Wartenberg's coefficient

Wartenberg's coefficient is a measure of correlation developed by epidemiologist Daniel Wartenberg.^[1] This coefficient is a multivariate extension of spatial autocorrelation that aims to account for spatial dependence of data while studying their covariance.^[2] A modified version of this statistic is available in the R package adespatial.^[3]

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

${\displaystyle I=\sum _{ij}w_{ij}z_{i}z_{j}}$

Wartenberg generalized this by letting ${\displaystyle z_{i}}$ be a vector of ${\displaystyle M}$ observations at ${\displaystyle i}$ and defining where:

${\displaystyle I=Z^{T}WZ}$

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

For two variables ${\displaystyle x}$ and ${\displaystyle y}$ the bivariate correlation is

${\displaystyle I_{xy}={\frac {\sum _{ij}w_{ij}(x_{i}-{\bar {x}})(y_{j}-{\bar {y}})}{{\sqrt {\sum _{i}(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i}(y_{i}-{\bar {y}})^{2}}}}}}$

For ${\displaystyle M=1}$ this reduces to Moran's ${\displaystyle I}$. For larger values of ${\displaystyle M}$ the diagonals of ${\displaystyle I}$ are the Moran indices for each of the variables and the off-diagonals give the corresponding Wartenberg correlation coefficients. ${\displaystyle I}$ is an example of a Mantel statistic and so its significance can be evaluated using the Mantel test.^[4]

Criticisms

Lee^[5] points out some problems with this coefficient namely:

• There is only one factor of ${\displaystyle W}$ in the numerator, so the comparison is between the raw ${\displaystyle x}$ data and the spatially averaged ${\displaystyle y}$ data.
• ${\displaystyle I_{xy}\neq I_{yx}}$ for non-symmetric spatial weight matrices.

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

See also

• Tjøstheim's Coefficient