*The concept of total inertia does not exist in DCA. Alternative software use the total inertia from other ordination methods such as orthogonal correspondence analysis. Just call cca() for your data to get the total inertia of orthogonal CA. However, that really has no relevance for DCA, although that statistics is commonly used and ritually reported in papers.*

### Table of Contents

Section: Ordination analysis

## CA & DCA (unimodal unconstrained ordination)

### Example 1: CA on Ellenberg's Danube meadow dataset

This simple example shows correspondence analysis of Danube meadow dataset, collected by Heinz Ellenberg and used in number of methodological studies.
First, import data, initiate vegan library and calculate CA using the function `cca`

from vegan:

danube.spe <- read.delim ('https://raw.githubusercontent.com/zdealveindy/anadat-r/master/data/danube.spe.txt', row.names = 1) library (vegan) CA <- cca (danube.spe) CA

Call: cca(X = danube.spe) Inertia Rank Total 3.043 Unconstrained 3.043 24 Inertia is mean squared contingency coefficient Eigenvalues for unconstrained axes: CA1 CA2 CA3 CA4 CA5 CA6 CA7 CA8 0.5718 0.4944 0.2950 0.2472 0.2057 0.1764 0.1528 0.1418 (Showed only 8 of all 24 unconstrained eigenvalues)

The total inertia (heterogeneity of the dataset) is 3.043, and the first axis captures 18.8% of total variation in species composition (0.5718 / 3.043 = 0.1879, where 0.5718 is eigenvalue of the first axis CA1, and 3.043 is total inertia).

Ordination diagram reveals the pattern of samples and species in ordination diagram:

ordiplot (CA)

It is evident that sample 19 is quite different from the rest of data, and correspondence analysis even greatly exaggerates this difference.

### Example 2: DCA on Vltava river valley dataset to decide whether linear or unimodal ordination method should be used

To decide whether the compositional data are homogeneous or heterogeneous, respectively (and thus more suitable for linear or unimodal ordination methods, respectively), we may calculate detrended correspondence analysis (DCA) first and check the length of the first ordination axis (in units of S.D.) to decide.

vltava.spe <- read.delim ('https://raw.githubusercontent.com/zdealveindy/anadat-r/master/data/vltava-spe.txt', row.names = 1) DCA <- decorana (log1p (vltava.spe)) DCA

Call: decorana(veg = log1p(vltava.spe)) Detrended correspondence analysis with 26 segments. Rescaling of axes with 4 iterations. DCA1 DCA2 DCA3 DCA4 Eigenvalues 0.5338 0.3956 0.2488 0.2457 Decorana values 0.5533 0.3677 0.2410 0.1961 Axis lengths 4.5446 3.5426 2.9208 2.9726

The length of first axis is 4.5 S.D. units, which means that (according to Lepš & Šmilauer 2003) unimodal ordination methods are preferable.

I won't draw an ordination diagram of the result here - since this example is focused only on the decision of whether the `vltava`

dataset is homogeneous or heterogeneous, it is not relevant here. However, note that the section Ordination diagrams is devoted to drawing ordination diagrams using this dataset.

You may be surprised that you haven't gotten any **total inertia** values when printing decorana results, although in other software (e.g. CANOCO) these are available, together with percentage variance explained by particular axes. The reason for this is that DCA does not support the concept of total inertia values (also, it produces only four axes, i.e. four eigenvalues). However, if you insist, you may get total inertia applying correspondence analysis (CA) on your data:

cca (log1p (vltava.spe))

Call: cca(X = log1p(vltava.spe)) Inertia Rank Total 7.372 Unconstrained 7.372 96 Inertia is mean squared contingency coefficient Eigenvalues for unconstrained axes: CA1 CA2 CA3 CA4 CA5 CA6 CA7 CA8 0.5533 0.4594 0.4131 0.3083 0.2951 0.2576 0.2147 0.2032 (Showed only 8 of all 96 unconstrained eigenvalues)

As you can see, total inertia is 7.372, and if needed, variation captured by particular axes can be calculated as eigenvalue/total inertia (e.g., for the first axis, 0.553/7.372*100 = 7.50%)^{1)}

^{1)}