ape (version 5.8)

Moran.I: 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")

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.

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.

Author

Julien Dutheil and Emmanuel Paradis

Details

The matrix weight is used as ``neighbourhood'' weights, and Moran's I coefficient is computed using the formula: $$I = \frac{n}{S_0} \frac{\sum_{i=1}^n\sum_{j=1}^n w_{i,j}(y_i - \overline{y})(y_j - \overline{y})}{\sum_{i=1}^n {(y_i - \overline{y})}^2}$$ with

• \(y_i\) = observations

• \(w_{i,j}\) = distance weight

• \(n\) = number of observations

• \(S_0\) = \(\sum_{i=1}^n\sum_{j=1}^n wij\)

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.

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