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Statistical Tool for Soil Biology : 11. Autocorrelogram and Mantel Test

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Statistical Tool for Soil Biology : 11. Autocorrelogram and Mantel Test ‘i Pl w f ’* Em J. 1996, (4), i Soil BioZ., 32 195-203 Statistical tool for soil biology. XI. Autocorrelogram and Mantel test Jean-Pierre Rossi Laboratoire d Ecologie des Sols Trofiicaux, ORSTOM/Université Paris 6, 32, uv. Varagnat, 93143 Bondy Cedex, France. E-mail: rossij@[email protected]. fr. Received October 23, 1996; accepted March 24, 1997. Abstract Autocorrelation analysis by Moran’s I and the Geary’s c coefficients is described and illustrated by the analysis of the spatial pattern of the tropical earthworm Clzuniodrilus ziehe (Eudrilidae). Simple and partial Mantel tests are presented and illustrated through the analysis various data sets. The interest of these methods for soil ecology is discussed. Keywords: Autocorrelation, correlogram, Mantel test, geostatistics, spatial distribution, earthworm, plant-parasitic nematode. L’outil statistique en biologie du sol. X.?. Autocorrélogramme et test de Mantel. Résumé L’analyse de l’autocorrélation par les indices I de Moran et c de Geary est décrite et illustrée à travers l’analyse de la distribution spatiale du ver de terre tropical Chuniodi-ibs zielae (Eudrilidae). Les tests simple et partiel de Mantel sont présentés et illustrés à travers l’analyse de jeux de données divers. L’intérêt de ces méthodes en écologie du sol est discuté. Mots-clés : Autocorrélation, corrélogramme, test de Mantel, géostatistiques, distribution spatiale, vers de terre, nématodes phytoparasites. INTRODUCTION the variance that appear to be related by a simple power law. The obvious interest of that method is Spatial heterogeneity is an inherent feature of that samples from various sites can be included in soil faunal communities with significant functional the analysis. It leads to a general and representative implications. In order to quantify that heterogeneity, index that is considered as species-specific by Taylor . many aggregation indices have been proposed and (Taylor et al., 1988). applied to various soil fauna taxa (Cancela Da Fonseca, 1966 ; Cancela Da Fonseca & Stamou, 1982 ; Whatever the index used, it is obvious that one Campbell & Noe, 1985). Most of them are directly cannot restrict the study of the spatial distribution derived from the estimation of the population variance of a population to the computation of any of the and mean from samples and are often sensitive to available dispersion indices. Indeed, dispersion indices the average density of the population (Elliot, 1971). are limited to the description of the kind of distribution Thus, they may give varying index values for samples encountered and to a certain extent to the quantification with different density while the true aggregation is of the degree of clustering (Nicot et al., 1984). They the same. Among the available indices, the index of do not talce into account the spatial position of the Taylor’s Power Law (Taylor, 1961) is commonly used - sampling-points, neither do they furnish information in soil biology (Boag & Topham, 1984; Ferris et on the true patte? of the variable (Liebhold et al., al., 1990; Boag et al., 1994). The method i,s based 1993 ; Rossi et al.: 1996). In a previous paper of this on the empirical relationship between the mean and series (Rossi et aZ., 1995) we introduced the use and I a 196 J.-P. Rossi -:r( interest of the geostatistical tool in soil Ecology. We spatial autocorrelation coefficient for qualitative data. showed how geostatistics made it possible to look A variable is said to be autocorrelated - or regionalized for the presence of spatial autocorrelation in the data - when the measure made at one sampling site brings and how variogram analysis (viz. structural analysis information on the values recorded at a point located or variography) could quantify it. In addition, we a given distance apart. The autocorrelation coefficient introduced the lcriging procedure, an optimal mapping measures the degree of autocorrelation i.e. lil-Leness method. between couples of values recorded at sampling points Ecologists often study complex ecological systems separated by a given distance. The principle is the same and thus face the problem of assessing species- as for the semi-variance analysis (Rossi et al., 1995). environment relationships. Since most of the eco- Moran’s I and Geary’s c are respectively: logical variables are spatially structured at various scales, it is necessary to take into account the presence of spatial autocorrelation in the data sets. In that case, maps are particularly relevant to the problem. However, assessing relationships between spatially autocorrelated variables brings statistical problems as autocorrelation impairs the standard statistical tests e.g. correlation coefficient, ANOVA (Legendre et al., 1990 ; Fortin & Gurevitch, 1993 ; Legendre, 1993). Moreover, imagine a significant common pattern is found shall we consider the variables to be correlated ? In the case of spatially structured variables