Wartenberg's coefficient

Wartenberg's coefficient is a measure of correlation developed by epidemiologist Daniel Wartenberg. This coefficient is a multivariate extension of spatial autocorrelation that aims to account for spatial dependence of data while studying their covariance. A modified version of this statistic is available in the R package adespatial.

For data $$x_i$$ measured at $$N$$ spatial sites Moran's I is a measure of the spatial autocorrelation of the data. By standardizing the observations $$z_i = (x_i - \bar{x})/s$$ by subtracting the mean and dividing by the variance as well as normalising the spatial weight matrix such that $$\sum_{ij} w_{ij} = 1$$ we can write Moran's I as


 * $$I = \sum_{ij} w_{ij} z_i z_j$$

Wartenberg generalized this by letting $$z_i$$ be a vector of $$M$$ observations at $$i$$ and defining where:
 * $$I = Z^T W Z$$


 * $$W$$ is the N \times N spatial weight matrix
 * $$Z$$ is the N \times M standardized data matrix
 * $$Z^T$$ is the transpose of Z
 * $$I$$ is the M \times M spatial correlation matrix.

For two variables $$x$$ and $$y$$ the bivariate correlation is
 * $$I_{xy} = \frac{ \sum_{ij} w_{ij} (x_i - \bar{x}) (y_j - \bar{y})}{ \sqrt{ \sum_i (x_i -\bar{x})^2} \sqrt{ \sum_i (y_i -\bar{y})^2} }$$

For $$M=1$$ this reduces to Moran's $$I$$. For larger values of $$M$$ the diagonals of $$I$$ are the Moran indices for each of the variables and the off-diagonals give the corresponding Wartenberg correlation coefficients. $$I$$ is an example of a Mantel statistic and so its significance can be evaluated using the Mantel test.

Criticisms
Lee points out some problems with this coefficient namely: He suggests an alternative coefficient which has two factors of $$W$$ in the numerator and is symmetric for any weight matrix.
 * There is only one factor of $$W$$ in the numerator, so the comparison is between the raw $$x$$ data and the spatially averaged $$y$$ data.
 * $$I_{xy} \neq I_{yx}$$ for non-symmetric spatial weight matrices.