Effects of plot size on the ordination of vegetation samples

Questions: Do ordination patterns differ when based on vegetation samples recorded in plots of different size? If so, how large is the effect of plot size relative to the effects of data set heterogeneity and of using presence/absence or cover‐abundance data? Can we combine plots of different size in a single ordination? Methods: Two homogeneous and two heterogeneous data sets were sampled in Czech forests and grasslands. Cover‐abundances of plant species were recorded in series of five or six nested quadrats of increasing size (forest 49‐961 m2; grassland 1‐49 m²). Separate ordinations were computed for plots of each size for each data set, using either species presences/absences or cover‐abundances recorded on an ordinal scale. Ordination patterns were compared with Procrustean analysis. Also, ordinations of data sets jointly containing plots of different size were calculated; effects of plot size were evaluated using a Monte Carlo test in constrained ordination. Results: The results were consistent between forest and grassland data sets. In homogeneous data sets, the effect of presence/absence vs. cover‐abundance was similar to, or larger than, the effect of plot size; for presence/absence data the differences between ordinations of differently sized plots were smaller than for cover‐abundance data. In heterogeneous data sets, the effect of plot size was larger than the effect of presence‐absence vs. cover‐abundance. The plots of smaller size (= 100 m2 in forests, = 4 m² in grasslands) yielded the most deviating ordination patterns. Joint ordinations of differently sized plots mostly did not yield patterns that would be artifacts of different plot size, except for plots from the homogeneous data sets that differed in size by a factor of four or higher. Conclusions: Variation in plot size does influence ordination patterns. Smaller plots tend to produce less stable ordination patterns, especially in data sets with low ß‐diversity and species cover‐abundances. Data sets containing samples from plots of different sizes can be used for ordination if they represent vegetation with large ß‐diversity. However, if data sets are homogeneous, i.e. with low ß‐diversity, the differences in plot sizes should not be very large, in order to avoid the danger of plot size differences distorting the real vegetation differentiation in ordination patterns.     

Finding a common ordination for several data sets by individual differences scaling

Individual differences scaling is a multidimensional scaling method for finding a common ordination for several data sets. An individual ordination for each data set can then be derived from the common ordination by adjusting the axis lengths so as to maximize the correlations between observed proximities and individual ordination distances. The importance of the various axes for each data set and the mutual similarities and goodness of fit for the individual data sets are described by weight plots. As an example, 46 soft-water lakes in eastern Finland are ordinated on two dimensions according to 3 chemical data sets (water in summer and autumn, sediment) and 4 biological sets (major phytoplankton groups, phytoplankton, surface sediment diatom and cladoceran assemblages). The method seems to be effective as a means of ordination for obtaining the common ordination for the data sets. The major taxonomic groups gave the ordination which differed most clearly from the ordinations of the other data sets. Phytoplankton was most poorly ordinated in all the analyses. The other data sets were fairly coherent. When only biological data sets were ordinated, the diatoms and cladocerans showed rather different patterns. It seems that the cladocerans are best correlated with water chemistry, both according to weights in the joint analysis, and according to correlation between the axes from the biological data sets and the chemical variables.Abbreviations CCA = Canonical correspondence analysis - IDS = Individual differences scaling - MDS = multidimensional scaling - PCA = Principal components analysis     

Matching data sets from two different spatial samples

Abstract. Methods for coupling two data sets (species composition and environmental variables for example) are well known and often used in ecology. All these methods require that variables of the two data sets have been recorded at the same sample stations. But if the two data sets arise from different sample schemes, sample locations can be different. In this case, scientists usually transform one data set to conform with the other one that is chosen as a reference. This inevitably leads to some loss of information. We propose a new ordination method, named spatial‐RLQ analysis, for coupling two data sets with different spatial sample techniques. Spatial‐RLQ analysis is an extension of co‐inertia analysis and is based on neighbourhood graph theory and classical RLQ analysis. This analysis finds linear combinations of variables of the two data sets which maximize the spatial cross‐covariance. This provides a co‐ordination of the two data sets according to their spatial relationships. A vegetation study concerning the forest of Chizé (western France) is presented to illustrate the method.     

