XLSTAT: Redundancy Analysis (RDA) in Excel

 

Background

Redundancy Analysis (RDA) is a multivariate extension of linear models, where multiple response variables are explained by a set of explanatory variables. It performs PCA on the response variables (Y matrix), constrained by the explanatory variables (X matrix), assuming linear relationships.

RDA vs. CCA:

• RDA assumes linear relationships between Y and X.

• CCA assumes unimodal relationships, making it more suitable for ecological niche modeling where environmental gradients are fully represented.

Partial RDA and Conditioning Variables:

• Partial RDA removes the influence of unwanted explanatory variables (conditioning variables) to isolate effects of interest.

• Conditioning variables might include block effects or environmental noise.

 

Dataset for Example

The attached data corresponds to the abundances of 30 plant species (in columns) at 20 sites (in rows) in a dune environment. Species names are the combinations of the first 4 letters of their genus and the first four letters of their species names in Latin. Sites are also described by two environmental variables in columns: 

• Soil thickness : quantitative variable, depth of the A1 soil horizon.

• Management : qualitative variable with levels BF (Biological farming), HF (Hobby farming), NM (Nature Conservation Management) and SF (Standard Farming).

The data are extracted from [Jongman, R.H.G, ter Braak, C.J.F & van Tongeren, O.F.R. (1987). Data Analysis in Community and Landscape Ecology. Pudoc, Wageningen]. Other environmental variables from the same data source have been excluded for this tutorial.

Using the dataset, we will investigate the linear relationships between management type and A1 soil horizon depth on dune plant communities using Redundancy Analysis.

 

Setting Up

• Open XLSTAT.

• Click on the Marketing/ Multiblock Data Analysis menu and select the Redundancy Analysis feature.

  

In the General tab:

• Select the Site / Species matrix in the Response Variables field.

• Under Explanatory Variables, select Soil Thickness in the Quantitative field and Management type in the Qualitative field.

In the Options tab:

• Activate the Permutation test option, and set the number of permutations to 1000. If your response variables or your quantitative explanatory variables are not on the same scale, consider activating the Reduce options for the corresponding matrix. 

In the Charts tab:

• Deactivate the Observations option under Display.

• Click on the OK button to launch the computations.

• The results are displayed on a new Excel sheet.

 

Interpreting the Results

One first dialog box pops up allowing you to select the axes to represent in the RDA charts. With axes F1 & F2, we are able to represent 80% of the constrained inertia (explanation further below). Select those two axes and click Done.

  

After descriptive statistics on both the response and the explanatory variables, the  Inertia summary table appears:

  

Here we are investigating the inertia or variability of the response variables. It is split into: 

• Constrained inertia  , which is the part that is explained by the explanatory variables matrix.

• Unconstrained inertia which is the remaining part.

Around 39% of the species community variability can be explained by the two environmental variables under investigation. Then a permutation test allows to test the null hypothesis that the response and explanatory variables are not linearly related. Here the p-value is far lower than the risk threshold alpha. We may thus reject the null hypothesis while taking a very small risk of being wrong. This is an important step as it validates the reliability of subsequent RDA results.

Then the details of the constrained inertia carried by each RDA axis are provided in a table: 

Axis F1 carries 45% of the constrained inertia, which is 18% of the total inertia. Together, as we have seen, axes F1 & F2 carry 80% of the constrained inertia, which corresponds to 32% of the total inertia. An equivalent table with the unconstrained inertias  is displayed after:

The  standardized canonical coefficients allow to assess the effect strength of each coefficient from the explanatory variables on every axis.

  

 Observation scores are the coordinates of observations (sites) on the RDA chart.

  

  Contributions (Observations) are the extent to which every observation contributed to the construction of every axis.

 Squared cosines (Observations) are the representation quality of each observation on each axis.

Observations 15, 16 and 20 seem to have widely contributed to the construction of axis F1 and are well represented on it. Scores and squared cosines are also shown for the response variables (species in this example).

Likewise, species squared cosines reflect the representativeness of each species on each axis. Species such as  Elymus repens , Poa pratensis , Ranunculus flammula and other species seem to be well presented on axis F1. Scores of the explanatory variables coefficients are also provided. At the bottom of the report, we see the RDA chart  , also called RDA triplot if it also includes observations (which have been removed here to ensure a better readability).

Interpret only observations and response variables with high squared cosines on an RDA axis.

• Axis 1 contrasts Nature Conservation Management and high soil depth (left) with Biological and Hobby Farming and shallow soils (right).

  • Deep soils and conservation areas are associated with species like Eleocharis palustris, Ranunculus flammula, and Salix repens.

  • Shallow soils under biological/hobby farming link to species like Lolium perenne and Poa pratensis.

• Axis 2 also reflects soil depth and highlights Standard Farming, associated with Alopecurus geniculatus.