Multiscale Ordination Research Articles

Multi-Scale Altitudinal Patterns of Soil and Litter Invertebrate Communities in a Warm Temperate Deciduous Broadleaf Forest

The diversity and spatial patterns of soil invertebrates are the cornerstones for understanding their ecological functions, which are crucial to maintaining nutrient cycling and soil health in forest ecosystems. Based on a continuous altitudinal gradient (1020–1770 m) composed of 119 plots, this study analyzed the spatial patterns of invertebrate communities in temperate mountain forest litter and soil using multi-scale ordination. The results indicate that along the altitudinal gradient, the invertebrate communities in both litter and soil layers exhibit “patches” at a scale of approximately 33 plots that are mainly composed of Mesostigmata and Apterogasterine oribatid mites. In the litter layer, at the 11-plot scale, an aggregation of Onychiuridae is also formed, while in the soil layer, there are “patches” represented by Diadocidiidae at the 33-plot scale. The positive–negative associations among invertebrate taxa also shift between the litter and soil layers, as well as among “patches”. Our study confirms that the richness of invertebrates in temperate-forest litter is higher and forms multi-scale assembly “patches” despite the higher abundance of invertebrates in the soil layer. Future studies should delve deeper into the aggregation mechanisms of these specific taxa and may require higher sampling densities to reveal the multi-scale spatial patterns of soil invertebrates.

The relationship between observational scale and explained variance in benthic communities.

This study addresses the impact of spatial scale on explaining variance in benthic communities. In particular, the analysis estimated the fraction of community variation that occurred at a spatial scale smaller than the sampling interval (i.e., the geographic distance between samples). This estimate is important because it sets a limit on the amount of community variation that can be explained based on the spatial configuration of a study area and sampling design. Six benthic data sets were examined that consisted of faunal abundances, common environmental variables (water depth, grain size, and surficial percent cover), and sonar backscatter treated as a habitat proxy (categorical acoustic provinces). Redundancy analysis was coupled with spatial variograms generated by multiscale ordination to quantify the explained and residual variance at different spatial scales and within and between acoustic provinces. The amount of community variation below the sampling interval of the surveys (< 100 m) was estimated to be 36–59% of the total. Once adjusted for this small-scale variation, > 71% of the remaining variance was explained by the environmental and province variables. Furthermore, these variables effectively explained the spatial structure present in the infaunal community. Overall, no scale problems remained to compromise inferences, and unexplained infaunal community variation had no apparent spatial structure within the observational scale of the surveys (> 100 m), although small-scale gradients (< 100 m) below the observational scale may be present.

Is the Mantel correlogram powerful enough to be useful in ecological analysis? A simulation study

The Mantel correlogram is an elegant way to compute a correlogram for multivariate data. However, recent papers raised concerns about the power of the Mantel test itself. Hence the question: Is the Mantel correlogram powerful enough to be useful? To explore this issue, we compared the performances of the Mantel correlogram to those of other methods, using numerical simulations based on random, normally distributed data. For a single response variable, we compared it to the Moran and Geary correlograms. Type I error rates of the three methods were correct. Power of the Mantel correlogram was nearly as high as that of the univariate methods. For the multivariate case, the test of the multivariate variogram developed in the context of multiscale ordination is in fact a Mantel test, so that the power of the two methods is the same by definition. We devised an alternative permutation test based on the variance, which yielded similar results. Overall, the power of the Mantel test was high, the method successfully detecting spatial correlation at rates similar to the permutation test of the variance statistic in multivariate variograms. We conclude that the Mantel correlogram deserves its place in the ecologist's toolbox.

Effects of study design and analysis on the spatial community structure detected by multiscale ordination

