wasserstein.sc | R Documentation |

Two-sample test for single-cell RNA-sequencing data to check for differences between two distributions using the 2-Wasserstein distance: Semi-parametric implementation using a permutation test with a generalized Pareto distribution (GPD) approximation to estimate small p-values accurately

```
wasserstein.sc(x, y, method = c("TS", "OS"), permnum = 10000, seed = NULL)
## S4 method for signature 'matrix,vector'
wasserstein.sc(x, y, method = c("TS", "OS"), permnum = 10000, seed = NULL)
## S4 method for signature 'SingleCellExperiment,SingleCellExperiment'
wasserstein.sc(x, y, method = c("TS", "OS"), permnum = 10000, seed = NULL)
```

`x` |
matrix of single-cell RNA-sequencing expression data with genes in
rows and cells (samples) in columns [alternatively, a |

`y` |
vector of condition labels [alternatively, a |

`method` |
method employed in the testing procedure: if "OS", a one-stage test is performed, i.e. the semi-parametric test is applied to all (zero and non-zero) expression values; if "TS", a two-stage test is performed, i.e. the semi-parametric test is applied to non-zero expression values only and combined with a separate test for differential proportions of zero expression using logistic regression |

`permnum` |
number of permutations used in the permutation testing procedure |

`seed` |
number to be used to generate a L'Ecuyer-CMRG seed, which itself seeds the generation of an nextRNGStream() for each gene to achieve reproducibility; default is NULL, and no seed is set |

Details concerning the testing procedure for
single-cell RNA-sequencing data can be found in Schefzik et al.
(2020). Corresponds to the function `.testWass`

when identifying the argument
`inclZero=TRUE`

in `.testWass`

with the argument `method="OS"`

and the argument
`inclZero=FALSE`

with the argument `method="TS"`

.

The input data matrix `x`

[alternatively, the input data matrices to form the `SingleCellExperiment`

objects `x`

and `y`

, respectively] as the starting point of the test is supposed to contain the single-cell RNA-sequencing expression data after several pre-processing steps. In particular, 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.

For the two-stage approach (`method="TS"`

) according to Schefzik et al. (2021), two separate tests for differential proportions of zero expression (DPZ) and non-zero differential distributions (non-zero DD), respectively, are performed. In the DPZ test using logistic regression, the null hypothesis that there are no DPZ is tested against the alternative that there are DPZ. In the non-zero DD test using the semi-parametric 2-Wasserstein distance-based procedure, the null hypothesis that there is no difference in the non-zero expression distributions is tested against the alternative that the two non-zero expression distributions are differential.

The current implementation of the test assumes that the expression data matrix is based on one replicate per condition only. For approaches on how to address settings comprising multiple replicates per condition, see Schefzik et al. (2021).

Matrix, where each row contains the testing results of the respective gene from `dat`

. The corresponding values of each row (gene) are as follows, see Schefzik et al. (2021) for details.
In case of `inclZero=TRUE`

:

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: p-value of the semi-parametric 2-Wasserstein distance-based test when applied to all (zero and non-zero) respective gene expression values

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)

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

`\geq 0.05`

(otherwise NA)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

pval.adj: adjusted p-value of the semi-parametric 2-Wasserstein distance-based test according to the method of Benjamini-Hochberg (i.e. adjusted p-value corresponding to pval)

In case of `inclZero=FALSE`

:

d.wass: 2-Wasserstein distance between the two samples computed by quantile approximation (based on non-zero expression only)

d.wass^2: squared 2-Wasserstein distance between the two samples computed by quantile approximation (based on non-zero expression only)

d.comp^2: squared 2-Wasserstein distance between the two samples computed by decomposition approximation (based on non-zero expression only)

d.comp: 2-Wasserstein distance between the two samples computed by decomposition approximation (based on non-zero expression only)

location: location term in the decomposition of the squared 2-Wasserstein distance between the two samples (based on non-zero expression only)

size: size term in the decomposition of the squared 2-Wasserstein distance between the two samples (based on non-zero expression only)

shape: shape term in the decomposition of the squared 2-Wasserstein distance between the two samples (based on non-zero expression only)

rho: correlation coefficient in the quantile-quantile plot (based on non-zero expression only)

p.nonzero: p-value of the semi-parametric 2-Wasserstein distance-based test when applied to non-zero respective gene expression values

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)

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

`\geq 0.05`

(otherwise NA)perc.loc: fraction (in %) of the location part with respect to the overall squared 2-Wasserstein distance obtained by the decomposition approximation (based on non-zero expression only)

perc.size: fraction (in %) of the size part with respect to the overall squared 2-Wasserstein distance obtained by the decomposition approximation (based on non-zero expression only)

perc.shape: fraction (in %) of the shape part with respect to the overall squared 2-Wasserstein distance obtained by the decomposition approximation (based on non-zero expression only)

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 (based on non-zero expression only)

p.zero: p-value of the test for differential proportions of zero expression (logistic regression model)

p.combined: combined p-value of p.nonzero and p.zero obtained by Fisher's method

p.adj.nonzero: adjusted p-value of the semi-parametric 2-Wasserstein distance-based test based on non-zero expression only according to the method of Benjamini-Hochberg (i.e. adjusted p-value corresponding to p.nonzero)

p.adj.zero: adjusted p-value of the test for differential proportions of zero expression (logistic regression model) according to the method of Benjamini-Hochberg (i.e. adjusted p-value corresponding to p.zero)

p.adj.combined: adjusted combined p-value of p.nonzero and p.zero obtained by Fisher's method according to the method of Benjamini-Hochberg (i.e. adjusted p-value corresponding to p.combined)

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.

```
#simulate scRNA-seq data
set.seed(24)
nb.sim1<-rnbinom(n=(750*250),1,0.7)
dat1<-matrix(data=nb.sim1,nrow=750,ncol=250)
nb.sim2a<-rnbinom(n=(250*100),1,0.7)
dat2a<-matrix(data=nb.sim2a,nrow=250,ncol=100)
nb.sim2b<-rnbinom(n=(250*150),5,0.2)
dat2b<-matrix(data=nb.sim2b,nrow=250,ncol=150)
dat2<-cbind(dat2a,dat2b)
dat<-rbind(dat1,dat2)*0.25
#randomly shuffle the rows of the matrix to create the input matrix
set.seed(32)
dat<-dat[sample(nrow(dat)),]
condition<-c(rep("A",100),rep("B",150))
#call wasserstein.sc with a matrix and a vector including conditions
#set seed for reproducibility
#two-stage method
wasserstein.sc(dat,condition,method="TS",permnum=10000,seed=24)
#one-stage method
wasserstein.sc(dat,condition,method="OS",permnum=10000,seed=24)
#alternatively, call wasserstein.sc with two SingleCellExperiment objects
#note that the possibly pre-processed and normalized expression matrices need to be
#included using the "counts" argument
sce.A <- SingleCellExperiment::SingleCellExperiment(
assays=list(counts=dat[,1:100]))
sce.B <- SingleCellExperiment::SingleCellExperiment(
assays=list(counts=dat[,101:250]))
#set seed for reproducibility
#two-stage method
wasserstein.sc(sce.A,sce.B,method="TS",permnum=10000,seed=24)
#one-stage method
wasserstein.sc(sce.A,sce.B,method="OS",permnum=10000,seed=24)
```

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