On the variation explained by ordination and constrained ordination axes

Abstract. Total inertia (TI), the sum of eigenvalues for all ordination axes, is often used as a measure of total variation in a data set. By use of simulated data sets, I demonstrate that lack-of-fit of data to the response model implicit in any eigenvector ordination method results in polynomial distortion ordination axes, with eigenvalues that normally contribute 30–70% to TI (depending on data set properties). The amount of compositional variation extracted on ecologically interpretable ordination axes (structure axes) is thus underestimated by the eigenvalue-to-total-inertia ratio. I recommend that the current use of total inertia as a measure of compositional variation is discontinued. Eigenvalues of structure axes can, however, be used with some caution to indicate their relative importance. I also demonstrate that when the total inertia is partitioned on different sets of explanatory variables and unexplained variation by use of (partial) constrained ordination, (35) 50–85% of the variation ‘unexplained’ by the supplied explanatory variables represents lack-of-fit of data to model. Thus, the common interpretation of ‘unexplained variation’ as random variation (‘noise’) or coenoclinal variation caused by unmeasured explanatory variables, is generally inappropriate. I recommend a change of focus from the variation-explained-to-total inertia ratio and ‘unexplained’ variation to relative amounts of variation explained by different sets of explanatory variables.     

A SOM projection technique with the growing structure for visualizing high-dimensional data

The Self-Organizing Map (SOM) is an efficient tool for visualizing high-dimensional data. In this paper, an intuitive and effective SOM projection method is proposed for mapping high-dimensional data onto the two-dimensional grid structure with a growing self-organizing mechanism. In the learning phase, a growing SOM is trained and the growing cell structure is used as the baseline framework. In the ordination phase, the new projection method is used to map the input vector so that the input data is mapped to the structure of the SOM without having to plot the weight values, resulting in easy visualization of the data. The projection method is demonstrated on four different data sets, including a 118 patent data set and a 399 checical abstract data set related to polymer cements, with promising results and a significantly reduced network size.     

Partitioning the variation in a plot‐by‐species data matrix that is related to n sets of explanatory variables

Abstract. Variation partitioning by (partial) constrained ordination is a popular method for exploratory data analysis, but applications are mostly restricted to simple ecological questions only involving two or three sets of explanatory variables, such as climate and soil, this because of the rapid increase in complexity of calculations and results with an increasing number of explanatory variable sets. The existence is demonstrated of a unique algorithm for partitioning the variation in a set of response variables on n sets of explanatory variables; it is shown how the 2n– 1 non‐overlapping components of variation can be calculated. Methods for evaluation and presentation of variation partitioning results are reviewed, and a recursive algorithm is proposed for distributing the many small components of variation over simpler components. Several issues related to the use and usefulness of variation partitioning with n sets of explanatory variables are discussed with reference to a worked example.     

Herb-rich spruce forests in W Nordland, N Norway: an ecological and methodological study

The vegetation of herb-rich spruce forests in three localities in Brønnøy municipality, W Nordland, N Norway, has been analysed using 120 sample plots, each 25 m², distributed by a restricted random method. In connection with every sample plot a set of ecological variables have been measured. The most important gradients for the differentiation of the vegetation were identified by DCA ordination and statistical analysis of the vegetational and the ecological data sets. The gradients were: (1) the nutrient gradient, (2) the soil moisture gradient and (3) the microclimate gradient. The importance of choice of ordination technique (DCA or LNMDS) relative to the importance of the choice of some parameters in DCA and LNMDS has been evaluated. Indicating from this evaluation were (1) the choice of weighting function prior to DCA ordination can be as important as the choice of ordination technique when the data set is small; (2) choice of dimensionality in LNMDS is normally not as decisive for the ordination result as the choice of ordination technique and (3) when the data set is larger, the choice of scale range is less decisive for the ordination result than the choice of dimensionality in LNMDS.     

The bootstrapped ordination re-examined

Abstract. A method is described to determine the number of significant dimensions in metric ordination of a sample. The method is probabilistic, based on bootstrap resampling. An iterative algorithm takes bootstrap samples with replacement from the sample. It finds in each bootstrap sample ordination coordinates and computes, after Procrustean adjustments, the correlation between observed and bootstrap ordination scores. It compares this correlation to the same parameter generated in a parallel bootstrapped ordination of randomly permuted data, which upon many iterations will generate a probability. The method is assessed in principal coordinates analysis of simulated data sets that have varying number of variables and correlation levels, uniform or patterned correlation structure. The results suggest the method is more reliable than other available methods in recovering the true intrinsic dimensionality. Examples with grassland data illustrate utility.     