AbstractQuestion:What are the effects of different aspects of data collection and analysis on the spatial partitioning of ordination results by multiscale ordination?Locations:Heterogeneous pasture at Marchairuz in the Jura mountains, Switzerland and Mixed hardwood‐pine forest in Oostings Natural Area, NC, USA.Methods:We evaluated the efficiency of different sampling designs for identifying the spatial structure of plant communities and analysed two datasets with multiscale ordination. We compared the effects of quadrat size, the number of species included in the analysis, data type, detrending and ordination method on the shape and precision of the community variogram summarizing spatial community structure.Results:A three‐block sampling design provided a more even distribution of the number of pairs of observation per distance class than random, transect or grid designs. The precision of variogram estimates depended more strongly on the number of species than on the number of quadrats. In contrast, the choice of data type (abundance transformation) had little influence on the shape of the variogram. Detrending reduced the range of spatial autocorrelation. An increase in quadrat size resulted in a smoother variogram and stronger spatial autocorrelation. Principal components analysis (PCA) and Redundancy analysis (RDA) resulted in a larger range of spatial autocorrelation values than did Correspondence analysis (CA) and Canonical correspondence analysis (CCA), but the shape of the variograms was similar.Conclusion:Random samples as well as transects and regular grids may not be efficient sampling designs for spatial analysis of community structure. While the number of species considered strongly affects the precision of a community variogram, its shape depends on the size of the sampling units. Earlier studies may have overestimated the spatial scale of internal organization in plant communities.

An operational, additive framework for species diversity partitioning and beta‐diversity analysis

Summary An important goal of community ecology is the assessment of factors that are likely to influence the spatio‐temporal distribution of species assemblages and diversity. Surprisingly, most statistical methods devoted to this have remained poorly interconnected, as well as poorly connected with the popular metrics of diversity estimation. In the present paper we show that important questions related to determinants of species diversity can be specified through a simple multivariate linear model and explored, in common diversity metrics, using standard methods and routines of variance/covariance decomposition. Thanks to an unusual form of presentation of taxonomic data into a table of species occurrences, which considers the individuals as data units, Shannon and Simpson indices as well as species richness can all be expressed as a (weighted) sum of squares. Subsequent apportionments into explained and residual sum of squares provide direct estimates of the beta‐ and alpha‐diversity components with respect to either categorical habitat types or continuous gradient variables. Appropriate statistics and non‐parametric tests are available to assess the significance of these components. Explicit analytical relationships exist between the linear approximation of the table of species occurrences by sampling sites, and the more classical table of species abundances by sites. Therefore, direct links with methods of ordination in reduced space, such as correspondence analysis and canonical correspondence analysis, provide opportunities for partitions that preserve consistency with usual diversity indices. The sum of squares of the approximated occurrence table provides measures of intersites beta‐diversity, from which measures of dissimilarity with explicit references to diversity indices can be derived. Such measures are amenable to distance‐based apportionments through multivariate variograms and multiscale ordination. What are the relative effects of the biological, environmental and anthropogenic factors and of their potential interactions on species diversity? Are these effects stable across scales, from landscape to region, between regions and across ecosystems? The methodological integration proposed in our analytical framework enables one to address these questions using standard statistical tools, and opens new prospects for quantitative biodiversity studies. This also paves the way towards refined models for predicting species diversity at unsampled locations.

Using multivariate factorial kriging for multiscale ordination: a case study

Vegetation ordination is usually based on classical data reduction techniques such as principal components analysis, correspondence analysis, or multidimensional scaling. The usual methods do not account for multiscale correlations among species. In this paper, we use a geostatistical method, known as multivariate factorial kriging, for studying multiple-scale correlations. The case study was carried out in a mixed broadleaf forest of central Spain. Six tree species were included in the analysis. Data analysis included (i) experimental variogram calculation and modeling with the use of the linear model of coregionalization, (ii) principal components analysis, and (iii) cokriging. The results indicate that correlations among species are different depending on the spatial scale. We conclude that competition for light is the main factor controlling the spatial distribution of species at the plot-level scale of variation. At larger scales of variation, soil conditions and (or) human intervention are the key factors in determining the observed vegetation pattern. Based on the factor scores for the largest scale of variation, we conducted a cluster analysis to identify plots with similar characteristics. The resulting clusters have the remarkable property of being spatially continuous.

A GENERALIZED, VARIOGRAM-BASED FRAMEWORK FOR MULTI-SCALE ORDINATION

Multi-scale ordination (MSO) deals with potential scale dependence in spe- cies assemblages by studying how results from multivariate ordination may be different at different spatial scales. MSO methods were initially based on two-term local covariances between species and, therefore, required sampling designs composed of adjacent quadrats. A variogram-based MSO, recently introduced by H. H. Wagner, is applicable to very diverse sampling designs and for use with principal-components analysis, correspondence analysis, and derived ''two-table'' (also called ''direct'') ordination methods, i.e., redundancy anal- ysis and canonical correspondence analysis. In this paper we put forward an enlarged framework for variogram-based MSO that relies on a generalized definition of inter-species covariance and on matrix expression of spatial contiguity between sampling units. This enables us to provide distance-explicit decompositions of variances and covariances (in their generalized meaning) that are con- sistent with many ordination methods in both their single- and two-table versions. A spatially explicit apportioning of diversity indices is proposed for some particular definitions of variance. Referring to two-table ordination methods allowed the multi-scale study of re- sidual spatial patterns after factoring out available environmental variables. Some aspects of the approach are briefly illustrated with vegetation data from a Neotropical rain forest in French Guiana.

