János Podani

RANDOMIZATION OF PRESENCE–ABSENCE MATRICES: COMMENTS AND NEW ALGORITHMS

Randomization of presence–absence data matrices with fixed row and column totals is an important tool in ecological research wherever the significance of data-based statistics (e.g., species association measures) is to be evaluated. In the current literature of numerical ecology, however, there has been no algorithm that randomizes moderately large matrices in short time such that equidistribution of results is guaranteed. We show that a simple modification of the swap algorithm, called here the “trial-swap method,” satisfies the requirement for equidistribution. Since this is relatively slow, we suggest two fast algorithms that, combined with the trial-swap method, produce all possible results with equal chance. The three procedures are illustrated using actual examples taken from bird biogeography and vegetation ecology.

EXTENDING GOWER'S GENERAL COEFFICIENT OF SIMILARITY TO ORDINAL CHARACTERS

Summary The possibilities of calculating similarity based on ordinal characters are evaluated by distinguishing subtypes of the ordinal scale. Multivariate analysis is most problematic when ordinal variables appear together with other scale types in the data. This difficulty is solved by extending Gowers general coefficient of similarity to ordinal data types, facilitating cluster analysis and multidimensional scaling. Two alternatives, a non-metric and a metric version, are offered. The modified formula implies that ordinal variables are equally weighted with the others, and that partially and fully ranked data are both applicable, due to the inherent standardisation procedure. A morphological data set derived for the moss genus Tortula illustrates the new approach.

On sampling procedures in population and community ecology

In this paper we emphasize that sampling decisions in population and community ecology are context dependent. Thus, the selection of an appropriate sampling procedure should follow directly from considerations of the objectives of an investigation. We recognize eight sampling alternatives, which arise as a result of three basic dichotomies: parameter estimation versus pattern detection, univariate versus multivariate, and a discrete versus continuous sampling universe. These eight alternative sampling procedures are discussed as they relate to decisions regarding the required empirical sample size, the selection or arrangement of sampling units, and plot size and shape. Our results indicate that the decision-making process in sampling must be viewed as a flexible exercise, dictated not by generalized recommendations but by specific objectives: there is no panacea in ecological sampling. We also point to a number of unresolved sampling problems in ecology.

Braun-Blanquet's legacy and data analysis in vegetation science

Abstract This article investigates whether the Braun-Blanquet abundance/dominance (AD) scores that commonly appear in phytosociological tables can properly be analysed by conventional multivariate analysis methods such as Principal Components Analysis and Correspondence Analysis. The answer is a definite NO. The source of problems is that the AD values express species performance on a scale, namely the ordinal scale, on which differences are not interpretable. There are several arguments suggesting that no matter which methods have been preferred in contemporary numerical syntaxonomy and why, ordinal data should be treated in an ordinal way. In addition to the inadmissibility of arithmetic operations with the AD scores, these arguments include interpretability of dissimilarities derived from ordinal data, consistency of all steps throughout the analysis and universality of the method which enables simultaneous treatment of various measurement scales. All the ordination methods that are commonly used, for example, Principal Components Analysis and all variants of Correspondence Analysis as well as standard cluster analyses such as Wards method and group average clustering, are inappropriate when using AD data. Therefore, the application of ordinal clustering and scaling methods to traditional phytosociological data is advocated. Dissimilarities between relevés should be calculated using ordinal measures of resemblance, and ordination and clustering algorithms should also be ordinal in nature. A good ordination example is Non-metric Multidimensional Scaling (NMDS) as long as it is calculated from an ordinal dissimilarity measure such as the Goodman & Kruskal γ coefficient, and for clustering the new OrdClAn-H and OrdClAn-N methods. Abbreviations: AD = Abundance/dominance; NMDS = Non-metric Multidimensional Scaling.

A measure for assessing functional diversity in ecological communities

Functional diversity is regarded as a key in understanding the link between ecosystem function and biodiversity, but its measurement is rather problematic. The two widely used continuous measures are the dendrogram-based measure (DBM) and the functional attribute diversity (FAD). In contrast to DBM, FAD does not require the knowledge of the entire species pool before the analysis, and hence FAD is a more ideal tool for measuring functional diversity. However, the original form of FAD and its variants have several undesirable properties. Here, we suggest a modified FAD (denoted by MFAD), which—as illustrated by artificial and actual data sets—allows calculating functional diversity without violating the twinning and monotonicity criteria such that the number of species collected is compensated for. These requirements are met by replacing the original species by so-called functional species and then by dividing FAD by the number of functional units. Accordingly, MFAD measures the dispersion of species in the functional traits space so that MFAD values for different communities can directly be compared if the same set of functional traits is used. Finally, using data of two freshwater communities (caddisfly and riverine fish), we evaluate the change of species richness and functional diversity in relation to sampling effort (sample unit size). We found that functional diversity is a better and more reliable community descriptor than species richness in a sense that it converges to the maximum faster in the function of sampling effort.

