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:

$I=Z^{T}WZ$

• $W$ 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:

• 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.

He suggests an alternative coefficient which has two factors of $W$ in the numerator and is symmetric for any weight matrix.

See also

• Tjøstheim's Coefficient