. Recently, Tiberi and others (2020) presented distinct, a hierarchical non-parametric permutation approach using empirical cumulative distribution functions comparisons. The method requires biological replicates (at least 2 samples per group) and allows adjustment for covariates but only tackles the comparison between two conditions to our knowledge.
However, the limitations of these state-of-the-art methods for scRNA-seq DEA are many. The approaches based on a distributional assumption face methodological issues. They rely on strong distributional assumptions that are difficult to test in practice. A deviation from the hypothesized distribution will translate into erroneous p-values and may lead to inaccurate results. While the increasing complexity of clinical and biological studies calls for greater tools versatility, the majority of existing methods, whether parametric or non-parametric, cannot handle data sets with a complex design, making them very restrictive. In fact, the most commonly used methods remain in the traditional framework of DEA restricted to the comparison between two conditions only. One might be interested in the genes differentially expressed across several conditions (e.g. more than two cell groups, multi-arm...) or in testing the genes differentially expressed according to a continuous variable (e.g. cell surface markers measured by flow cytometry...). In particular, the identification of surrogate biomarkers from transcriptomic measurements is becoming an emerging field of interest, especially in cancer therapy (Wang and others, 2007) or in new immunotherapeutic vaccines (Cliff and others, 2004). Gene expression could be used to compare treatments in observational settings. Yet in such cases, adjusting for some technical covariates or some confounding factors is paramount to ensure the validity of an analysis, as this external influence can impact the outcome as well as the dependent variables and thus generates spurious results by suggesting a non-existent link between variables.
Overall, the need of testing the association between gene expression and the variables of interest, potentially adjusted for covariates, makes an additional motivation for developing suitable tools. The complex hypothesis we aim to test consists in performing a conditional independence test (CIT). A CIT broadens the classical independence test by testing for independence between two variables given a third one, or a set of additional variables (see Figure 8). Two random variables X and Y are conditionally independent given a third variable Z if, and only if, P (X, Y | Z) = P (X | Z)P (Y | Z). As described in Li and Fan (2020), many CIT have been developed previously and are readily available such as discretization-based tests (Huang and others, 2010), metric-based tests (Runge, 2018;Su and White, 2007;Huang and others, 2016), permutation-based two-sample tests (Doran and others, 2014;Sen and others, 2017), kernel-based tests (Muandet and others, 2017;Li and others, 2009) and regression-based tests (see Li and Fan (2020) for a short review). Yet, these CIT either suffer from the curse of dimensionality or are hardly applicable to a large number of observations (Muandet and others, 2017). Zhou and others (2020) have converted the conditional independence test into an unconditional independence test and then used the Blum-Kiefer-Rosenblatt correlation (Blum and others, 1961) to develop an asymptotic test. Yet, the latter cannot be applied when X is discrete. Those limitations make these tests impractical in our context of scRNA-seq DEA and thus require adaptation.
Performing DEA necessarily involves performing as many independent tests as there are genes.
The variables of interest may be either discrete or continuous, while the number of covariates to condition upon may also increase. Consequently there is an urgent need for a CIT that is both flexible and fast. Here, we propose a novel, distribution-free, and flexible approach, called ccdf, to test the association of gene expression to one or several variables of interest (continuous or discrete) potentially adjusted for additional covariates. Because of the current limitations of existing CIT and the growing interest in testing differences in distribution, we make use of a CIT based on the conditional cumulative distribution function (CCDF), estimated by a regression technique. We derive the asymptotic distribution of the ccdf test statistic, which does not rely on the underlying distribution of the data, as well as a permutation test to ensure a good control of type I error and FDR, even with a limited sample size.
Section 2 describes our proposed method with both asymptotic and permutation tests. Section 3 presents several simulation scenarios to illustrate the good performances of ccdf when we consider complex designs (i.e. beyond the two condition comparison), while a benchmark in the two condition case shows our method retains similar performance in terms of statistical power compared to competitive state-of-the-art methods. Section 4 compares the performances of our method with several methods on a "positive data set" that included differentially expressed genes as well as a "negative data set" without any differentially expressed gene. Section 5 illustrates the application of ccdf to a scRNA-seq study in COVID-19 patients. Section 6 discusses the strengths and limitations of our proposed approach.

