Moran.I: Moran's I Autocorrelation Index

Description

Computes Moran's I autocorrelation index of a given vector x according to a distance weights matrix dist using the method described by Gittleman and Kot (1990).

Usage

Moran.I(x, dist, scaled = FALSE)

Arguments

a numeric vector.

dist

a distance weigths matrix.

scaled

logical, indicate whether the index should be scaled to allow comparisons between indices (default to FALSE).

Value

A list containing the attributes:
observedMoran's I index of x.
expectedExpected value of I under the null hypothesis.
sdThe standard deviation of I under the null hypothesis.
p.valueThe p-value of having the observed value under the null hypothesis.

Details

The dist matrix is used as distance weights, and Moran's I indice 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 value of I may be evaluated under two null hypotheses, normality and randomization. Only the randomization hypothesis is implemented for now.

References

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

Examples

Run this code

### This will draw the correlogram from Gittleman and Kot 1990
### This is only for example, consider function plot.correlogram() to draw correlograms.
library(ape)
data(carnivora)
attach(carnivora)
# Compute distance matrix for each taxonomic level:
dG  <- dist.taxo(Genus)
dF  <- dist.taxo(Family)
dSF <- dist.taxo(SuperFamily) 
dO  <- dist.taxo(Order)
# We draw the correlogram of average body weights, 
# with log10-transformed variable:
IG  <- Moran.I(log10(SW), dG        , scale=TRUE)
IF  <- Moran.I(log10(SW), dF  & !dG , scale=TRUE)
ISF <- Moran.I(log10(SW), dSF & !dF , scale=TRUE)
IO  <- Moran.I(log10(SW), dO  & !dSF, scale=TRUE)
# All Moran's I indices:
i <- c(IG$obs, IF$obs, ISF$obs, IO$obs)
# With their corresponding p-values:
p <- c(IG$p.v, IF$p.v, ISF$p.v, IO$p.v)
# Here's the legend:
l <- c("G", "F", "SF", "O")
# Draw the correlogram (using lattice library):
library(lattice)
# New Black & White device:
trellis.device(device=X11, new=FALSE, color=FALSE)
# Black circles are significant at the 5% level:
pch <- ifelse(p < 0.05, 19, 21)
# Plot it!
xyplot(i~ordered(l,levels=l), type="b", xlab="Rank", ylab="I / Imax",
    lty=2, lwd=2, cex=1.5, pch=pch, ylim=c(-0.75,0.75))

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