# wasserstein.test: Two-sample test to check for differences between two... In goncalves-lab/waddR: Statistical tests for detecting differential distributions based on the 2-Wasserstein distance

## Description

Two-sample test to check for differences between two distributions using the 2-Wasserstein distance, either using the semi-parametric permutation testing procedure with a generalized Pareto distribution (GPD) approximation to estimate small p-values accurately or the test based on asymptotic theory

## Usage

 `1` ```wasserstein.test(x, y, method = c("SP", "ASY"), permnum = 10000) ```

## Arguments

 `x` sample (vector) representing the distribution of condition A `y` sample (vector) representing the distribution of condition B `method` testing procedure to be employed: "SP" for the semi-parametric permutation testing procedure with GPD approximation, "ASY" for the test based on asymptotic theory; if no method is specified, "SP" will be used by default. `permnum` number of permutations used in the permutation testing procedure (if `method="SP"` is performed); default is 10000

## Details

Details concerning the two testing procedures (i.e. the semi-parametric permutation testing procedure with GPD approximation and the test based on asymptotic theory) can be found in Schefzik et al. (2020).

Note that the asymptotic theory-based test (`method="ASY"`) should only be employed when the samples x and y can be assumed to come from continuous distributions. In contrast, the semi-parametric test (`method="SP"`) can be used for samples coming from continuous or discrete distributions.

## Value

A vector, see Schefzik et al. (2020) for details:

• d.wass: 2-Wasserstein distance between the two samples computed by quantile approximation

• d.wass^2: squared 2-Wasserstein distance between the two samples computed by quantile approximation

• d.comp^2: squared 2-Wasserstein distance between the two samples computed by decomposition approximation

• d.comp: 2-Wasserstein distance between the two samples computed by decomposition approximation

• location: location term in the decomposition of the squared 2-Wasserstein distance between the two samples

• size: size term in the decomposition of the squared 2-Wasserstein distance between the two samples

• shape: shape term in the decomposition of the squared 2-Wasserstein distance between the two samples

• rho: correlation coefficient in the quantile-quantile plot

• pval: The p-value of the semi-parametric or the asymptotic theory-based test, depending on the specified method

• p.ad.gpd: in case the GPD fitting is performed: p-value of the Anderson-Darling test to check whether the GPD actually fits the data well (otherwise NA). This output is only returned when performing the semi-parametric test (method="SP")!

• N.exc: in case the GPD fitting is performed: number of exceedances (starting with 250 and iteratively decreased by 10 if necessary) that are required to obtain a good GPD fit, i.e. p-value of Anderson-Darling test ≥q 0.05 (otherwise NA). This output is only returned when performing the semi-parametric test (method="SP")!

• perc.loc: fraction (in %) of the location part with respect to the overall squared 2-Wasserstein distance obtained by the decomposition approximation

• perc.size: fraction (in %) of the size part with respect to the overall squared 2-Wasserstein distance obtained by the decomposition approximation

• perc.shape: fraction (in %) of the shape part with respect to the overall squared 2-Wasserstein distance obtained by the decomposition approximation

• decomp.error: relative error between the squared 2-Wasserstein distance computed by the quantile approximation and the squared 2-Wasserstein distance computed by the decomposition approximation

## References

Schefzik, R., Flesch, J., and Goncalves, A. (2020). waddR: Using the 2-Wasserstein distance to identify differences between distributions in two-sample testing, with application to single-cell RNA-sequencing data.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```set.seed(24) x<-rnorm(100) y1<-rnorm(150) y2<-rexp(150,3) y3<-rpois(150,2) #for reproducibility, set a seed for the semi-parametric, permutation-based test set.seed(32) wasserstein.test(x,y1,method="SP",permnum=10000) wasserstein.test(x,y1,method="ASY") set.seed(33) wasserstein.test(x,y2,method="SP",permnum=10000) wasserstein.test(x,y2,method="ASY") set.seed(34) #only consider SP method, as Poisson distribution is discrete wasserstein.test(x,y3,method="SP",permnum=10000) ```

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