Analysis→Point Data→Dispersion
This procedure takes point counts from quadrats and calculates a number of indices that can be used to identify the spatial patterning of the points. Although many of these indices were originally designed to be used with scattered (thrown) quadrats, they could be calculated over contiguous quadrats as well.
| Menu: | Analysis→Point Data→Dispersion |
| Button: | |
| Batch: | DispersionIndices |
The input data consists of a column from a data matrix in which each
cell represents the count from a single quadrat. These analyses are based
on the mean (
) and variance
(s2)
of the quadrat counts and the number of quadrats (n).
Dispersion Indices window.
PASSaGE calculates the mean and variance of the counts, as well as seven closely related indices: Index of Dispersion (ID), Index of Cluster Size (ICS), Green’s Index (GI), Index of Cluster Frequency (ICF), Index of Mean Crowding (IMC), Index of Patchiness (IP), and Morisita’s Index (IM).
These indices primarily examine the deviation from a random (Poisson) distribution. Some of these indices are dependent on n and quadrat size; all should be used with caution. See Hurlbert (1990) for interesting criticisms and Pielou (1969), Upton and Fingleton (1985), and Ludwig and Reynolds (1988) for general information.
The Index of Dispersion is also known as the Variance-to-Mean Ratio. Under a random distribution of points, ID is expected to equal 1. ID × (n – 1) follows a χ2 test with n – 1 degrees of freedom.
PASSaGE reports ID and the results of the χ2 test.
This index is also known as the Index of Clumping (IC) (David and Moore 1954) and is a direct function of the Index of Dispersion. Under a random distribution of points, ICS is expected to equal 0. Positive values indicate a clumped distribution; negative values a regular distribution.
GI (Green 1966) is a modification of the Index of Cluster Size that is independent of n. It varies between 0 for random distributions and 1 for maximally clumped distributions.
The Index of Cluster Frequency (Douglas 1975) is a measure of aggregation and is equal to k of the negative binomial distribution. ICF is proportional to the quadrat area and is related to the Index of Cluster Size.
The Index of Mean Crowding (Lloyd 1967) is the average number of other points contained in the quadrat that contains a randomly chosen point. It is related to the Index of Cluster Size.
The Index of Patchiness (Lloyd 1967) is related to the Index of Cluster Frequency and the Index of Mean Crowding, and is similar to Morisita’s Index. It is a measure of pattern intensity that is unaffected by thinning (the random removal of points).
Morisita’s Index (Morisita 1959) is related to the Index of Patchiness. It is the scaled probability that two points chosen at random from the whole population are in the same quadrat. The higher the value, the more clumped the distribution.
As an example, we took the 355 cancer registration locations and counted the number of points found in each of thirty 5° × 5° cells (using PASSaGE’s gridding function). These counts were then used as the basis for dispersion indice calculations. Not surprisingly, the points come out as extremely non-random since many of the grid cells used to generate the counts sit on the ocean or countries not included in the study.
Dispersion Indices
Data matrix: Grid Data 1
Column of counts: Count
# of counts = 30
Mean count = 11.83333
Variance = 204.14368
Index of Dispersion = 17.25158 (p = 0.00000)
Index of Cluster Size = 16.25158
Green's Index = 0.56040
Index of Cluster Frequency = 0.72813
Index of Mean Crowding = 28.08491
Index of Patchiness = 2.37337
Morisita's Index = 2.33134
Example of dispersion index output for the cancer registration district point locations, gridded into 5° × 5° cells.