7.2.6 Testing for the Presence of Spatial Correlation:
Conditions
As shown above, spatial correlation coefficients can be tested for significance.
However, conditions of application must be respected. The condition of normality
can be relaxed if the test is carried out by permutations. To test the significance of
coefficients of spatial correlation, however, the condition of second-order
stationarity must be met. That condition states that the mean of the variable and
its spatial covariance (numerator of Eq. 7.3) are the same over the study area, and
that its variance (denominator of Eq. 7.3) is finite. This condition tells us, in other
words, that the spatial variation of the data should be adequately described by the
same single spatial correlation function in all portions of the study area. Spatial
correlation coefficients cannot be tested for significance if an overall trend is present
in the data, or if the variable has been measured in a region where several distinct
structures should be modelled by different spatial correlation functions. Data
displaying simple trends can often be detrended by means of a first-degree function
of the site geographical coordinates (Sect. 7.3), as we did above before computing
the Mantel correlogram of the mite data.
Another, relaxed form of stationarity is called the intrinsic assumption, a short
form for “hypothesis of intrinsic stationarity of order 2” (Wackernagel 2003). This
condition considers only the increments of the values of the variable; it states that the
differences (y h À y i ) for any distance d in the numerator of Eq. 7.5 have zero mean
and constant and finite variance over the study area, independently of the location
(Legendre and Legendre 2012). This condition allows one to compute and examine
correlograms but without tests of significance.
Legendre and Legendre (2012, p. 800) show how to interpret all-directional
correlograms (i.e., correlograms built on distance classes defined in the same way
in all directions) as well as directional correlograms.
A word is needed here about multiple testing. In Sect. 7.2.5 several spatial
correlation values were tested simultaneously for significance. In such cases the
probability of type I error increases with the number of tests. If k tests are carried out,
the binomial law tells us that the overall probability of type I error (technically called
the “experimentwise error rate”) is equal to 1 À (1 À α)
k where α is the nominal
value for a single test. For instance, in the Mantel correlogram shown in Fig. 7.1,
seven tests are carried out simultaneously. Without correction, the overall probability of obtaining at least one type I error is equal to 1 À (1–0.05)
7
¼ 0.302 instead of
the nominal α ¼ 0.05. Several methods have been proposed to achieve a correct level
of type I error in multiple tests (reviewed in Legendre and Legendre 2012; Wright
1992). The most conservative solution for k independent tests is to divide the
significance level by the number of simultaneous tests: α’ ¼ α / k and compare the
p-values to α’. Conversely one can multiply the p-values by k (i.e., p’ ¼ kp) and
compare the resulting values to the unadjusted α. For non-independent tests, Holm‘s
procedure (Holm 1979) is more powerful. The reason is that Holm’s correction
consists in applying Bonferroni’s correction sequentially as follows. First, order the
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7 Spatial Analysis of Ecological Data
Conditions
As shown above, spatial correlation coefficients can be tested for significance.
However, conditions of application must be respected. The condition of normality
can be relaxed if the test is carried out by permutations. To test the significance of
coefficients of spatial correlation, however, the condition of second-order
stationarity must be met. That condition states that the mean of the variable and
its spatial covariance (numerator of Eq. 7.3) are the same over the study area, and
that its variance (denominator of Eq. 7.3) is finite. This condition tells us, in other
words, that the spatial variation of the data should be adequately described by the
same single spatial correlation function in all portions of the study area. Spatial
correlation coefficients cannot be tested for significance if an overall trend is present
in the data, or if the variable has been measured in a region where several distinct
structures should be modelled by different spatial correlation functions. Data
displaying simple trends can often be detrended by means of a first-degree function
of the site geographical coordinates (Sect. 7.3), as we did above before computing
the Mantel correlogram of the mite data.
Another, relaxed form of stationarity is called the intrinsic assumption, a short
form for “hypothesis of intrinsic stationarity of order 2” (Wackernagel 2003). This
condition considers only the increments of the values of the variable; it states that the
differences (y h À y i ) for any distance d in the numerator of Eq. 7.5 have zero mean
and constant and finite variance over the study area, independently of the location
(Legendre and Legendre 2012). This condition allows one to compute and examine
correlograms but without tests of significance.
Legendre and Legendre (2012, p. 800) show how to interpret all-directional
correlograms (i.e., correlograms built on distance classes defined in the same way
in all directions) as well as directional correlograms.
A word is needed here about multiple testing. In Sect. 7.2.5 several spatial
correlation values were tested simultaneously for significance. In such cases the
probability of type I error increases with the number of tests. If k tests are carried out,
the binomial law tells us that the overall probability of type I error (technically called
the “experimentwise error rate”) is equal to 1 À (1 À α)
k where α is the nominal
value for a single test. For instance, in the Mantel correlogram shown in Fig. 7.1,
seven tests are carried out simultaneously. Without correction, the overall probability of obtaining at least one type I error is equal to 1 À (1–0.05)
7
¼ 0.302 instead of
the nominal α ¼ 0.05. Several methods have been proposed to achieve a correct level
of type I error in multiple tests (reviewed in Legendre and Legendre 2012; Wright
1992). The most conservative solution for k independent tests is to divide the
significance level by the number of simultaneous tests: α’ ¼ α / k and compare the
p-values to α’. Conversely one can multiply the p-values by k (i.e., p’ ¼ kp) and
compare the resulting values to the unadjusted α. For non-independent tests, Holm‘s
procedure (Holm 1979) is more powerful. The reason is that Holm’s correction
consists in applying Bonferroni’s correction sequentially as follows. First, order the
308
7 Spatial Analysis of Ecological Data