# gowdis: Gower Dissimilarity In FD: Measuring functional diversity (FD) from multiple traits, and other tools for functional ecology

## Description

`gowdis` measures the Gower (1971) dissimilarity for mixed variables, including asymmetric binary variables. Variable weights can be specified. `gowdis` implements Podani's (1999) extension to ordinal variables.

## Usage

 `1` ```gowdis(x, w, asym.bin = NULL, ord = c("podani", "metric", "classic")) ```

## Arguments

 `x` matrix or data frame containing the variables. Variables can be `numeric`, `ordered`, or `factor`. Symmetric or asymmetric binary variables should be `numeric` and only contain 0 and 1. `character` variables will be converted to `factor`. `NA`s are tolerated. `w` vector listing the weights for the variables in `x`. Can be missing, in which case all variables have equal weights. `asym.bin` vector listing the asymmetric binary variables in `x`. `ord` character string specifying the method to be used for ordinal variables (i.e. `ordered`). `"podani"` refers to Eqs. 2a-b of Podani (1999), while `"metric"` refers to his Eq. 3 (see ‘details’); both options convert ordinal variables to ranks. `"classic"` simply treats ordinal variables as continuous variables. Can be abbreviated.

## Details

`gowdis` computes the Gower (1971) similarity coefficient exactly as described by Podani (1999), then converts it to a dissimilarity coefficient by using D = 1 - S. It integrates variable weights as described by Legendre and Legendre (1998).

Let X = {Xij} be a matrix containing n objects (rows) and m columns (variables). The similarity Gjk between objects j and k is computed as

Gjk = sum(Wijk * Sijk) / sum(Wijk)

,

where Wijk is the weight of variable i for the j-k pair, and Sijk is the partial similarity of variable i for the j-k pair,

and where Wijk = 0 if objects j and k cannot be compared because Xij or Xik is unknown (i.e. `NA`).

For binary variables, Sijk = 0 if Xij is not equal to Xik, and Sijk = 1 if Xij = Xik = 1 or if Xij = Xik = 0.

For asymmetric binary variables, same as above except that Wijk = 0 if Xij = Xik = 0.

For nominal variables, Sijk = 0 if Xij is not equal to Xik and Sijk = 1 if Xij = Xik.

For continuous variables,

Sijk = 1 - [ |Xij - Xik| / (Xi.max - Xi.min) ]

where Xi.max and Xi.min are the maximum and minimum values of variable i, respectively.

For ordinal variables, when `ord = "podani"` or `ord = "metric"`, all Xij are replaced by their ranks Rij determined over all objects (such that ties are also considered), and then

if `ord = "podani"`

Sijk = 1 if Rij = Rik, otherwise

Sijk = 1 - [ |Rij - Rik| - (Tij - 1) / 2 - (Tik - 1) / 2 / Ri.max - Ri.min - (Ti.max - 1) / 2 - (Ti.min - 1) / 2 ]

where Tij is the number of objects which have the same rank score for variable i as object j (including j itself), Tik is the number of objects which have the same rank score for variable i as object k (including k itself), Ri.max and Ri.min are the maximum and minimum ranks for variable i, respectively, Ti.max is the number of objects with the maximum rank, and Ti.min is the number of objects with the minimum rank.

if `ord = "metric"`

Sijk = 1 - [ |Rij - Rik| / (Ri.max - Ri.min) ]

When `ord = "classic"`, ordinal variables are simply treated as continuous variables.

## Value

an object of class `dist` with the following attributes: `Labels`, `Types` (the variable types, where 'C' is continuous/numeric, 'O' is ordinal, 'B' is symmetric binary, 'A' is asymmetric binary, and 'N' is nominal), `Size`, `Metric`.

## Author(s)

Etienne Laliberté etiennelaliberte@gmail.com http://www.elaliberte.info, with some help from Philippe Casgrain for the C interface.

## References

Gower, J. C. (1971) A general coefficient of similarity and some of its properties. Biometrics 27:857-871.

Legendre, P. and L. Legendre (1998) Numerical Ecology. 2nd English edition. Amsterdam: Elsevier.

Podani, J. (1999) Extending Gower's general coefficient of similarity to ordinal characters. Taxon 48:331-340.

`daisy` is similar but less flexible, since it does not include variable weights and does not treat ordinal variables as described by Podani (1999). Using `ord = "classic"` reproduces the behaviour of `daisy`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```ex1 <- gowdis(dummy\$trait) ex1 # check attributes attributes(ex1) # to include weights w <- c(4,3,5,1,2,8,3,6) ex2 <- gowdis(dummy\$trait, w) ex2 # variable 7 as asymmetric binary ex3 <- gowdis(dummy\$trait, asym.bin = 7) ex3 # example with trait data from New Zealand vascular plant species ex4 <- gowdis(tussock\$trait) ```

### Example output

```Loading required package: ade4
This is vegan 2.4-4
sp1       sp2       sp3       sp4       sp5       sp6       sp7
sp2 0.2181884
sp3 0.5240052 0.6678082
sp4 0.6737443 0.5610028 0.8225701
sp5 0.5291113 0.8145699 0.4862253 0.4843264
sp6 0.6100161 0.5932587 0.2784736 0.7073925 0.6067323
sp7 0.4484235 0.6863374 0.4848663 0.5575126 0.3023416 0.6187844
sp8 0.4072834 0.2039443 0.5958904 0.2390962 0.5585525 0.4470207 0.7030186
\$Labels
[1] "sp1" "sp2" "sp3" "sp4" "sp5" "sp6" "sp7" "sp8"

\$Size
[1] 8

\$Metric
[1] "Gower"

\$Types
[1] "C" "C" "N" "N" "O" "O" "B" "B"

\$class
[1] "dist"

sp1       sp2       sp3       sp4       sp5       sp6       sp7
sp2 0.1190154
sp3 0.4156230 0.5584826
sp4 0.7157249 0.7541962 0.7478800
sp5 0.6538987 0.8231658 0.3994155 0.3880160
sp6 0.5074762 0.4422926 0.2767123 0.7753720 0.6317876
sp7 0.5006495 0.6116622 0.4934116 0.4642192 0.3773199 0.5997205
sp8 0.2813567 0.2730468 0.3359142 0.3658275 0.4977458 0.3645410 0.6129055
sp1       sp2       sp3       sp4       sp5       sp6       sp7
sp2 0.2545531
sp3 0.5240052 0.6678082
sp4 0.7699935 0.6545032 0.8225701
sp5 0.5291113 0.8145699 0.4862253 0.4843264
sp6 0.6100161 0.5932587 0.2784736 0.7073925 0.6067323
sp7 0.4484235 0.6863374 0.4848663 0.5575126 0.3023416 0.6187844
sp8 0.4751640 0.2447331 0.5958904 0.2789456 0.5585525 0.4470207 0.7030186
```

FD documentation built on May 1, 2019, 9:27 p.m.