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‘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 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 sets. The interest of these methods for soil ecology is discussed.

Keywords: Autocorrelation, , Mantel test, , 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 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 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 - -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 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. , 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 H1 “There is a spatial 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 . 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

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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 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 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. - 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 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). The was collected in an African 5.30 m width. Figure 2 shows the distribution of the grass savanna in July 1994. 100 sampling points were point pairs among the distance classes. The number of located at regular intervals of 5 m on a lox 10 points data couples involved in the computation is small for grid. At each sample location a 25x25~10 cm soil the last 3 distance classes (table I), and autocoirelation monolith was talcen and earthworms were handsorted. coefficients cannot be interpreted because they are Data were log, transformed before analysis to reduce computed upon too few pairs of point. That is why the asymmetry of the frequency distribution. Here the dividing distances into classes with equal frequencies normalisation of data is obtained through the Box- is sometimes preferred upon forming equal distance Cox transformation (Sokal & Rohlf, 1995) which is: classes. y = (d - l)/S. The S parameter was estimated as Autocorrelation coefficients of Moran and Geary S = 0.32606 using the program VerNorm 3.0 from the were estimated for each distance class and the “R package” developed by Legendre & Vaudor (1991). probability for obtaining such values under the After transformation, the frequency distribution met null hypothesis were computed using the program normality as confirmed by a Kolmogorov-Smirnov Autocorrelation from the “R pacltage” written by test of normality (done, again, using the program Legendre & Vaudor (1991). Table 1 gives the VerNorm 3.0 from the “R package”). Figure 1 shows coefficient values and associated probability. The

Table 1. - Moran and Geary autocorrelation coefficient values for the earthworm Clzuiiiodrilus ïielae density. The width of the distance classes is 5.3 m, p(H0) indicates the probability to obtain the coefficient value under the null hypothesis.

Distance Lower limit Upper limit I (Moran) P(H0) c (Geary) classes (m) (m) O+ 1 O 5.3 0.4870 0.4665 0tO+ 2 5.3 10.6 0.2579 0t 0.6493 3 10.6 15.9 -0.0010 0.613 0.8790 0.016 4 15.9 21.2 -0.1412 O+0t 1.0499 0.161 5 21.2 26.5 -0.1352 1.0665 0.069 6 26.5 31.8 -0.0397 0.193 0.9749 0.263 7 31.8 37.1 0.0898 0.003t 0.9330 0.079 8 37.1 42.4 - 0.03 16 0.313 1.1727 0.014 9 42.4 41.7 0.0253 0.229 1.2257 0.028 10 47.7 53 -0.1910 0.035 , 1.5046 0.003* 11 53 58.3 -0.5917 O+0t 2.1275 0t 12 58.3 63.6 - 1.1141 2.7608 0.001t Significant probabilities at the Bonferroni-corrected probability level of 0.05/12 = 0.00416.

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12 3 4 5 6 7 8 9101112 Distance classes 0.5 Figure 2. - Number of pairs of points in each distance class. 123 4 5 6 7 S 9 101112 resulting correlograms are showed in figure 3 where Distance classes open circles represent non-significant individual Figure 3. - Moran’s I and Geary’s c spatial correlograms for the values of the coefficients at the statistical level earthworm Clmniodrihs zielae density. a = 0.05. Before examining the correlogram it is necessary to ensure that it is globally significant at the corrected probability level a‘ = 0.05/12 = coefficient. This approach may be impaired if the 0.00416. This condition is actually verified for variables are non-linearly related, and variables several autocorrelation values in our example (table I). must be transformed or non- used. Correlograms indicate the presence of a positive However, these approaches are based on the autocorrelation at short distance classes, which reveals assumption that observations are independent. In that the presence of a contagious distribution. In the Moran case, each of the observations brings one degree of correlogram, the autocorrelation coefficient is positive freedom. In turn, the sum of the degrees of freedom at distance class 1 and decreases up to distance is used to compute the of the correlation class 4. From distance class 5 to 7 it increases again. coefficient and thus check for its significance. Significant positive autocorrelation values for short If the variables under study are positively spatially distance classes indicates a contagious distribution. autocorrelated and if the sampling scale matches with The similarity between samples decreases up to the scale at which the spatial structure is expressed, distance class 4 and then increases up to distance the assumption of independence of observations is class 7. The distance class 7 ranges between 37.1 clearly violated. Since the information brought by and 42.4 m which corresponds approximately to the a value is partly brought by another value as a interpatch distance in the map of the variable (Rossi sequel of autocorrelation, the degrees of freedom et al. (1995), fig. 6a). Points separated by a distance are overestimated and testing correlation coefficient falling within the bounds of distance class 7 are similar becomes impossible. The overestimation of degrees because the corresponding samples are taken in the of freedom leads to conclude that a correlation patches. The analysis of Geary’s e correlogram leads is significant when it is not, but conversely, if a to the same conclusions. For small distance classes correlation coefficient is declared non-significant the the coefficient is lower than 1 indicating a positive result is valid. This problem has been solved by autocorrelation. From distance class 7 it becomes Clifford et al. (1989) who proposed to estimate significantly greater than 1 which corresponds to a the autocorrelation of the variables and using these negative autocorrelation. estimates obtain a reduced number of degrees of freedom which can then be used to test the significance of the correlation coefficient. RELATIONSHIPS BETWEEN AUTOCORRE- In geostatistics, the relationship between two LATED DATA autocorrelated variables is assessed by the cross- variogram (Isaaks & Srivastava, 1989; Rossi et al., The assessment of the relationship between variables 1995). A structure function (the cross-variogram) is generally made by computing their correlation describes the variation of the cross semi-variance in Autocorrelogram and Mantel test 199 U