Method
In this section, we propose a new, easy-to-use, fast, and flexible test for scRNA-seq DEA. We give both its asymptotic distribution as well as a permutation approach to obtain valid p-values in small samples.

Conditional independence test
2.1.1 Null hypothesis. Testing the association of Y , namely the expression of a gene, with a factor or a group of factors of interest X either discrete (multiple comparisons) or continuous given covariates Z is equivalent to test conditional independence between Y and X knowing Z: " H 0 : Y ⊥ X | Z ". Several statistical tools can be used to characterize the probability law of a random variable, such as the characteristic function, the probability density distribution or the cumulative distribution function. The characteristic function is not often used in practice, due to its relative analytical complexity. The probability density distribution, while more popular, remains difficult to estimate in practice when the number of variables increases due to the increasing number of bandwidths to be optimized. This curse of dimensionality quickly leads to classical computational problems because of complexity growth. As for the cumulative distribution function, its estimation does not require any parameter akin to these bandwidths, making it an efficient tool in highdimensional non-parametric statistics. From this point, we built a general DEA method including an estimation of the CCDF based on regression technique.
The conditional independence test we propose is based on CCDFs. Indeed, if a group of factors is associated with the expression of a gene, the immediate consequence is that the CCDF of the gene expression would be significantly different from the marginal cumulative distribution, which overlooks this conditioning. Thus, the null hypothesis can be written as: If there are no covariates, the conditional independence test turns into a traditional independence test as we test the null hypothesis Y ⊥ X which is equivalent to test 2.1.2 Test statistic. In this section, we propose a general test statistic for testing the null hypothesis of conditional independence that is easy to compute. We denote outcome vector (i.e. normalized read counts for gene g in n cells) and X g = (X g 1 , ..., X g n ) a s × n matrix encoding the condition(s) to be tested that can be either continuous or discrete. One may want to add exogenous variables, which are not to be tested but upon which it is necessary to adjust the model. Let Z g = (Z g 1 , ..., Z g n )) be a r × n matrix for continuous or discrete covariates to take into account. For the sake of simplicity, we drop the notation g in the remainder as we refer to gene-wise DEA.
We have Y ∈ [ζ min , ζ max ] for some known constants ζ min , ζ max . Let ζ min ω 1 < ω 2 < ... < ω p < ζ max be a sequence of p ordered and regular thresholds. For each ω j with j = 1, ..., p, variable that is 1 if Y i ω j and 0 otherwise. We propose to estimate these conditional expectations through a sequence of p working models: where β 1j = (β 1j1 , ..., β 1js ) is the vector of size s referring to the regression of Y ij onto X i and β 2j is the vector of size r referring to the regression of Y ij onto Z i , for the fixed thresholds ω 1 , ω 2 , ..., ω p . If X has no link with Y given Z, then we expect that β 1j will be null. So, we aim to test: Although, there are many different test statistics associated with this null hypothesis , we propose to use the following test statistic that can be written as D = n p j=1 s k=1 β 2 1jk .