if we find a correlation, does it mean there is a true correlation between them or are they simply following the same gradient (or any other kind of pattern) due to unknown Data are grouped by distance classes (d)which are a common driving factor(s) ? function of the separating distance between sampling The aims of this paper are twofold. First points, y; and yj are the values of the variables with i methods alternative to the semi-variogram analysis and j varying from 1 to n the number of data points. are presented. They allow overall statistical testing jj the mean of the y’s, wzj is a weighting factor taking for the presence of spatial autocorrelation. Second, the value of 1 if the points belong to the same distance matrix methods for assessing and testing for the class and zero otherwise. W is the sum of the 711’s i.e. presence of a common spatial pattern between two the number of data pairs involved in the estimation of autocorrelated variables are showed. The different the coefficient for the distance class d. Positive values methods introduced in this paper have been used of Moran’s I and value smaller than 1 for Geary’s in various fields of the life sciences but are still c coefficients correspond to positive autocorrelation. sparsely applied in soil biology and ecology. This Notice that autocorrelation analysis requires at least paper aims to introduce these approaches and provide 30 sampling localities to produce significant results relevant literature while emphasizing the potential (Legendre & Fortin, 1989). interest in soil ecology with examples taken from various unpublished studies. The plot of the autocorrelation coefficient against the distance classes is called the correlogram. Each of the coefficient values can be tested for statistical SPATIAL STRUCTURE AND AUTOCORRE- significance. Formulas can be found in Sokal & Oden LATION TESTS (1978), Cliff & Ord (1981) and Legendre & Legendre (1984). The test is based on the null hypothesis HO In a previous paper (Rossi et al., 1995), we “There is no spatial autocorrelation” tested against showed the use of the semi-variance and semi- the alternative hypothesis H1 “There is a spatial variograms to identify spatial autocorrelation. In the autocorrelation”. Under HO, the value of Moran’s I variogram analysis however, semi-variance values and coefficient is E (I) = -(n - l)-’ M O with E(1) variograms are not tested for statistical significance. the expectation of I and the n number of data points. The presence of a consistent pattern is indicated by Geary’s c coefficient equals E (c) = 1. Under H1, the the shape of the variogram as well as the ratio of value of Moran’s I coefficient is significantly different total heterogeneity that can be ascribed to the spatial from O and Geary’s c coefficient is significantly structure. Correlogram analysis allows tests for the different from 1. presence of autocorrelation in data. However, the correlogram must be checked for The method is based on the use of spatial global significance. A correction is thus to be used autocorrelation coefficients, Moran’s I (Moran, 1950) as we perform simultaneously k statistical tests (one or Geary’s c (Geary, 1954) coefficients, that are for each coefficient). The test is made according to used to analyse quantitative variables. Note that the Bonferroni method of correction. It consists in Sokal & Oden (1978) proposed a special form of checking if at least one of the coefficients is significant Eur. J. Soil Biol. n i Autocorrelogram and Mantel test 197 u at the statistical level a’ = a/k with Q = 5% and k 60 35 the number of distance classes (Oden, 1984). A/ The shape of the correlograms gives information 30 on the type of spatial structure encountered. Some 25 characteristic shapes are associated with specific patterns, for instance the alternation of significant 20 positive and negative autocorrelation is typical of a patchy distribution. Sokal & Oden (1978) 15 and Legendre & Fortin (1989) gave a series of typical correlograms and the corresponding patterns. 10 However, the shape of the correlogram is not 5 always specific of a given distribution type, e.g. data presenting a sharp step and a gradient, lead to O quite the saine correlogram shape. Therefore maps are necessary to fully describe the spatial patterns. Among Number of individuals Number of individuals the various mapping methods, the kriging procedure per sampling unit per sampling unit (after is particularly useful since it is optimal and unbiased B ox-Cox transformation) & & (Burgess Webster, 1980a, b; Isaaks Srivastava, Figure 1. - Frequency distribution of the earthworm Chuniodrilus 1989; Webster & Oliver, 1990). ïielue: (A) Raw data and (B) Data transformed according to the Example 1 : Spatial autocorrelation of the earthworm Box-Cox transformation. Clzurziodrilus ziehe (Eudrilidae). In order to illustrate the use of correlograms we the frequency distribution of the raw and transformed shall apply the method to the earthworm Clzuniodrihs data. zielae (Eudrilidae) data we presented in Rossi et al. Data were allocated to 12 distance classes with (1995).
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