194
Elements of Spatial Data Analysis in Ecological Assessments
ables, whereas negative values indicate a negative
interaction (e.g., spatial segregation).
Surface Pattern Methods
Correlogram-based Methods
A correlogram is a graph in which spatial autocorrelation values are plotted against a measure of distance among sites or samples. Correlograms (Cliff
and Ord, 1981) can be produced for qualitative data
(Join-Count) or quantitative data (Moran's I,
Geary's c, Mantel test, among others) and for single variables (Join-Count, Moran's I, Geary's c) or
multivariate data (Mantel test). In all cases, a test
of significance is available for each individual
value plotted (Legendre and Legendre, 1998).
When the data are qualitative (categorical), the
Join-Count statistic can be calculated (Upton and
Fingleton, 1985). The Join-Count statistic tests
whether neighboring samples are more likely to be
in the same categorical state or in different states
than would be expected for a random process. Spatial correlograms based on the Join-Count statistic
are graphed using neighbor networks (Upton and
Fingleton, 1985), rather than the Euclidean distances that are calculated for the surface pattern
methods discussed later (Moran'S I, Geary's c,
Mantel test, and semivariogram-based methods).
The significance of the J oin-Count statistic is obtained by computing a standard normal deviate, using a two-sided test to detect both positive and negative spatial autocorrelations. Originally developed
for binary data, the method has been extended to
multicategorical data (Cliff and Ord, 1981; Upton
and Fingleton, 1985).
For quantitative variables, spatial autocorrelation
is measured using either Moran's I or Geary's c
(Cliff and Ord, 1981). Moran's I is the degree of
correlation between the values of a variable as a
function of spatial locations. It is related to the
Pearson's correlation coefficient, in that it represents the deviations between the values of a variable and its mean. Like Pearson's correlation coefficient, Moran's I varies between -1 (negative
autocorrelation) and + 1 (positive autocorrelation),
with a value of 0 in the absence of spatial autocorrelation. In contrast, Geary's c is a distance-type
function that measures the difference (distance)
among values of the variable at nearby locations.
It varies from 0 (positive autocorrelation) to some
unspecified larger value close to 2 (strong negative
autocorrelation). Both coefficients can be graphed
against distance to produce a correlogram. To interpret the correlogram, coefficients can be tested
for significance using the Bonferroni method
(Oden, 1984), but the test requires limiting the correlogram to a small number of distance classes. To
alleviate this problem, the spline correlogram technique (Bj!3rnstad and Falck, 1997) has been suggested.
The Mantel test (Legendre and Fortin, 1989)
quantifies the degree of relationship between two
distance matrices based on data sets (e.g., environmental variables and species abundance data) obtained at the same sampling locations. The normalized Mantel statistic, which is the sum of all
products between corresponding elements of the
distance matrices, behaves like a product-moment
correlation coefficient, r, varying from -1 to + 1.
When one of the distance matrices contains the Euclidean distances between the sample locations, the
resulting Mantel test is a measure of the overall
spatial pattern of the data. Significance is assessed
by using a randomization test under the null hypothesis of no relationship between the two distance matrices. The observed Mantel test is expected to have a value near the mode of a reference
distribution obtained by randomization of the data.
Semivariogram-based Methods
Methods based on the analysis of the semivariogram, developed by geological engineers and soil
scientists, quantify spatial pattern and allow modeling of this pattern by interpolation (Cressie, 1991;
Rossi et aI., 1992). Interpolation uses only the data
that are spatially dependent to calculate estimated
values of a variable, in contrast to regression-based
methods (e.g., trend surface analysis), which use
coefficients calibrated over the entire study region
(see the discussion in Burrough, 1995). The semivariance is defined as one-half the variance in the
differences between the values of a variable at two
locations. The plot of semi variance against sampling interval or distance is called a semivariogram.
Semivariance is relatively easy to calculate and is
robust to deviation in non stationarity of the variance; therefore, semivariograms have been favored
over correlograms as a basis for predictive modeling of regionalized variables (Burrough, 1995). For
a second-order stationary process, the semivariance
levels out at some distance in the semivariogram.
This level is called the sill. The distance at which
the sill appears is the distance within which sample points are spatially dependent. Beyond that distance, the samples can be treated as independent.
The semivariogram does not always pass through
the (0, 0) point. This indicates that there is residual random variation, called the nugget variance,
that is not spatially correlated. The semivariogram
Elements of Spatial Data Analysis in Ecological Assessments
ables, whereas negative values indicate a negative
interaction (e.g., spatial segregation).
