gowdis {FD}

Gower Dissimilarity

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

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. NAs 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 \mathbf{X} = \{x_{ij}\} be a matrix containing n objects (rows) and m columns (variables). The similarity G_{jk} between objects j and k is computed as

G_{jk} = \frac{\sum_{i=1}^{n} w_{ijk} s_{ijk}}{\sum_{i=1}^{n} w_{ijk}}

,

where w_{ijk} is the weight of variable i for the j-k pair, and s_{ijk} is the partial similarity of variable i for the j-k pair,

and where w_{ijk} = 0 if objects j and k cannot be compared because x_{ij} or x_{ik} is unknown (i.e. NA).

For binary variables, s_{ijk} = 0 if x_{ij} \neq x_{ik}, and s_{ijk} = 1 if x_{ij} = x_{ik} = 1 or if x_{ij} = x_{ik} = 0.

For asymmetric binary variables, same as above except that w_{ijk} = 0 if x_{ij} = x_{ik} = 0.

For nominal variables, s_{ijk} = 0 if x_{ij} \neq x_{ik} and s_{ijk} = 1 if x_{ij} = x_{ik}.

For continuous variables,

s_{ijk} = 1 - \frac{|x_{ij} - x_{ik}|} {x_{i.max} - x_{i.min}}

where x_{i.max} and x_{i.min} are the maximum and minimum values of variable i, respectively.

For ordinal variables, when ord = "podani" or ord = "metric", all x_{ij} are replaced by their ranks r_{ij} determined over all objects (such that ties are also considered), and then

if ord = "podani"

s_{ijk} = 1 if r_{ij} = r_{ik}, otherwise

s_{ijk} = 1 - \frac{|r_{ij} - r_{ik}| - (T_{ij} - 1)/2 - (T_{ik} - 1)/2 }{r_{i.max} - r_{i.min} - (T_{i.max} - 1)/2 - (T_{i.min}-1)/2 }

where T_{ij} is the number of objects which have the same rank score for variable i as object j (including j itself), T_{ik} is the number of objects which have the same rank score for variable i as object k (including k itself), r_{i.max} and r_{i.min} are the maximum and minimum ranks for variable i, respectively, T_{i,max} is the number of objects with the maximum rank, and T_{i.min} is the number of objects with the minimum rank.

if ord = "metric"

s_{ijk} = 1 - \frac{|r_{ij} - r_{ik}|}{r_{i.max} - r_{i.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 [e-mail] https://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.

See Also

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

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)