Permutational MANOVA or adonis

Adonis is a technique used in ecology to explain communities with environmental variables.

Permutational multivariate analysis of variance (PERMANOVA)

Permutational multivariate analysis of variance (PERMANOVA),also called Non Parametric MANOVA (NPMANOVA), is a model used in ecology and biology to explain communities with environmental variables. Its goal is similar to Multivariate Analysis of Variance (MANOVA) but as opposed to MANOVA, PERMANOVA is based on permutations of distance matrices.

The adonis function developed in XLSTAT-R calls the adonis function from the vegan package in R (Martin Henry H. Stevens and Jari Oksanen), which proposes several options that will let you gain a deep insight on your data:

Run your permutational MANOVA on quantitative and/or qualitative variables
Constrain permutations within each group (strata)
Include interactions among your data
Choose among several distances such as the Bray-Curtis distance, the Canberra distance, the Euclidean distance, and many other dissimilarity measures
Use square root dissimilarities to avoid negative eigenvalues
Choose between a marginal test of terms and a sequential test of terms
Choose how to handle missing data

What is the difference between PERMANOVA, MANOVA and ANOVA?

The regular MANOVA uses the same conceptual framework as the ANOVA. The main difference comes from the nature of the dependent variables: instead of a single variable, we can study many of them. With the MANOVA, explanatory variables are often called factors. Effects of factors are estimated on a combination of several response variables.

Even though MANOVA and ANOVA are widely used models in statistics, their stringent assumptions make them hard to use when it comes to analyzing ecological data, as well as environmental and biological data. For example in some ecological datasets, species with low abundances are unlikely to fill the normality assumption. In these cases it is suggested to use a more suitable method like PERMANOVA, which adresses the limits of these assumptions.