Geary’s C

Introduction

Geary’s C is a measure of clustering (or spatial autocorrelation) of a variable, and is computed by (Cliff and Ord, 1981):

\(C = (N – 1){{\sum\nolimits_{i=1}^N \sum\nolimits_{j=1}^N w_{ij}(X_{i}-X_{j})^2}\over{2W\sum\nolimits_{i=1}^n(X_{i}-\overline{X})^2}}\)

Where \(X_{i}\) and \(X_{j}\) are the values for regions i and j respectively, \(\overline{X}\) is the mean of the variable, \(W\) is the global sum of the weights, i.e. The spatial weight matrix that is employed by this component must be symmetric. Hence if the selected spatial weight matrix is not already symmetric, part of the process will include converting the spatial weight matrix so it becomes symmetric.

Given the null hypothesis is one of no global spatial autocorrelation, the expected value of Geary’s C equals 1. If C is larger than 1, the distribution of the variable being tested is characterised by negative spatial autocorrelation. If C is smaller than 1, then the distribution of the variable displays positive spatial autocorrelation. As with Moran’s I, inference is based on z-values:

\(Z_{c} = {{C – 1}\over{sd(C)}}\)

While Moran’s I is based on the cross-products of the deviations from the mean, Geary’s C is based on the deviations in responses of each observation with one another. This in effect means that Moran’s I is more sensitive to extreme values and is more a global measure, whereas Geary’s C is more sensitive to differences between values in neighbouring areas.

Inputs

To compute Geary’s C, we will look at socio-economic data in Melbourne to examine the extent of spatial-autocorrelation.

To do this:

  • Select Melbourne GCCSA as your area
  • Select SA2 SEIFA 2011 – The Index of Relative Socio-Economic Advantage and Disadvantage (IRSAD)  as your dataset, selecting all variables
  • Spatialise the dataset, naming it something like SPATIALISED SEIFA IRSAD Melbourne
  • Generate a Contiguous Spatial Weights Matrix for the spatialised dataset, using 1st order Queen contiguity. Name it something like Contig SWM Melbourne SA2s

Once you have done this, open the Geary’s C tool (Tools → Spatial Statistics → Geary’s C) and enter the parameters as they appear in the image below. These are also explained underneath the image

[Click to Enlarge]

[Click to Enlarge]

  • Dataset Input: the dataset that contains the variable(s) to be tested. Here we use the dataset named SPATIALISED SEIFA IRSAD Melbourne
  • Spatial Weights Matrix: the spatial weight matrix to be used (described here). In this instance we use the one name Contig SWM Melbourne SA2s 
  • Variable: the variable(s) to be tested. Here we use Score
  • Alternative Hypothesis: indicates the alternative hypothesis; can be two.sided, greater. Here we use two.side
  • Inference: indicates the inference under which the variance of is calculated. Checked is randomisation, unchecked is normality. Here we select randomisation (the default).

Once you have entered the parameters, click Add and Run to execute the tool

Outputs

Once your tool has run, click the Display button on the pop dialogue box. This will open a text editor with the outputs of the tool, which should look something like the image below

[Click to Enlarge]

[Click to Enlarge]

These outputs are:

  • Geary’s C with the corresponding z-score and p-value
  • Geary’s C with its expectation and variance
  • The alternative hypothesis
  • The assumption under which the variance was calculated