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If there is no significant difference between two groups of metagenomic samples according to "Multivariate differential abundance tests" for example PERMANOVA and ANOSIM, does it makes sense to use "Univariate differential abundance testing" to see, for example, what species have significantly different abundances between groups?

The issue is that I found species that have significantly different abundances among samples of two groups using Wald test in DESeq2, but there is no significant difference between the two groups according to PERMANOVA and ANOSIM.

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Yes, it does make sense to perform differential abundance testing after one doesn't find compositional differences with PERMANOVA / ANOSIM.

It seems to me you are drawing an (implicit) analogy to ANOVA and post-hoc tests like the Tukey Honest Significant Difference test. Intuitively, the analogy makes sense, however, these situations are quite different. PERMANOVA / ANOSIM and DESeq2 are operating on different aspects of the data, with different transformations and under different assumptions.

PERMANOVA and ANOSIM do not operate on the raw counts directly, rather, they test differences between groups by calculating some distance or dissimilarity between the samples. One can choose between several distance / dissimilarity measures, and the test result may be impacted by the chosen measure.

DESeq2 (and other packages), on the other hand, test for species abundances under a parametric model, operating on the raw counts directly.

Thus (sadly) one is not a natural, optional follow-up of the other, they are complementary and will tell different stories about the data.

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