Estimation and asymptotic distribution.
In this section, we describe how to estimate β 1j , which allows the computation of the test statistic D.
While in principle any link function g(·) could be selected for the models (2.1), we select the identity link g(y) = y for its computational simplicity, and we compute coefficient estimates using ordinary least squares (OLS). Because our approach requires p models for each of possibly thousands of genes, speed is of utmost importance, so we use OLS. Other selections of g(·) are of course possible and could be explored at the cost of additional computation time.
We show in the Supplementary Material that using OLS to estimate (2.1), β 1j can be ex- These estimates may be plugged in to obtain the estimated test statistic . We furthermore show in the Supplementary Material that the asymptotic distribution of the test statistic may be approximated by a mixture of χ 2 1 random variables: ..,ũ ps ) ∼ N (0, I) are standard multivariate normal random variables and a j are the eigenvalues of Σ = cov( √ n γ 1 ) where γ 1 is a vectorized version of β 1 = ( β 11 , β 12 , ..., β 1p ) (the s × p matrix) concatenating the s rows of β 1 one after another.
We may then compute p-values by comparing the observed test statistic D n to the distribution of ps j=1 a j χ 2 1 , where a j is an estimate of a j based on a consistent estimator for Σ (see Supple-mentary Material for details). Therefore, this allows us to derive a p-value for the significance of a gene with regards to the variable(s) to be tested. In practice, we make use of saddlepoint approximation for distributions of quadratic forms (Kuonen, 1999;Chen and Lumley, 2019) to compute p-values for the mixture of χ 2 s, implemented in the survey R package (Lumley, 2004).
Note that we obtain a simple limiting distribution without relying on any distributional assumptions on the gene expression. In fact, based on the results in Li and Duan (1989), our test will be asymptotically valid as long as there exist any g(·) and any β 0j , β 1j , β 2j such that (2.1) holds.
Lastly, the great number of tests requires the Benjamini-Hochberg (Benjamini and Hochberg, 1995) correction afterwards, which is automatically applied to the raw p-values in the R package of ccdf.

Permutation test Permutation tests are recommended when the number of observations is
too small, so that the asymptotic distribution can not be assumed to hold. When the sample size n is low, we propose to perform permutations to estimate the empirical distribution of D n under the null hypothesis. The test is described in the Supplementary Material, along with practical considerations for computational speed up.

Comparisons with state-of-the-art methods in the two conditions case
We compared the performance of our method ccdf with three state-of-the-art methods, MAST, scDD and SigEMD, to find differentially distributed (DD) genes. We have selected methods implemented in R that have shown good results in Wang and others (2019) benchmarks and that have been especially designed for single-cell data (excluding methods for bulk RNA-seq like edgeR and DESeq2). Also, we considered methods that use normalized (and therefore continuous) counts as input, so that the results of our simulations are comparable. We generated simulated count data from negative binomial distributions and mixtures of negative binomial distributions. Since MAST, scDD and SigEMD require continuous input, we transformed the counts into continuous values while preserving as much as possible the count nature of the data (i.e. negative binomial assumption). To do so, we added a small Gaussian noise with mean equal to 0 and variance equal to 0.01 to the simulated counts. 500 simulated datasets were generated including 10,000 genes each, of which 1,000 are differentially expressed under a two conditions setting for 7 different sample sizes n (20,40,60,80,100,160,200). The observations were equally divided into the two groups. Korthauer and others (2016) classified four different patterns of unimodal or multi-modal distributions: differential expression in mean (DE), differential proportion (DP), differential modality (DM) and both differential modality and different component means within each condition (DB). We used these four differential distributions to create our own simulations.
Specifically, we simulated 250 DD genes, 250 DM genes, 250 DP genes and 250 DB genes. Plus, 9,000 non-differentially expressed genes are simulated according to two non-differential scenarios (see Supplementary Materials for the simulation settings). MAST is designed to detect difference in the overall mean (traditional differential expression), which is absent in DB scenario. The difference in modality and in different component means is then overlooked as expected with its parametric model. SigEMD, scDD and both of ccdf's tests present competitive performances. Generally speaking, ccdf retains competitive performance with the state-of-the-art in this two condition benchmark, which makes it a method particularly adapted to traditional DEA. Even though ccdf is a non-parametric method, the asymptotic test is relatively reasonable in terms of computation times, especially compared to SigEMD (see Table 1). If the computation times seem too large, we recommend to use the adaptive thresholds strategy explained in the Supplementary Materials and in particular for the permutation test, we recommend to switch to the adaptive permutations also described in the Supplementary Materials.  Figure 5 shows that both versions of ccdf have great power to identify DE and DM genes as the sample size increases. DP and DB genes require larger number of cells to achieve sufficient power (from n > 200).

