# Moran.I

##### Moran's I Autocorrelation Index

This function computes Moran's I autocorrelation coefficient of
`x`

giving a matrix of weights using the method described by
Gittleman and Kot (1990).

- Keywords
- models, regression

##### 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:
$$I = \frac{n}{S_0} \frac{\sum_{i=1}^n\sum_{j=1}^n w_{i,j}(y_i -
\overline{y})(y_j - \overline{y})}{\sum_{i=1}^n {(y_i -
\overline{y})}^2}$$
{```
I = n/S0 * (sum{i=1..n} sum{j=1..n} wij(yi - ym))(yj - ym)
/ (sum{i=1..n} (yi - ym)^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.

##### 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`

.

##### 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

##### 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
```

*Documentation reproduced from package ape, version 2.1-1, License: GPL (>= 2)*