Evolution and character congruence in some western Indian Ocean Phelsuma: Numerical analysis of biochemistry, shape and scalation

The evolution of five island populations of Green gecko, representing inter- and intra-specific divergence, was studied using biochemical data, scalation and shape. The data were numerically analysed using ordination analyses for the phenetic classification and Wagner trees to hypothesize the phylogeny. These studies revealed three phenetic groups corresponding to three mono-phyletic lineages. The numerical analysis of morphological data agreed with the numerical analysis of biochemical data. It is concluded that the classification based on biochemical affinities differed from the previous classification based on conventional analysis of morphology due to methodological and philosophical differences rather than differences between morphological and biochemical evolution.
The ordination analyses were very congruent between data sets (biochemical, shape, scalation, total) and the Wagner trees were generally congruent between data sets. Some Wagner trees based on scalation data were incongruent. The phenetic and cladistic classifications corresponded to each other but differed from the conventional classification. The phylogenetic analysis of the total data set indicated that the three specific lineages showed relatively equal anagenesis. However, anagenic divergence differed markedly between character types. It is suggested that a range of character types be used when studying anagenesis.     

Local and global multivariate analysis of geographical mitochondrial DNA variation in Leuciscus cephalus L. 1758 (Pisces: Cyprinidae) in the Balkan Peninsula

Geographical variation of cytochrome b mitochondrial DNA (mtDNA) in chub (Leuciscus cephalus L.) was analysed in 31 samples from 20 different river basins in the Balkan Peninsula and Danube catchment. Multivariate methods of ordination were used to analyse variation of the data sets. The results were interpreted in the context of the proposed ichthyogcographic districts separating the Balkan Peninsula into two main ichthyogeographic divisions (Eastern Greece/Ponto-Acgcan and Western Greece/South Adriatic-Ionian). Boundary detection supported these two ichthyogeographic districts for L. cephalus , revealing a boundary that ran from north to south through the Balkan Peninsula and the middle of Greece. The results also revealed the existence of a third division in Central Greece. The results of ordination techniques on homogeneous zones and analysis of the molecular variance confirmed the results obtained in studying local variability (boundaries). They also allowed us to test the existence of possible subdivisions proposed by different authors inside the two main ichthyogeographic districts. These subdivisions were not supported. The multivariate methods used in this study allowed us to propose a coherent picture of chub ichthyogeographic districts in terms of boundary detection and maximal autocorrelation between populations and to explain the patterns of chub mtDNA variation. A complete interpretation of results concerning L. cephalus requires careful consideration of both boundary analysis and autocorrelative approach. Results from an the autocorrclativc approach alone could lead to substantial misinterpretations.     

Ordination on the basis of fuzzy set theory

Fuzzy set theory is an extension of classical set theory where elements of a set have grades of membership ranging from zero for non-membership to one for full membership. Exactly as for classical sets, there exist operators, relations, and mappings appropriate for these fuzzy sets. This paper presents the concepts of fuzzy sets, operations, relations, and mappings in an ecological context. Fuzzy set theory is then established as a theoretical basis for ordination, and is employed in a sequence of examples in an analysis of forest vegetation of western Montana, U.S.A. The example ordinations show how site characteristics can be analyzed for their effect on vegetation composition, and how different site factors can be synthesized into complex environmental factors using the calculus of fuzzy set theory.In contrast to current ordination methods, ordinations based on fuzzy set theory require the investigator to hypothesize an ecological relationship between vegetation and environment, or between different vegatation compositions, before constructing the ordination. The plotted ordination is then viewed as evidence to corroborate or discredit the hypothesis.I am grateful to Dr R. D. Pfister (formerly USDA Forest Service) for kind permission to publish data from a Forest Service study.I would like to gratefully acknowledge the helpful comments and criticisms of Drs. G. Cottam, J. D. Aber, T. F. H. Allen, E. W. Beals, I. C. Prentice, C. G. Lorimer, and two anonymous reviewers.Taxonomic nomenclature follows Hitchcock & Cronquist (1973).I would like to thank the Dean of the College of Letters and Sciences, University of Wisconsin—Madison, for a fellowship which supported this research, and the Department of Botany for computer funds to perform the analyses.     

A phytoecological study of the mire Northern Kisselbergmosen, SE Norway. II. Identification of gradients by detrended (canonical) correspondence analysis

The vegetation of a poor mire is sampled by two procedures; 800 randomly placed sample plots made up the R data set, 765 subjectively selected plots in 153 sample plot series made up the S data set. DCA ordination and constrained ordination by DCCA of the data sets and subsets showed the existence of three coenoclines in the material: (1) the coenocline along the mire expanse: low to high median depth to the water table—mire margin gradient, (2) the poor-rich coenocline, dependent on a complex-gradient in substrate chemistry, and (3) a coenocline attributed to variation in peat productivity. Thus the assumption of Fennoscandian mire scientists embedded in numerous systems for classifying mire vegetation, that three gradients are the most important in the mire ecosystem, is partly confirmed. In the investigated area, two of the gradients normally considered make up one complex coenocline (1), and a fourth coenocline (3) has to be added. The effects of sampling techniques on correlations between coenoclines and on ordination results are discussed, and an improved sampling technique is suggested. The major faults of DCA: (1) the tongue effect, and (2) the instability, are described and discussed. It is concluded that if due attention is taken to reveal effects of the faults of the method, DCA is among the best ordination methods currently available.     