Two conditions comparison given a covariate Z
As represented in Figure 8, a potential confounding covariate Z can interfere the test of the link between the outcome Y and the variable X. Then, it is needed to adjust for the confounding variable by carrying out a CIT. We aim to emphasize the importance of taking into account Z, thanks to our approach, by showing the erroneous results obtained by not doing so. For this purpose, we simulated a confounding variable Z from a Normal distribution. The values of the variable to be tested X depends on the quartile of Z, creating a strong link between X and Z.
Y is constructed from X for DE genes and from Z for non-DE genes, the last case is particularly misleading if one does not take into account Z (see simulation settings in Supplementary Material).
We simulated 10,000 genes of which 1,000 are differentially expressed for several sample sizes n (20,40,60,80,100,160,200). ccdf permutation test was excluded in this simulation scheme because of the large amount of time to compute. In fact, when we have to adjust for a covariate, the permutation test is not suited to such large sample sizes and number of genes because of the underlying permutation strategy.
MAST is able to adjust for covariates like ccdf so we expect good performances from both of them. Conversely, scDD and SigEMD can not control for confounding variables that might impact the detected DE genes. The results of the benchmark between MAST, scDD, SigEMD and ccdf are depicted in Figure 6. Under the alternative hypothesis, we created a large difference in the mean of the normal distributions between the two conditions in order to make DE genes easier to identify. We see therefore that scDD, SigEMD and ccdf exhibit high power at all sample sizes whereas MAST tends to be less powerful for a given size. As expected, the link between Y and Z is very confusing for scDD, SigEMD which interpret it as a link between Y and X, since X is constructed from Z. Consequently, we observe a consistent rise of Type-I error. ccdf and MAST perfectly control the Type-I error. In addition, scDD and SigEMD suffer greatly in terms of TDR as the number of cells is increasing. In fact, fewer real discoveries are found meaning that more genes are identified as significant while being actually false positive, which is in line with the drastically inflated FDR. Dealing with this complex design, ccdf outperforms the leading methods by controlling the Type-I error as well as the FDR and by providing a powerful test.
This simulation study highlights the importance of taking into account the confounding variables that may exist. Otherwise, DEA may lead to inaccurate results and a potential huge amount of false positives. It is worth mentioning that ccdf can adjust for more than one covariate using the asymptotic test while preserving relatively fast calculation times. The permutation test is for now limited to one adjustment variable and the computation times are obviously increased due to the permutation strategy.

Comparisons using real data benchmarks
To be in a more realistic context with a greater number of zeros, a positive control data set and a negative control data set described in Wang and others (2019) were used in order to compare the performances of several methods. The genes with a variance equal to zero were removed from the datasets and counts were converted into log-transformed count per millions (CPM) values.

Positive control dataset
The dataset from Islam and others (2011) Table 2. ccdf leads to the highest true positive rate (0.734) and enables to identify 24.6% more genes in common with the top 1,000 genes compared to SigEMD (0.488). scDD shows a true positive rate of 0.351 and MAST exhibits the lowest rate (0.198) of all the tools.

Negative control dataset
To get false positive rate, we used the dataset from Grün and others (2014). We selected 80 samples under the same condition. To create 10 datasets, we randomly divided these 80 cells into 2 groups of 40 cells. As there is no difference between the two groups, not a single DE gene is to be found in each dataset. Performance evaluations are compared across methods in Table 3.
ccdf and MAST do not detect any genes which was expected. Although all the cells are under the same condition, scDD and SigEMD identified respectively 5 and 50 DE genes.

