Statistical tests for detecting differential distributions based on the 2-Wasserstein distance

For more details on the methods and further information, please have a look at our paper: Schefzik, R., Flesch, J., and Goncalves, A. (2021). Fast identification of differential distributions in single-cell RNA-sequencing data with waddR. Bioinformatics, 37, 3204-3211. DOI:

The waddR package offers statistical tests based on the 2-Wasserstein distance for detecting and characterizing differences between two distributions given in the form of samples. Functions for calculating the 2-Wasserstein distance and testing for differential distributions are provided, as well as specifically tailored test for differential expression in single-cell RNA sequencing data.

The package provides tools to address the following tasks: 1. Computation of the 2-Wasserstein distance 2. Two-sample tests to check for differences between two distributions 3. Detection of differential gene expression distributions in single-cell RNA sequencing data



Via Package Repository

Available on Bioconductor:

if (!requireNamespace("BiocManager", quietly = TRUE))

From Github

The latest package version can be installed from Github using BiocManager:

if (!requireNamespace("BiocManager"))

Running Tests

Tests can be run by calling test() from the devtools package. All tests are implemented using the testthat package and reside in tests/testhat

Using waddR

2-Wasserstein distance functions

The 2-Wasserstein distance is a metric to quantify the distance between two distributions, representing two different conditions A and B. The waddR package specifically considers the squared 2-Wasserstein distance, which offers a decomposition into location, size, and shape terms, thus providing a characterization of potential differences.

The waddR package offers three functions to calculate the 2-Wasserstein distance, which are implemented in C++ and exported to R with Rcpp for faster computation. The function wasserstein_metric is a C++ reimplementation of the function wasserstein1d from the R package transport. The functions squared_wass_approx and squared_wass_decomp compute approximations of the squared 2-Wasserstein distance, with squared_wass_decomp also returning the decomposition terms for location, size, and shape.

See ?wasserstein_metric, ?squared_wass_aprox, and ?squared_wass_decomp, as well as the accompanying paper Schefzik et al. (2021).

Testing for differences between two distributions

The waddR package provides two testing procedures using the 2-Wasserstein distance to test whether two distributions given in the form of samples are different by specifically testing the null hypothesis of no difference against the alternative hypothesis that the two distributions are different.

The first, semi-parametric (SP), procedure uses a permutation-based test combined with a generalized Pareto distribution approximation to estimate small p-values accurately.

The second procedure uses a test based on asymptotic theory (ASY) which is valid only if the samples can be assumed to come from continuous distributions.

See ?wasserstein.test for more details.

Testing for differences between two distributions in the context of single-cell RNA sequencing data:

The waddR package provides an adaptation of the semi-parametric testing procedure based on the 2-Wasserstein distance which is specifically tailored to identify differential distributions in single-cell RNA-seqencing (scRNA-seq) data. In particular, a two-stage (TS) approach is implemented that takes account of the specific nature of scRNA-seq data by separately testing for differential proportions of zero gene expression (using a logistic regression model) and differences in non-zero gene expression (using the semi-parametric 2-Wasserstein distance-based test) between two conditions.

Note that as input for scRNA-seq analysis, waddR expects a table of pre-filtered and normalised count data. As filtering and normalisation are important steps that can have a profound impact in a scRNA-seq workflow (Cole et al., 2019), these should be tailored to the specific question of interest before applying waddR. waddR is applicable to data from any scRNA-seq platform (demonstrated in our paper for 10x Genomics and Fluidigm C1 Smart-Seq2) normalised using most common methods, such as those implemented in the Seurat (Butler et al., 2018) or scran (Lun et al., 2016) packages. Normalisation approaches that change the shape of the gene distributions (such as quantile normalisation) and gene-wise scaling or standardizing should be avoided when using waddR.

See ? and ?testZeroes for more details.


We have included detailed examples of how to use the functions provided with waddR in our vignettes. They are available online here (update this link once it is final) or from an R session with the following command: browseVignettes("waddR")


Butler, A., Hoffman, P., Smibert, P., Papalexi, E., and Satija, R. (2018). Integrating single-cell transcriptomic data across different conditions, technologies, and species. Nature Biotechnology, 36, 411–420.

Cole, M. B., Risso, D., Wagner, A., De Tomaso, D., Ngai, J., Purdom, E., Dudoit, S., and Yosef, N. (2019). Performance assessment and selection of normalization procedures for single-cell RNA-seq. Cell Systems, 8, 315–328.

Lun, A. T. L., Bach, K., and Marioni, J. C. (2016). Pooling across cells to normalize single-cell RNA sequencing data with many zero counts. Genome Biology, 17, 75.

Schefzik, R., Flesch, J., and Goncalves, A. (2021). Fast identification of differential distributions in single-cell RNA-sequencing data with waddR. To appear in Bioinformatics. DOI:

goncalves-lab/waddR documentation built on Oct. 28, 2021, 5:50 a.m.