R: Moran's I Autocorrelation Index

Moran.I {ape} | R Documentation

## Moran's I Autocorrelation Index

### Description

This function computes Moran's I autocorrelation coefficient of `x` giving a matrix of weights using the method described by Gittleman and Kot (1990).

### Usage

      Moran.I(x, weight, scaled = FALSE, na.rm = FALSE,
              alternative = "two.sided")

### Arguments

`x` | a numeric vector.
`weight` | a matrix of weights.
`scaled` | a logical indicating whether the coefficient should be scaled so that it varies between -1 and +1 (default to `FALSE`).
`na.rm` | a logical indicating whether missing values should be removed.
`alternative` | a character string specifying the alternative hypothesis that is tested against the null hypothesis of no phylogenetic correlation; must be of one "two.sided", "less", or "greater", or any unambiguous abbrevation of these.

### Details

The matrix `weight` is used as “neighbourhood” weights, and Moran's I coefficient is computed using the formula:

\code{I = n/S0 * (sum{i=1..n} sum{j=1..n} wij(yi - ym))(yj - ym) / (sum{i=1..n} (yi - ym)^2)}

with

- yi = observations

- wij = distance weight

- n = number of observations

- S0 = \code{sum_{i=1..n} sum{j=1..n} wij}

The null hypothesis of no phylogenetic correlation is tested assuming normality of I under this null hypothesis. If the observed value of I is significantly greater than the expected value, then the values of `x` are positively autocorrelated, whereas if Iobserved < Iexpected, this will indicate negative autocorrelation.

### Value

A list containing the elements:

`observed` | the computed Moran's I.
`expected` | the expected value of I under the null hypothesis.
`sd` | the standard deviation of I under the null hypothesis.
`p.value` | the P-value of the test of the null hypothesis against the alternative hypothesis specified in `alternative`.

### Author(s)

Julien Dutheil julien.dutheil@univ-montp2.fr and Emmanuel Paradis

### References

Gittleman, J. L. and Kot, M. (1990) Adaptation: statistics and a null model for estimating phylogenetic effects. Systematic Zoology, 39, 227–241.

### See Also

`weight.taxo`

### Examples

    tr <- rtree(30)
    x <- rnorm(30)
    ## weights w[i,j] = 1/d[i,j]:
    w <- 1/cophenetic(tr)
    ## set the diagonal w[i,i] = 0 (instead of Inf...):
    diag(w) <- 0
    Moran.I(x, w)
    Moran.I(x, w, alt = "l")
    Moran.I(x, w, alt = "g")
    Moran.I(x, w, scaled = TRUE) # usualy the same

[Package ape version 4.1 Index]