function of the sample spacing. In soil biology, this the null hypothesis by repeated permutations of the method has been applied for example to study the lines and columns in one of the matrix A and B relationships between the density of various plant- and recomputing the Mantel statistic. The result is a parasitic nematode species (Delaville et al., 1996 ; of the Mantel statistic under the Rossi et al., 1996). We will now examine another null hypothesis. If there is no relationship between method for assessing the relationship between two the matrices, the observed r value is near the centre variables and checking its significance : the Mantel the sampling distribution, while if a relationship is statistic (Mantel, 1967). present, one would expect the observed value to be The Mantel test allows to look for correlation more extreme than most of the values obtained by between two proximity or distance matrices (say, permutation (Legendre & Fortin, 1989). A and B) by computing the cross-product of the The Mantel test is primarily used to search for linear corresponding values in A and B. The matrices trend in the data e.g. linear gradient conesponding to describe the relationships among the IZ sampling some kind of underlying diffusive processes. However, sites. A and B can be formed by using one of the linear trends are not the most frequent spatial pattern many distance or dissimilarity indices available in encountered in soil ecology. Organisms frequently the literature (Legendre & Legendre, 1984; Gower display complex spatial distributions in patches of & Legendre, 1986). The method can be applied to different sizes (Wallace & Hawkins, 1994 ; Robertson different cases, the multivariate and the univariate & Freckman, 1995; Rossi et al., 1996, 1997). Thus if approaches. In the multivariate case matrices are the trend in the data is linear, the geographic distance formed using many variables describing the sampling is to be used. If some other relationship prevails, stations. One can then look for relationships between one may use some other function of the geographic a distance matrix representing the environmental distance D such as I/D or 1/D2. It is recommended variables and a second matrix describing the to use large sample size (n>20) to detect significant community composition either in terms of density, spatial pattern (Fortin & Gurevitch, 1993). biomass or presencehbsence. In the univariate case, each matrix is formed from one variable with the Exainple 2: Application of the Mantel test to the Euclidean distance coefficient : the distance between Clzuniodrilus zielae data. two sampling stations is computed as the unsigned The Mantel test was applied to look for difference among values of the variable. One of the relationship between two dissimilarity matrices the two matrices can be directly built from the corresponding to the variables Chuiiiodrilus zielae geographic distances among sampling stations. In that density and the geographic distance between the case, comparing this matrix and another one derived sampling points. The data set is the same as in from a given variable constitutes a way to look for a exainple 1. The matrix A was formed by taking the spatial trend in the data. unsigned difference among values of C. zielae density The Mantel statistic tests the null hypothesis HO for all possible pairs of station while the matrix B was “Distances among points in the matrix A are not formed by taking the geographic distances among the linearly related to the corresponding distances in sampling localities. The distance matrix computations the matrix B” against the alternative hypothesis H1 were carried out using the program Simil 3.01 from “Distances among points in matrix B are linearly the “R package” (Legendre & Vaudor, 1991). correlated to the corresponding distances in the matrix The Mantel statistic and permutation test were B”. carried out with the program Mantel 3.0 from The Mantel statistic is: the “R package” (Legendre & Vaudor, 1991). T Z = xij yij with i # j and i and j the row The standardised Mantel r was = O. 12230 with ij a probability to reach such a value under the and column indices. null hypothesis of p=O.OOl (obtained with 1000 The Mantel statistic can be normalised to permutations). Since the Mantel statistic is significant between -1 and +1. we reject the null hypothesis that there is no [lh 1)1 L>/s,l relationships between the matrices A and B, in 7- = - c c [(Xij - :>/szI [(Yyij - other words, the earthworm density displays a spatial ij pattern. It is important to notice that the Mantel with j and and j the row and column indices i # i statistic r is a correlation between two distance and n the number of distances in one of the matrix matrices and is not equivalent to the correlation without accounting for the diagonal. between the variables used to form these matrices. The statistical significance of the Mantel coefficient can be tested in two ways i) by computing the The example above shows that the Mantel test is expected value and variance under the null hypothesis able to detect the spatial pattern of C. zielae although and performing a z-test (Legendre & Fortin, 1989) it is not a simple gradient as there are two patches provided the size of the matrix tested is large i.e. (see jig. 6a in Rossi et al. (1995)). n>40, and ii) application of a permutation test. Example 3 : Assessing the relationships between a The latter consists in simulating the realisations of plant-parasitic nematode and soil clay content. 