DIRECT MULTI-SCALE ORDINATION WITH CANONICAL CORRESPONDENCE ANALYSIS

Gradient analysis uses ordination methods to study the structure of biotic communities caused by biotic processes operating in a heterogeneous environment. This structure has two spatial components: spatial processes within the community create au- tocorrelation, and the spatial structure of environmental factors creates spatial dependence. Ordination methods, however, do not make use of spatial information. Spatial alternatives are available in multivariate geostatistics, but are not compatible with important ordination methods used in gradient analysis, correspondence analysis and canonical correspondence analysis (CA, CCA). This paper shows how CA and CCA can be partitioned by distance (indirect and direct multi-scale ordination) and integrated with geostatistics. A diagnostic tool enables ecologists to partition ordination results by distance, to distinguish between components of spatial dependence and of spatial autocorrelation, and to check assumptions of independent residuals, stationarity, and scale-invariant correlation. The application is illustrated with a well-known data set of oribatid mites. Empirical chi-square variograms of individual species, their pair-wise cross variograms, and the variogram of the total inertia are defined and summarized in a variogram matrix, which leads to a spatial partitioning of the eigenvalues. The empirical variogram matrix provides a link to coregionalization anal- ysis that may be used to simultaneously model spatial dependence and spatial autocorre- lation. This will be useful for answering questions about the organism-specific scale of response to the environment, the optimal spacing of sampling units, or the scale-dependent effect of environmental factors.

SPATIAL COVARIANCE IN PLANT COMMUNITIES: INTEGRATING ORDINATION, GEOSTATISTICS, AND VARIANCE TESTING

Spatial structure in plant communities occurs in the forms of (1) single- species aggregation and dispersion patterns, (2) distance-dependent interactions between species, and (3) the response to the spatial structure of environmental conditions. Different methods deal with these components of spatial variation: geostatistical analysis reveals autocorrelation in a spatial sample; the variance of species richness has been used as an indicator for interspecific interactions due to niche limitation; and ordination techniques describe multispecies responses to environmental factors. Based on the mathematical prop- erties of presence-absence data, it is shown how variogram modeling, the testing of in- terspecific associations, and multiscale ordination can be integrated using the same set of distance-dependent variance-covariance matrices (variogram matrix). The variogram matrix partitions the variance of community data into spatial components at the levels of the individual species, species composition, and species richness. It can be used to factor out the effects of single-species aggregation patterns, interspecific interactions, or environ- mental heterogeneity. The mathematical integration of traditionally unrelated methods in- creases the interpretability of variograms of plant communities, provides a spatial extension and an empirical null model for the variance test of species richness, and extends multiscale ordination to nonsystematic spatial samples. Beyond the individual applications, the var- iogram matrix provides a framework for a mathematical unification of geostatistics, mul- tivariate data analysis, and the analysis of variance that may enable ecologists from a broad range of fields to incorporate spatial effects into their research and to integrate analyses across different levels of biological organization.

Composition and spatial structure of plant communities on southeastern Victoria Island, arctic Canada

We used multivariate methods to investigate the composition and spatial organisation of tundra plant communities in the Wellington Bay region of southeastern Victoria Island (69°N, 106°W). Ordination and classification of sites at an intermediate scale (≈ 1 ha) were conducted using principal components analysis and sums of squares agglomerative clustering on a matrix of standardized chord distances. The findings suggested eight vegetation classes. These communities are described floristically. At this spatial scale, the vegetation showed correspondence to elevation, slope, and thickness of soil, but not to aspect. The spatial patterns of multiple plant species and physical variables (i.e., slope of terrain and thickness of soil) were examined using multiscale ordination and double logarithmic regressions of variance on distance, respectively. Multiscale ordination revealed ever-increasing plant heterogeneity with distance (0.25 – 1600 m) and suggested only weak general patterns at scales ≤ 200 m. Similarly, variance in the physical factors tended to increase continually with distance. Both the vegetation and physical environment thus appeared to be organised on gradients. Key words: Arctic, classification, ordination, spatial scale, tundra.

