Wartenberg's coefficient
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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 measured at spatial sites Moran's I is a measure of the spatial autocorrelation of the data. By standardizing the observations by subtracting the mean and dividing by the variance as well as normalising the spatial weight matrix such that we can write Moran's I as
Wartenberg generalized this by letting be a vector of observations at and defining where:
- is the spatial weight matrix
- is the standardized data matrix
- is the transpose of
- is the spatial correlation matrix.
For two variables and the bivariate correlation is
For this reduces to Moran's . For larger values of the diagonals of are the Moran indices for each of the variables and the off-diagonals give the corresponding Wartenberg correlation coefficients. is an example of a Mantel statistic and so its significance can be evaluated using the Mantel test.[4]
Criticisms
[edit]Lee[5] points out some problems with this coefficient namely:
- There is only one factor of in the numerator, so the comparison is between the raw data and the spatially averaged data.
- for non-symmetric spatial weight matrices.
He suggests an alternative coefficient which has two factors of in the numerator and is symmetric for any weight matrix.
See also
[edit]References
[edit]- ^ Burger, J; Gochfeld, M (2020). "In Memoriam: Daniel Wartenberg (1952–2020)". Environ Health Perspect. 128 (11): 111601. doi:10.1289/EHP8405. PMC 7641299. PMID 33147071.
- ^ Wartenberg, D (1985). "Multivariate spatial correlation: a method for exploratory geographical analysis". Geographical Analysis. 17 (4): 263–283. Bibcode:1985GeoAn..17..263W. doi:10.1111/j.1538-4632.1985.tb00849.x.
- ^ "Adespatial: Multivariate Multiscale Spatial Analysis". 18 October 2023.
- ^ Dale, Mark R. T.; Fortin, Marie-Josée (2014). Spatial Analysis: A Guide For Ecologists. Cambridge University Press. p. 428. ISBN 978-0-521-14350-9.
- ^ Lee, Sang-Il (2001). "Developing a bivariate spatial association measure: an integration of Pearson's r and Moran's I.". Journal of Geographical Systems. 3 (4): 369–385. Bibcode:2001JGS.....3..369L. doi:10.1007/s101090100064.