Add example on PERMANOVA (#634)

inst/pages/beta_diversity.qmd

@@ -104,6 +104,53 @@ tse <- addDivergence(
     FUN = getDissimilarity)
 ```
 
+## Permutational Multivariate Analysis of Variance
+
+PERMANOVA (Permutational Multivariate Analysis of Variance) can be seen as a
+non-parametric, multivariate extension of ANOVA (Analysis of Variance). It is
+used to compare groups in multivariate data and to assess whether a variable
+explains differences in the data. PERMANOVA first calculates the dissimilarities
+between samples, then compares the centroids (multivariate means) of each group
+to determine if they differ significantly.
+
+PERMANOVA determines statistical significance by permuting (randomly shuffling)
+group labels many times. By comparing the observed test statistic to those from
+the permutations, it assesses if the observed group differences are likely due
+to chance, without relying on parametric assumptions.
+
+Since PERMANOVA relies on comparing centroids, it assumes that the variation
+within groups is smaller than the variation between groups, meaning the groups
+are distinct. If this assumption isn’t met, it can affect the interpretation of
+the results, so it’s important to check this assumption before drawing
+conclusions.
+
+```{r}
+#| label: permanova
+
+res <- getPERMANOVA(
+    tse,
+    assay.type = "relabundance",
+    formula = x ~ SampleType
+    )
+res
+```
+
+The output includes both PERMANOVA and homogeneity results. When PERMANOVA
+results are significant, we check the homogeneity results to ensure reliability.
+A non-significant homogeneity result indicates that group dispersions are
+similar, meaning the group dispersion is homogeneous, and the PERMANOVA results
+are reliable.
+
+In our case, PERMANOVA results are significant, and the homogeneity test shows
+that group dispersion is indeed homogeneous. This suggests that _SampleType_
+explains the variance in the microbial profile.
+
+PERMANOVA and distance-based redundancy analysis (dbRDA) are closely related,
+and often they give similar results. However, their assumptions are different;
+while PERMANOVA is non-parametric method, dbRDA assumes linear relationships in
+dissimilarity matrix. However, dbRDA can be used more broadly as the ordination
+can also be visualized.
+
 ## Unsupervised ordination {#sec-unsupervised-ordination}
 
 Unsupervised ordination methods analyze variation in the data without additional