Surface Pattern Methods
Correlogram-based Methods
A correlogram is a graph in which spatial autocorrelation values are plotted against a measure of distance among sites or samples. Correlograms (Cliff
and Ord, 1981) can be produced for qualitative data
(Join-Count) or quantitative data (Moran's I,
Geary's c, Mantel test, among others) and for single variables (Join-Count, Moran's I, Geary's c) or
multivariate data (Mantel test). In all cases, a test
of significance is available for each individual
value plotted (Legendre and Legendre, 1998).
When the data are qualitative (categorical), the
Join-Count statistic can be calculated (Upton and
Fingleton, 1985). The Join-Count statistic tests
whether neighboring samples are more likely to be
in the same categorical state or in different states
than would be expected for a random process. Spatial correlograms based on the Join-Count statistic
are graphed using neighbor networks (Upton and
Fingleton, 1985), rather than the Euclidean distances that are calculated for the surface pattern
methods discussed later (Moran'S I, Geary's c,
Mantel test, and semivariogram-based methods).
The significance of the J oin-Count statistic is obtained by computing a standard normal deviate, using a two-sided test to detect both positive and negative spatial autocorrelations. Originally developed
for binary data, the method has been extended to
multicategorical data (Cliff and Ord, 1981; Upton
and Fingleton, 1985).
For quantitative variables, spatial autocorrelation
is measured using either Moran's I or Geary's c
(Cliff and Ord, 1981). Moran's I is the degree of
correlation between the values of a variable as a
function of spatial locations. It is related to the
Pearson's correlation coefficient, in that it represents the deviations between the values of a variable and its mean. Like Pearson's correlation coefficient, Moran's I varies between -1 (negative
autocorrelation) and + 1 (positive autocorrelation),
with a value of 0 in the absence of spatial autocorrelation. In contrast, Geary's c is a distance-type
function that measures the difference (distance)
among values of the variable at nearby locations.
It varies from 0 (positive autocorrelation) to some
unspecified larger value close to 2 (strong negative
autocorrelation). Both coefficients can be graphed
against distance to produce a correlogram. To interpret the correlogram, coefficients can be tested
for significance using the Bonferroni method
(Oden, 1984), but the test requires limiting the correlogram to a small number of distance classes. To
alleviate this problem, the spline correlogram technique (Bj!3rnstad and Falck, 1997) has been suggested.
The Mantel test (Legendre and Fortin, 1989)
quantifies the degree of relationship between two
distance matrices based on data sets (e.g., environmental variables and species abundance data) obtained at the same sampling locations. The normalized Mantel statistic, which is the sum of all
products between corresponding elements of the
distance matrices, behaves like a product-moment
correlation coefficient, r, varying from -1 to + 1.
When one of the distance matrices contains the Euclidean distances between the sample locations, the
resulting Mantel test is a measure of the overall
spatial pattern of the data. Significance is assessed
by using a randomization test under the null hypothesis of no relationship between the two distance matrices. The observed Mantel test is expected to have a value near the mode of a reference
distribution obtained by randomization of the data.
Semivariogram-based Methods
Methods based on the analysis of the semivariogram, developed by geological engineers and soil
scientists, quantify spatial pattern and allow modeling of this pattern by interpolation (Cressie, 1991;
Rossi et aI., 1992). Interpolation uses only the data
that are spatially dependent to calculate estimated
values of a variable, in contrast to regression-based
methods (e.g., trend surface analysis), which use
coefficients calibrated over the entire study region
(see the discussion in Burrough, 1995). The semivariance is defined as one-half the variance in the
differences between the values of a variable at two
locations. The plot of semi variance against sampling interval or distance is called a semivariogram.
Semivariance is relatively easy to calculate and is
robust to deviation in non stationarity of the variance; therefore, semivariograms have been favored
over correlograms as a basis for predictive modeling of regionalized variables (Burrough, 1995). For
a second-order stationary process, the semivariance
levels out at some distance in the semivariogram.
This level is called the sill. The distance at which
the sill appears is the distance within which sample points are spatially dependent. Beyond that distance, the samples can be treated as independent.
The semivariogram does not always pass through
the (0, 0) point. This indicates that there is residual random variation, called the nugget variance,
that is not spatially correlated. The semivariogram