Towards unification of national vegetation classifications: A comparison of two methods for analysis of large data sets

Abstract. In European phytosociology, national classifications of corresponding vegetation types show considerable differences even between neighbouring countries. Therefore, the European Vegetation Survey project urgently needs numerical classification methods for large data sets that are able to produce compatible classifications using data sets from different countries. We tested the ability of two methods, TWINSPAN and COCKTAIL, to produce similar classifications of wet meadows (Calthion, incl. Filipendulenion) for Germany (7909 relevés) and the Czech Republic (1287 relevés) in this respect. In TWINSPAN, the indicator ordination option was used for classification of two national data sets, and the extracted assignment criteria (indicator species) were applied crosswise from one to the other national data set. Although the data sets presumably contained similar community types, TWINSPAN revealed almost no correspondence between the groups derived from the proper classification of the national data set and the groups defined by the assignment criteria taken from the other national data set. The reason is probably the difference in structure between the national data sets, which is a typical, but hardly avoidable, feature of any pair of phytosociological data sets. As a result, the first axis of the correspondence analysis, and consequently the first TWINSPAN division, are associated with different environmental gradients; the difference in the first division is transferred and multiplied further down the hierarchy. COCKTAIL is a method which produces relevé groups on the basis of statistically formed species groups. The user determines the starting points for the formation of species groups, and groups already found in one data set can be tested for existence in the other data set. The correspondence between the national classifications produced by COCKTAIL was fairly good. For some relevé groups, the lack of correspondence to groups in the other national data set could be explained by the absence of the corresponding vegetation types in one of the countries, rather than by methodological problems.     

Vegetation changes along a precipitation gradient in Central Argentina

Changes in vegetation along a precipitation gradient in Central Argentina were studied. Floristic samples were taken along an east-west transect of about 300 km. Correlation analysis between precipitation and ordination axes was used to provide an environmental interpretation of vegetation variability.Floristic analysis produced an ordination of plant communities from evergreen forests (precipitation >500 mm) to desert shrublands and therophyte communities (precipitation <200 mm). Results showed a trend of floristic and structural impoverishment towards the west. There is a replacement of species along the transect and a shift in dominant growth forms. The first ordination axis is significantly, negatively correlated with annual precipitation.     

Reconstructing community relationships: the impact of sampling error, ordination approach, and gradient length

Effectively summarizing complex community relationships is an important feature in studies such as biodiversity, global change, and invasion ecology. The reliability of such community summaries depends on the degree of sampling variability that is present in the data, the structure of the data, and the choice of ordination method, but the relative importance of these factors is not understood. We compared the validity of results from different ordination methods by applying five levels of sampling error to a simulated coenoplane model at two gradient lengths using two types of data (abundance and presence–absence). The multivariate methods we compared were correspondence analysis (CA), detrended correspondence analysis (DCA), non-metric multidimensional scaling (NMDS), principal component analysis (PCA) and principal coordinates analysis (PCoA). Our results showed CA and PCA using presence–absence data were the most successful methods regardless of sampling error and gradient length, closely followed by the other methods using presence–absence data. With abundance data, PCA and CA were the most successful approaches with the short and long gradients, respectively. Approaches based on PCoA and NMDS using abundance data did not perform well regardless of the choice of distance measure used in the analysis. Both of these methods, along with the PCA using abundance data, were strongly affected by the longer gradient, leading to more distorted results.     

A comparison of detrended correspondence analysis and principal co-ordinates analysis using four sets of Tasmanian vegetation data

Indirect gradient analysis, which entails the elucidation of relationships between trends in community composition and underlying environmental or successional gradients, is a major objective of ordination in plant ecology. Two ordination techniques, detrended correspondence analysis (DCA) and principal co-ordinates analysis (PCOA), were compared using three sets of Tasmanian vegetation data having known gradients and one set where the vegetation was expected to respond to diverse environmental variables. In every case, the results obtained by DCA were considered superior to, or at least as good as, those of PCOA. Hence, DCA appears to be the more suitable of the two methods for indirect gradient analysis.     

