# Group Comparisons

This section explores concepts related to multivariate statistical tests of differences between groups. Concepts are presented in short chapters so that the reader can easily jump to the information they seek.

We begin by considering ANOVA / MANOVA, which illustrate how variation can be partitioned and how that partitioning can be expressed in a test statistic.

ANOVA and MANOVA assess the significance of the test statistic via its expected distribution – this is why they require parametric assumptions such as normality. The techniques we will cover are **permutation-based**, which means that they do not make these assumptions. We’ll consider:

- General principles about permutation tests,
- How to control permutations,
- How to restrict permutations.

The techniques we’ll cover as also **distance-based**, meaning that they are based on the distance matrix calculated from the response(s). The techniques that we’ll cover are:

- ANalysis Of SIMilarity (ANOSIM)
- Mantel tests
- Multi-Response Permutation Procedure (MRPP)
- PERMutational ANOVA/MANOVA (PERMANOVA)
- PERMutational analysis of DISPersion (PERMDISP)
- Randomization of Residuals in a Permutation Procedure (RRPP)
- MVABUND

We’ll illustrate each test using two datasets, a simple one that can be worked by hand and a larger one from our oak plant communities that requires computer calculations. Included is a script to load and process the larger dataset.

Most of the techniques are designed to test for differences among *a priori* groups such as experimental treatments. Some of the techniques are also appropriate for testing linear relationships with continuously distributed explanatory variables, as in regression. We’ll also consider how to analyze complex models.

Finally, I provide a comparison of the techniques.