Application to a scRNA-seq study in COVID-19 patients
Kusnadi and others (2021)  non-hospitalized were considered as mild, and they compared the gene expression across CD8+ T cells between them. Instead, we went further by analyzing the differences between the three COVID-19 categories, namely ICU, non-ICU hospitalized and not hospitalized to provide better insights between the different severity of COVID-19.
As is often done in scRNA-seq analyses, Kusnadi and others (2021) clustered the 84,140 cells, resulting in 8 different clusters. Table 8 presents the numbers of cells assigned to each cluster (Kusnadi and others (2021) excluded cluster 7 due to its small size). These clusters represent various CD8+ T cell sub-populations with different states of cell differentiation, with cluster 1 being composed of exhausted (dying) cells for instance. Some biological pathways, activated across some clusters, could also be associated to the disease severity (such as pro-survival features as reported in the original paper (Kusnadi and others, 2021)). Hence, it is not clear whether the association of the expression of some genes with the severity of the disease is either reflecting a difference in abundance of some clusters of CD8+ T cell populations, or due to the activation of some biological pathways independently of the clusters. We propose to compare the gene expression according to the severity of the disease while adjusting or not for the clusters, which might be acting as a confounding factor. Both the outcome and the potential confounding factor are categorical variables with more than 2 levels, making our analysis design relatively complex.
With our notations, Y is the gene expression, X the COVID-19 severity status and Z the cell cluster. We apply the asymptotic ccdf test for both analyses (with and without adjusting on Z) while testing each gene for differential expression across COVID-19 severity status. The number of thresholds to compute the marginal and conditional CDFs is set to 10. Following Kusnadi and others (2021), only transcripts expressed in at least 0.1% of the cells were included in the differential analyses, yielding 10,525 genes in total to be tested and UMI data are then converted to log 2 (CPM+1). 6,290 genes are found DE when adjusting for the clusters, while 8,181 genes are DE in unadjusted analyses (see Figure 7).
To understand where the difference of 2,129 DE genes comes from, we focused specifically on the 221 genes included in the IFN gene set, one of the 7 pathways highlighted by Kusnadi and others (2021) (see their supplementary material Table S5 for their definition). Among these genes, 188 were found significant regardless of the adjustment on the clusters while 7 become significant and 20 become not significant when adjusting for clusters. Figure 8 shows that some genes with low p-values in unadjusted DEA have much higher p-values after adjustment for clusters (e.g. IFIT3, IFI6, TDRD7, IFI44L, IFI44, IFITM2, OAS2, HERC6 and CD38) whereas others such as LCP2 do not exhibit a strong variation of the p-value. This could be expected when looking at the abundance of the genes according to the clusters (Table S4 in Kusnadi and others (2021)) as for instance IFIT3, IFI6, IFI44 were much more abundant in cluster 1 than in other clusters, whereas LCP2 abundance was more balanced across clusters. As a visualization example, Figure 9 displays the various conditional CDFs estimated through both ccdf analyses for TDRD7 gene. It illustrates the reduction of the difference between the conditional CDF on X and its marginal counterpart (when not conditioning on X) when additionally conditioning on Z the clusters. This is reflected by the p-value which increase from 1.796244e-06 to 0.4043629 with the conditioning on Z: while TDRD7 remains DE in both analyses, its significance is decreased when taking into account the different cell clusters. While the majority of IFN genes remain significant at a threshold of 5% in both DEA, there is a general increasing trend of the p-values in adjusted analyses. This supports some confounding effect of the clusters on the significance of some genes within the Interferon pathway. Thus, taking into account the clusters does not fundamentally change the biological interpretation that Interferon pathway plays an important role in the severity status. But a part of the difference in abundance of some genes is due to the difference in abundance of the identified clusters. See supplementary figures for additional visualizations of ccdf results.
An obvious limit of this real data analysis lies in the double use of the data. First Kusnadi and others (2021) performed a clustering to define cell populations that were not annotated beforehand. Then, we use once again those clusters of cells as a covariate in our DEA. The clustering is based on the cell gene expression, therefore the association between the clusters and the gene expression -needed for the clusters to play the role of a confounding variable -is constructed during the first step. Although this example was used as an illustration of the ccdf method, it is quite realistic thanks to high-throughput technologies that allow nowadays to measure cell characteristics (surface and intracellular proteins) as well as single-cell gene expression.

