### Introduction

### Theory, Examples & Exercises

en:varpart

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Note: variation partitioning is sometimes also called ** commonality analysis** in reference to the

In case we have two or more explanatory variables, one may be interested in variation in species composition explained by each of them. If some of these explanatory variables are correlated, one must expect that variation explained by the first or the other variable cannot be separated - it will be shared.

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The way how to approach this problem is variation partitioning, when variation explained by each variable (or set of variables) independently (marginal variation) is partitioned into variation attributable purely to given environmental variable (partial variation), and shared variation attributable to two or more variables.

Variation can be partitioned into individual variables (e.g. variation explained by soil pH vs variation explained by soil Ca) or into groups of variables (e.g. soil variables vs climatic variables).

Result can be visualized using Venn's diagram (see figure on the right). Meaning of the fractions in Venn's diagram is the following^{2)}:

- [a] - partial variation explained by variable 1 (i.e. variation explained by this variable after removing variation explained by the second variable);
- [c] - partial variation explained by variable 2;
- [b] - shared variation explained by both variables (cannot be decided to which of them should be attributed, and is a result of correlation between both variables);
- [a+b] - simple (or marginal) variation explained by variable 1 (i.e. variation this variable would explain if it is as the only explanatory variable in the model);
- [b+c] - marginal variation explained by variable 2;
- [d] - unexplained variation.

If the variation is partitioned among groups with the same number of variables (e.g. two soil and two climatic variables), than variation explained by each group is comparable without adjustement. However, if groups contain different numbers of variables, variation explained by not adjusted R^{2} is not comparable, since R2 tends to increase with the number of explanatory variables. Here, use of adjusted R2 is recommended.

The library `vegan`

offers function `varpart`

, which can partition variation among up to four variables (or groups of variables). Note that `varpart`

is based on redundancy analysis (`rda`

) and uses adjusted R^{2} to express explained variation. The reason for using only `rda`

is that in R, there is still no function available to calculate adjusted R^{2} for unimodal ordination methods (like `cca`

).

(library`varpart`

`vegan`

) - variation partitioning (using linear constrained ordinatino -`rda`

) among up to four matrices of environmental variables. First argument (`Y`

) is dependent (usually species composition) matrix (but could be also only one variable - in that case`varpart`

is conductin linear regression). Next arguments (up to four) are (groups of) explanatory variables. Uses either formula interface (with ~) or matrices.

Example: how much variation in species data (`vasc.hell`

) explains variables `Mg`

and `Ca`

(from `chem`

)?

# Carpathian wetlands - import data vasc <- read.delim ('http://www.davidzeleny.net/anadat-r/data-download/vasc_plants.txt', row.names = 1) chem <- read.delim ('http://www.davidzeleny.net/anadat-r/data-download/chemistry.txt', row.names = 1) # transform data using Hellinger transformation vasc.hell <- decostand (vasc, 'hell') # upload library vegan if not yet done library (vegan) # apply function varpart vp1 <- varpart (vasc.hell, ~ Mg, ~ Ca, data = chem)

In this function, the first is coming the species matrix, than explanatory matrices (or variables) - if using `formula`

interface, each has to start with tilda (~). If these variables are part of matrix of explanatory variables, you need to specify the environmental matrix in argument `data =`

.

Result is:

vp1

Partition of variation in RDA Call: varpart(Y = vasc.hell, X = ~Mg, ~Ca, data = chem) Explanatory tables: X1: ~Mg X2: ~Ca No. of explanatory tables: 2 Total variation (SS): 39.151 Variance: 0.56741 No. of observations: 70 Partition table: Df R.squared Adj.R.squared Testable [a+b] = X1 1 0.11987 0.10693 TRUE [b+c] = X2 1 0.13898 0.12632 TRUE [a+b+c] = X1+X2 2 0.16357 0.13860 TRUE Individual fractions [a] = X1|X2 1 0.01228 TRUE [b] 0 0.09465 FALSE [c] = X2|X1 1 0.03167 TRUE [d] = Residuals 0.86140 FALSE --- Use function 'rda' to test significance of fractions of interest

Alternative way how to use the function is not using `formula`

interface:

varpart (vasc.hell, chem$Mg, chem$Ca)

or, in case you use not-transformed data and you want them to be transformed (using `decostand`

function):

varpart (vasc, chem$Mg, chem$Ca, transfo = 'hell')

(argument `transfo`

is passed into function `decostand`

together with species data).

I can also plot directly so called Venn's diagram:

plot (vp1)

Note - the plotting function is called `plot.varpart`

, but if I use generic plot function on object vp1 (which is of class “varpart”), I don't have to specify the whole name.

As to the distinction between *variance* and *variation*, Legendre & Legendre (2012) note:
“The term variation, a less technical and looser term than variance, is used because one is partitioning the total sum of squared deviations of y from its mean (total SS). In variation partitioning, there is no need to divide the total SS of y by its degrees of freedom to obtain the variance s^{2}_{y}.”
The first edition of the book *Multivariate Analysis of Ecological Data using CANOCO* (Lepš & Šmilauer 2003) was using the term *variance partitioning*, while in the second edition (Šmilauer & Lepš 2014), authors adopted the term *variation partitioning*, noting:
“It was called **variance** partitioning in the original paper, but we prefer, together with Legendre & Legendre 2012, the more appropriate name referring to *variation*, as we include also unimodal ordination methods in our considerations.”

Note that in CANOCO 5, the coding of the fraction follows different logic - [c] is shared variation, and [a] or [b] are partial fractions; meaning of [d] remains the same (unexplained variation).

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