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The following example is taken from a study carried PARTIALLING OUT SPATIAL EFFECT out in the south-east of Martinique (Lesser Antilles : 14” 3‘ N and 62” 34’W) on a vertisol developed on Once the existence of a relationship between two volcanic ashes (Rossi and Quénéhervé, unpublished). variables has been demonstrated, one can wonder if In a 10-yr. old pasture regularly planted with the it is a true correlation or if it is only a spurious tropical grass Digitaria decuinbens both plant-parasitic correlation due to common spatial (or temporal) nematodes and some soil physico-chemical parameters pattern. In other words, two variables may appear were investigated by of 60 sampling points x to be related while they are only independently driven randomly distributed within a 60 25 m plot. Soil by a third common cause. In ecological studies, space samples with adhering roots for nematode analysis (sampling position) is likely to cause such spurious were removed from the 0-10 cm soil layer. The correlation and in that case determining whether the nematodes were extracted from the soil by the correlation is true or spurious requires partialling out elutriation-sieving technique (Seinhorst, 1962) and the spatial component of ecological variation. from the shredded roots in a mist chamber (Seinhorst, The partial Mantel test constitutes a way to assess 1950). Soil samples were removed from 0-10 cm soil the relationship between two dissimilarity matrices layer and soil texture (clay, silt and sand contents) while controlling for the effect of a third one (Smouse were determined by laser granulometry (Mastersizer et al., 1986). The third matrix is formed with the E, Malvern). geographic distances between sampling points if the The Mantel test was used to assess the relationships “space” effect is to be partialled out. A computation between density of the plant-parasitic nematode method has been developed by Smouse et al. (1 986) to HelicoQ1enchus retusus (Siddiqui & Brown, 1964) test the partial Mantel statistic between two matrices and soil clay content (9%). The clay content data met A and B while controlling for the matrix C. The normality whereas the Box-Cox procedure was used to Mantel test is applied to the matrices A’ and B’ normalise the nematode data. Both nematode density that respectively contain the residuals of the linear and clay content displayed significant spatial pattern regression of the values of A and B on the values as shown by Moran’s I correlogram (significant at of C. The statistical test of significance is the the Bonferroni corrected statistical level). A distance same as explained above: either Mantel’s normal matrix was formed for each variable (matrix CLAY approximation or the permutational test by permuting for clay content and matrix NEM for H. retusus either A’ or B’. The Mantel test between A’ and B’ density) as explained above. A third matrix was formed is the partial Mantel test between A and B whilst by taking the geographic distance among sampling controlling for the effect of C. points (matrix SPACE). The Mantel test was used to test for the correlation between the three possible If a correlation is spurious because of the effect of unknown factors causing a common spatial pattern, pairs of matrices (table 2). Since three tests are done simultaneously, it is necessary to use the Bonferroni- the partial Mantel test is expected not to be significant corrected probability level of 0.05/3 =0.01667 for an while controlling for the effect of space. Conversely, in the presence of true correlation, partialling out overall significance level of 0.05 (table 2). Table 2 a shows that all the simple standardised Mantel tests the effect of space still leads to a significant test are significant. Thus, the spatial patterns revealed (Legendre & Troussellier, 1988). by the correlograms analysis are confirmed by the Example 4: Common pattern among a plant- Mantel tests. The Mantel test between CLAY and parasitic nematodes and soil clay content. NEM indicates a significant correlation between the The data are those presented in example 3. matrices. From this result, shall we conclude that this The partial Mantel test was used to assess common pattern is due to a relationship between the correlation between the matrices NEM and CLAY variables or is it just a spurious correlation due to while controlling for the effect of the matrix the fact that both variables are independently driven SPACE. Computations were done with the software by a common cause? This topic is addressed in the Mantel 3.0 from the ‘IR package“ (Legendre & next section. Vaudor, 1991). The statistical significance of the Mantel statistic was checked using the permutation Table 2. - Simple standardised Mantel statistics and associated test with 1000 permutations. The test is denoted probabilities. Tests of significance are one-tailed (data from Rossi (CLAY.NEM).SPACE. The Y value was r=0.02550 and Quénéhervé, unpublished). with an associated probability p=O.19780. If the correlation were true, we would expect the test Mantel’s r Probability to be significant, a condition that is not met. So we can conclude that H. retusus and clay content HRET.CLAY 0.13078 0.00599t display similar spatial pattern without being truly HRETSPACE 0.16720 0.00200t CLAY SPACE 0.67063 0.00100~ correlated. Another factor, not explicitly mentioned, 7 Simple Mantel test significant at the Bonferroni-corrected independently drives the variables. probability level of (0.05/3 =0.01667) for an overall significance level This example shows that caution is needed when of 0.05 over three simultaneous tests. analysing autocorrelated data, first because specific J Autocorrelogram and Mantel test 201