Discussion
We propose a new framework for performing CIT, with an immediate application to differential expression analysis of scRNA-seq data. Our approach can accommodate complex designs, e.g.
with more than two experimental conditions or with continuous responses while adjusting for several additional covariates. ccdf is capable of distinguishing differences in distribution by using a CIT based on the estimation of CCDFs through a linear regression. The resulting asymptotic test is attractive due to the low computation times, especially dealing with a high number of observations. Yet, for small samples sizes (e.g. due to experimental design or cost limitations in data acquisition), we cannot always rely on an asymptotic test. Consequently, a permutation test is proposed in such cases. Performing permutations is obviously time consuming, but as it is necessary only for small sample sizes, computation times remain reasonable in such settings.
Nevertheless, easy parallelization of the permutation test can alleviate this problem. Furthermore, an adaptive procedure, for both the number of permutations and the number of thresholds, is implemented in order to accelerate ccdf while preserving a sufficient statistical power along with numerical precision for the lowest p-values. Per-gene asymptotic tests can also be computed in parallel to speed up computations. The proposed approach has been fully implemented in the userfriendly R package ccdf available on CRAN at https://CRAN.R-project.org/package=ccdf.
While ccdf can be applied to many types of data thanks to its flexibility, it has been specifically tailored for the need of scRNA-seq data DEA. In the simulation study, ccdf exhibits great results and versatility in complex designs such as multiple conditions comparison, and also allows to analyze data when the experiment design includes confounding variables while most of the competing methods cannot. Finally ccdf maintains great power while ensuring an effective control of FDR. The application to a real single-cell RNA-seq data set enhances the importance of being able to handle complex experimental designs, especially with a potential confounding variable. Of note, ccdf being a distribution-free approach, it can support any normalization method (and this choice is ultimately left to the responsibility of the user).
Tiberi and others (2020) recently proposed non-parametric permutation approach that also compares empirical CDF. While distinct shares some common ideas with ccdf, the two approaches widely differs in their test statistics and capabilities. On the one hand distinct requires multiple samples and only addresses the two groups comparison, while on the other hand ccdf provides an asymptotic test and can accommodate more complex experimental designs.
The number of evaluating thresholds considered in ccdf directly impacts both its computation time but also potentially its statistical power. To optimize this trade-off and to maintain the performances while reducing the computational cost, we propose to use a sequence of evenly spaced thresholds. Besides, instead of choosing a regular sequence of thresholds, one can prefer to manually chose the different thresholds at which CCDFs are computed and compared, e.g.
using prior knowledge to emphasize some areas of the distribution.
Although the linear regression estimation of the CCDF is efficient, different parameterizations can be considered (e.g. logistic regression which would be more natural). Currently ccdf cannot analyse multi-sample data, such as biological replicates or repeated measurements. This would consist in modifying the structure of the working model in equation (2.1). ccdf would thus be able to perform multi-sample analysis while keeping the advantage of its asymptotic test, at the cost of an increased computational burden. Besides, other test statistics could be investigated based on the same CIT.

Software
The proposed method has been implemented in an open-source R package called ccdf, available on CRAN at https://CRAN.R-project.org/package=ccdf and on Github at https://github. com/Mgauth/ccdf. All R scripts used for the simulations and the real data set analyses in this article were performed using R v3.6.3 and can be found on Zenodo at https://doi.org/10. 5281/zenodo.5699462.

Supplementary Material
Supplementary material is available online at http://biostatistics.oxfordjournals.org.  Fig. 4. Overall Type-I error, Power, FDR and TDR for ccdf under the 4 conditions case with increasing sample size. ccdf is the only method capable of dealing with more than 2 conditions. We perform the asymptotic test and the permutation test.  proportion. DB: both differential modality and different component means within each condition.ccdf is the only method capable of dealing with more than 2 conditions. We perform the asymptotic test and the permutation test.