A paired-quadrat method for use in multiscale ordination

Multiscale ordination is a technique for examining spatial patterns of several species at several scales. We present a paired-quadrat method (paired quadrat covariance; PQC) to be used in multiscale ordination and test it with artificial data. Multiscale ordination with PQC successfully extracted the salient features of the data set. The method appears to be more sensitive than blocked-quadrat techniques for extracting small-scale patterns. We suggest that PQC will be useful as a complement to existing procedures or as a tool for analysing data from scattered quadrat arrangements.

Multiscale ordination: a method for detecting pattern at several seales

Multivariate analyses of vegetation data have been restricted to a single scale of sampling, or multiscale sampling has been restricted to a single species. However, vegetation scientists need to be able to explore spatial relationships of many species over many scales. We present a modification of Noy-Meir & Anderson's (1971) method of multiscale ordination by summing two-term local covariance matrices and smoothing the component profiles. The advantages of our method are: 1) results are less subject to the starting position of the transect, 2) matrices may be added at any block size, and 3) plots of factor scores are smoothed by a moving weighted average to better reveal patterns at a prescribed scale. This procedure provides statistical associations of species over a range of scales. The scales which exhibit the association to the maximum extent are then determined from multiscale ordination. The relationships of different associations and their scales can then be examined. The application of the method to fabricated data proved successful in recovering the structure built into the data. When used on real vegetation data, from a community and a landscape, the method revealed the details of species associations over a range of scales, and of the relationships among associations.

The Effect of Sampling Scale on the Detection of Interspecific Patterns in a Hemlock-hardwood Forest Herb Stratum

The herb layer of a mature hemlock-hardwood forest was sampled with: (1) systematically located quadrats and (2) transects extending radially from herb-patch centers. Association analysis of the quadrats detected a different type of interspecific relationship than correlation analysis. Furthermore, herb-patch transect analysis revealed interspecific patterns which do not always appear in association or correlation analysis. These differences reflect primarily the numerous scales of pattern operating within the forest herb community. INTRODUCTION The measure of an interspecific association or correlation is expected to vary according to quadrat size and its relation to the scale of community pattern (GreigSmith, 1964; Pielou, 1977; Kershaw, 1973; Gauch, 1982). In recognition of this quadrat size dependence, pattern analysis traditionally has used a graduated set of quadrats or blocks which permit detection of the various scales of horizontal structure (Greig-Smith, 1961; Kershaw, 1973). In particular, two species can exhibit a different degree and direction of correlation in pattern analysis, depending on quadrat size (Greig-Smith, 1964). Other sampling designs also have been proposed for multiscale pattern analysis (Bouxin, 1983), as have methods for multiscale ordination in which the data from all species and all scales are integrated to summarize community pattern (Noy-Meir and Anderson, 1971). Association and correlation analyses have been used to describe temperate forest herb strata which have been sampled by systematically or randomly located quadrats of a single size (Bray, 1956; McIntosh, 1962; Struik and Curtis, 1962; Smith and Cottam, 1967; Gauch and Stone, 1979; Hicks, 1980). Despite the widely recognized dependence of pattern detection on sampling scale, few studies illustrate this phenomenon in forest understory vegetation by analyzing for two more scales of pattern with two or more quadrat sizes. Complexities resulting from several scales of pattern are presented here using data from two sampling schemes in a hemlock-hardwood forest herb community.

The Influence of Sheep on Pattern and Process in Atriplex vesicaria Populations From the Riverine Plain of New South Wales

The application of multiple pattern analysis (multiscale ordination) to a phase-differentiated sample of perennial saltbush (Atriplex vesicaria) rangeland, along with assessments of the vegetative and reproductive states of individual plants, served to indicate the importance of sheep in determining some scales of distributional pattern in the shrub population. By comparison with conventional pattern analysis, it is shown that multiple pattern analysis can be used to provide an economic description of 'total' pattern in temporally related phases of a monospecific plant population. It is demonstrated that the pattern-determining interaction between sheep and saltbush individuals is dependent on plant size, sex and location with respect of other shrub individuals, in particular due to the trampling of seedling and pioneer phases and the selective grazing of sexually mature shrubs by sheep.