Detection of multi-species patterns in plant populations

An analytical method for the detection of multi-species spatial patterns in grasslands was investigated. Several data sets of grasslands from the granitic pediment of the Sierra de Guadarrama (Central Spain) were used. The application of correspondence analysis to sequential abundance data of several species allowed the ordination of quadrats in an axis of floristic variation. Coordinate data were then subjected to pattern analysis through the use of variance tests, the results showing the existence of multi-species patches having variable dimensions.     

The effects of non-homology in RAPD bands on similarity and multivariate statistical ordination in Brassica and Helianthus

 In order to estimate the impact of mis-coding non-homologous, co-migrating DNA bands as homologous, two sets of data were utilized. Analyses were conducted using three Helianthus species in which each co-migrating band had previously been confirmed. Comparisons of the similarities between these three Helianthus species using the original 177 RAPD bands and the corrected, homology verified, 197 RAPD band data set revealed that the triangular relationship among these three species was almost identical in both data sets. The non-homology errors in the Helianthanus data sets were found to be random. These random errors merely reduced the absolute similarities, but not the relative similarities nor the relationships among the taxa, in principal-coordinate-analysis ordination. Analyses of RAPDs for the classical Brassica U triangle were made by inserting random non-homologies for 5, 10, 15 and 20% of the original 220 RAPD bands. These analyses revealed a progressive decrease in similarities and less loading on the first two axes in principal coordinate analysis (PCO). However, the basic U triangle of relationships among these six Brassica species was maintained. It appears that if errors in homology of co-migrating DNA bands are random, this will have little effect on the relative similarities and on PCO ordination. This helps explain the successful use of RAPDs at the specific level. Received: 6 December 1997 / Accepted: 11 December 1997     

Theoretical habitat templets, species traits, and species richness: aquatic macrophytes in the Upper Rhône River and its floodplain

• 1 The objective of this study, which is based on forty-two species of hydrophytes and helophytes, is to investigate: (i) relationships among species traits; (ii) habitat utilization by species; (iii) the relationship between species traits and habitat utilization; (iv) trends in species traits in the framework of spatial–temporal habitat variability, and if trends match predictions from the river habitat templet; and (v) trends in species richness in the framework of spatial–temporal habitat variability, and if trends match predictions of the patch dynamics concept.
• 2 Two data sets were used for this analysis: species traits (mainly reproductive and morphological characteristics) were documented from the literature; and species distribution across eight habitat types was from field surveys conducted in the floodplain of the Upper Rhone River, France. This information was structured by a fuzzy coding technique and analysed by ordination methods.
• 3 Several species traits, which are related to disturbances and reflect resistance (e.g. attachment to soil or substrate) or resilience (e.g. potential for regeneration of an individual), are closely related for aquatic macrophytes.
• 4 Habitat utilization by aquatic macrophytes separates the habitat types along a gradient of connectivity with the main channel, which corresponds to a gradient in flood disturbance frequency and the permanence of the different water-bodies.
• 5 The relationship between species traits and habitat utilization is highly significant, indicating that a particular set of habitat types is used by taxa with a particular set of species trait modalities.
• 6 Observations in one habitat templet (in which scaling of the templet is primarily based on water level fluctuations for the temporal variability axis and on substrate characteristics for the spatial variability axis) generally do not support predictions on trends in species traits but do support predictions on trends in species richness.
• 7 Observations in an alternative habitat templet (in which scaling of the templet is based on frequency of flood scouring for the temporal variability axis and on heterogeneity of the substrate for the spatial variability axis) support theoretical predictions on trends for about half of the species traits for which predictions were available. However, trends in species richness in this alternative habitat templet are only partly in agreement with predictions.

     

Bayesian multimodel inference for geostatistical regression models

The problem of simultaneous covariate selection and parameter inference for spatial regression models is considered. Previous research has shown that failure to take spatial correlation into account can influence the outcome of standard model selection methods. A Markov chain Monte Carlo (MCMC) method is investigated for the calculation of parameter estimates and posterior model probabilities for spatial regression models. The method can accommodate normal and non-normal response data and a large number of covariates. Thus the method is very flexible and can be used to fit spatial linear models, spatial linear mixed models, and spatial generalized linear mixed models (GLMMs). The Bayesian MCMC method also allows a priori unequal weighting of covariates, which is not possible with many model selection methods such as Akaike's information criterion (AIC). The proposed method is demonstrated on two data sets. The first is the whiptail lizard data set which has been previously analyzed by other researchers investigating model selection methods. Our results confirmed the previous analysis suggesting that sandy soil and ant abundance were strongly associated with lizard abundance. The second data set concerned pollution tolerant fish abundance in relation to several environmental factors. Results indicate that abundance is positively related to Strahler stream order and a habitat quality index. Abundance is negatively related to percent watershed disturbance.