tools are required and second because a common from O, a condition that is not met as r=0.1081 with spatial pattern may lead to the observation of spurious associated probability p = 0.005. correlation (Legendre & Troussellier, 1988). Finally the fourth model states that both clay and The simple Mantel and partial Mantel tests permit nematode spatial pattern are independently caused by segregation of different causal models (Legendre & unknown factors (Space) : CLAY t SPACE -+ NEM. Troussellier, 1988; Legendre & Fortin, 1989). In a If the model were supported by the data we would causal model, the ecologist includes some hypotheses expect the partial Mantel test (CLAY.NEM).SPACE about the factors determining the process at hand. not to be significantly different from O which condition The second step consist in checking whether the field was verified above (example 4). Before accepting the or laboratory data actually support predictions of the model, we must verify all the predictions that can be model. The investigations must not be restricted to derived from it: the initial model but have to be extended to all the alternative possible models. Each model leads to a group of causal predictions that must be checked. SPACE.CLAY#O (CLAY .NEM).SPACE=O Finally, a model is accepted if and only if all the SPACE.NEMf0 (SPACE.NEM).CLAY

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Mantel test to assess the correlation between matrices Although the Mantel test is devoted to the study corresponding to a complete set of environmental of linear gradients, the examples presented in this variables and a second formed with the abundance paper show that the test is efficient in assessing Of the ‘pecies making a community. In the data patchy distribution at least if the patches are smooth different kind of variables (quantitative, s“-I and relative]y large compared to the overall studied qualitative, qualitative) can be mixed up and an adapted coefficient of association be used (see Gower The test may be used to check & Legendre (1986) and Legendre & Legendre (1984) for the goodness-of-fit Of data to a model. In this case, for a review of the coefficients of association and the object is to compare a matrix formed by data to Legendre & Fortin (1989) for an example). that formed from predictions of a given model.

Acknowledgements I wish to thank Dr J. P. Cancela Da Fonseca who initiated the “Statistical Tool for Soil Biology” series and allowed me to publish under this heading. I am greatly indebted to Prof. P. Legendre for providing me with his computer program the “R package” and giving me statistical guidance. I also wish to thank J. E. Tondoh and R. Zouzou Bi Danko €or field assistance (it’s been no bed of roses !). Special thanks to Prof. P. Lavelle, L. Rousseaux and P. Quénéhervé for stimulating discussions and two anonymous referees for perceptive and detailed comments on an earlier draft.

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