$W_d^*$: distance-based multivariate analysis of variance for multivariate data

Introduction

WdStar is an R package for $W_d^*$, a method for multivariate analysis of variance based on Welch's MANOVA designed to address challenges of existing methods including PERMANOVA.

• Are you using PERMANOVA and concerned about how heteroscedasticity and unbalanced sample sizes may affect your analyses?
• Have you ever questioned the reliability of community-wide microbiome analysis results?
• Are you looking for a global test for your multivariate dataset?

$W_d^*$ addresses these concerns and is

• robust to heteroscedasticity;
• handles multi-level factors and stratification;
• allows for multiple post hoc testing scenarios;
• allows for adjustment of covariates; and
• can be used with any distance or dissimilarity matrix.

Peer-Reviewed Publications on the $W_d^*$-test Family

Preprint: Covariate-adjusted multivariate analysis for omics data

• Hamidi B, Fanning L, Wallace K, & Alekseyenko AV. Submitted to Briefings in Bioinformatics on June 23, 2026. Not currently accepted or under review. DOI forthcoming.
• Code repository

$W_d^*$-test: robust distance-based multivariate analysis of variance

• Hamidi B, Wallace K, Vasu C, & Alekseyenko AV. Microbiome. 2019.
• Code repository

$T_w^2$: multivariate Welch t-test on distances

• Alekseyenko AV. Multivariate Welch t-test on distances. Bioinformatics. 2016.
• Code repository

Installation

Source installation of the WdStar R package is available directly from GitHub using remotes for R 3.6 or later:

install.packages("remotes")
remotes::install_github("alekseyenko/WdStar", force = TRUE)
library(WdStar)
packageVersion("WdStar")

Until a CRAN release is available, install WdStar from GitHub as shown above.

Quick Start

For detailed and complex examples please refer to our publication repositories, which contain Markdown files with application datasets and code.

The following is a simple example using the mtcars dataset to assess the effect of gear on mpg, cyl, and disp (first three variables of the dataset):

# Load dataset
data(mtcars)

# The outcome could be a single variable or multiple variables (such as multidimensional omics data).  

### This is an example of outcome with a single variable (`mpg`):
dm <- dist(mtcars$mpg, method="euclidean")

### This is an example of outcome with multiple variables (`mpg`, `cyl`, and `disp`):
dm <- dist(mtcars[1:3], method="euclidean") 

# Grouping/independent variable. You could use multiple variables here too.
f <- factor(mtcars$gear)

# Basic multivariate test example ###########
#############################################
WdS.test(dm=dm, f=f)

# Stratified example ########################
#############################################
strata <- factor(mtcars$vs)
WdS.test(dm=dm, f=f, strata=strata)

# Covariate adjustment/elimination examples #
#############################################
## Right-hand side adjustment formula to specify adjustment covariates. 
formula <- ~ wt + as.factor(am) 

## Adjustment example 1: pass unadjusted `dm` and formula to WdS.test()
WdS.test(dm=dm, f=f, formula=formula, formula_data=mtcars) ## Perform adjusted test

## Note that the output includes goodness.of.fit computed from raw and adjusted  
##  distances, which can be used to assess the impact of adjustment on the data.  

## Adjustment example 2: Create the adjusted distance matrix `a.dm` outside the function
a.dm <- a.dist(dm=dm, formula=formula, formula_data=mtcars) 
WdS.test(dm=a.dm, f=f) ## Perform adjusted test with `a.dm`

## Goodness of fit can also be computed from raw and adjusted distances directly
dist.goodness.of.fit(dm=dm, adjusted_dm=a.dm)

Further examples are provided in the package documentation and may be accessed by running the following commands:

?WdS.test
?a.dist
?dist.goodness.of.fit

Feature Requests and Bugs

We welcome feature requests and bug reports and kindly ask you to submit them via our GitHub issue tracker.