The waddR package


waddR is an R package that provides a 2-Wasserstein distance based statistical test for detecting and describing differential distributions in one-dimensional data. Functions for wasserstein distance calculation, differential distribution testing, and a specialized test for differential expression in scRNA data are provided.

The package waddR provides three sets of utilities to cover distinct use cases, each described in a separate vignette:

These are bundled into the same package, because they are internally dependent: The procedure for detecting differential distributions in single-cell data is a refinement of the general two-sample test, which itself uses the 2-Wasserstein distance to compare two distributions.

Wasserstein Distance functions

The 2-Wasserstein distance is a metric to describe the distance between two distributions, representing two diferent conditions A and B. This package specifically considers the squared 2-Wasserstein distance d := W^2 which offers a decomposition into location, size, and shape terms.

The package waddR offers three functions to calculate the 2-Wasserstein distance, all of which are implemented in Cpp and exported to R with Rcpp for better performance. The function wasserstein_metric is a Cpp reimplementation of the function wasserstein1d from the package transport and offers the most exact results. The functions squared_wass_approx and squared_wass_decomp compute approximations of the squared 2-Wasserstein distance with squared_wass_decomp also returning the decomosition terms for location, size, and shape. See ?wasserstein_metric, ?squared_wass_aprox, and ?squared_wass_decomp.

Two-Sample Testing

This package provides two testing procedures using the 2-Wasserstein distance to test whether two distributions F_A and F_B given in the form of samples are different ba specifically testing the null hypothesis H0: F_A = F_B against the alternative hypothesis H1: F_A != F_B.

The first, semi-parametric (SP), procedure uses a test based on permutations combined with a generalized pareto distribution approximation to estimate small pvalues accurately.

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

See ?wasserstein.test for more details.

Single Cell Test: 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 has been 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.

See the documentation of the single cell procedure ? and the test for zero expression levels ?testZeroes for more details.


To install waddR from Bioconductor, use BiocManager with the following commands:

if (!requireNamespace("BiocManager"))

Using BiocManager, the package can also be installed from github directly:


The package waddR can then be used in R:


Session Info

#> R version 3.6.3 (2020-02-29)
#> Platform: x86_64-pc-linux-gnu (64-bit)
#> Running under: Ubuntu 18.04.4 LTS
#> Matrix products: default
#> BLAS:   /home/biocbuild/bbs-3.10-bioc/R/lib/
#> LAPACK: /home/biocbuild/bbs-3.10-bioc/R/lib/
#> locale:
#>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
#>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=C              
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#>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
#>  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> other attached packages:
#> [1] waddR_1.0.1
#> loaded via a namespace (and not attached):
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#> [67] compiler_3.6.3