Multivariate exploratory analysis of ordinal data in ecology: Pitfalls, problems and solutions

Abstract Questions: Are ordinal data appropriately treated by multivariate methods in numerical ecology? If not, what are the most common mistakes? Which dissimilarity coefficients, ordination and classification methods are best suited to ordinal data? Should we worry about such problems at all? Methods: A new classification model family, OrdClAn (Ordinal Cluster Analysis), is suggested for hierarchical and non-hierarchical classifications from ordinal ecological data, e.g. the abundance/dominance scores that are commonly recorded in relevés. During the clustering process, the objects are grouped so as to minimize a measure calculated from the ranks of within-cluster and between-cluster distances or dissimilarities. Results and Conclusions: Evaluation of the various steps of exploratory data analysis of ordinal ecological data shows that consistency of methodology throughout the study is of primary importance. In an optimal situation, each methodological step is order invariant. This property ensures that the results are independent of changes not affecting ordinal relationships, and guarantees that no illusory precision is introduced into the analysis. However, the multivariate procedures that are most commonly applied in numerical ecology do not satisfy these requirements and are therefore not recommended. For example, it is inappropriate to analyse Braun-Blanquet abudance/dominance data by methods assuming that Euclidean distance is meaningful. The solution of all problems is that the dissimilarity coefficient should be compatible with ordinal variables and the subsequent ordination or clustering method should consider only the rank order of dissimilarities. A range of artificial data sets exemplifying different subtypes of ordinal variables, e.g. indicator values or species scores from relevés, illustrate the advocated approach. Detailed analyses of an actual phytosociological data set demonstrate the classification by OrdClAn of relevés and species and the subsequent tabular rearrangement, in a numerical study remaining within the ordinal domain from the first step to the last. Abbreviations: AD = Abundance/Dominance; CL = Complete Link; DC = Coefficient of Discordance; ED = Euclidean distance; O = Ordinal; M = Metric; NMDS = Non-metric Multidimensional Scaling; OC = Ordinal Clustering; SL = Single Link; UPGMA = Unweighted Pair Group Method or Group Average Clustering.

RESEMBLANCE COEFFICIENTS AND THE HORSESHOE EFFECT IN PRINCIPAL COORDINATES ANALYSIS

Although principal coordinates analysis is one of the most widely used or- dination methods in ecology, no study had been undertaken as yet on the combined effect of gradient type and resemblance coefficient on the results. We examine the performance of principal coordinates analysis with different choices of the resemblance function and different types of a single underlying gradient. Whereas unimodal species response to long gradients always leads to horseshoe (or arch)-shaped configurations in the first two dimensions, the converse is not true; curvilinear arrangements cannot generally be explained by the Gaussian model. Several resemblance coefficients widely used in ecology produce paradoxical arches from perfectly linear data. Species richness changes alone may also lead to a horseshoe for even more distance functions, with the noted exception of Manhattan metric. The appearance of arches is a mathematical necessity in these cases; true artifacts are introduced only if distances are treated inappropriately before eigenanalysis. Examples illustrate that similar configurations (curves and even circles) may arise from very different data structures; there- fore the shape of the point scatter is insufficient by itself to identify background ecological phenomena. The horseshoe effect may be diminished and eigenvalue extraction may be made more efficient if input measures are raised to high powers; but this operation is recommended only in combination with standard analyses, as part of a comparative approach. We derive a new distance function, implying standardization by species totals, from the chi-square dis- tance. We found that this function improves gradient recovery when there is unimodal species response and some species have their optima outside the range of study.

The effect of the expansion of the clonal grass Calamagrostis epigejos on the species turnover of a semi-arid grassland

ABSTRACT Question: How does the dominance of Calamagrostis epigejos influence species turnover of a grassland? Location: Loess grassland at the foothills of Bükk Mountains, Hungary (47°54′ N, 20°35′ E). Methods: Presence/absence of vascular plants and different performance attributes of C. epigejos were recorded in a plot-subplot system between 2002 and 2005. Appearance and disappearance rates of grassland species were calculated for pairs of consecutive years. 1. Mean appearance and disappearance rates were compared in grassland plots dominated by C. epigejos and in plots free from this species, based on Monte Carlo randomization. 2. Mean appearance rates were assessed for categories of C. epigejos performance and their confidence intervals were calculated via Monte Carlo randomization. For two performance variables (percentage cover and shoot number) analyses were performed at two spatial scales. Results: 1. C. epigejos-dominated plots differed from unaffected ones by significantly lower appearance rates. 2. Change in appearance rates was best explained by differences in percentage cover of C. epigejos. Coarse-scale C. epigejos performance had a closer correspondence with appearance rate change than fine-scale performance. Low level C. epigejos performance enhanced appearance rate compared to intact stands, while high level performance decreased it, regardless of the choice of performance measure. Conclusions: C. epigejos lowers species number by hindering reappearance of species of the original grassland. This is best explained by the increased shading effect at the coarse scale. The marked non-linear initial enhancement in appearance rate, however, can also be taken as an early sign of future species loss. Nomenclature: Tutin et al. (1964–1993).

New combinatorial clustering methods

Sixteen clustering methods are compatible with the general recurrence equation of combinatorial SAHN (sequential, agglomerative, hierarchical and nonoverlapping) classificatory strategies. These are subdivided into two classes: the d-SAHN methods seek for minimal between-cluster distances the h-SAHN strategies for maximal within-cluster homogeneity. The parameters and some basic features of all combinatorial methods are listed to allow comparisons between these two families of clustering procedures. Interest is centred on the h-SAHN techniques; the derivation of updating parameters is presented and the monotonicity properties are examined. Three new strategies are described, a weighted and an unweighted variant of the minimization of the increase of average distance within clusters and a homogeneity-optimizing flexible method. The performance of d- and h-SAHN techniques is compared using field data from the rock grassland communities of the Sashegy Nature Reserve, Budapest, Hungary.

Simulation of Random Dendrograms and Comparison Tests: Some Comments

quasi-ultrametrics because they do not satisfy the identity axiom. The fifth descriptor considered is path difference which is not recommended for comparisons except for unrooted trees. Correlations among dendrogram descriptors are evaluated through simulation experiments, and it is shown that the significance of dendrogram comparisons is greatly influenced by the choice of the descriptor. The paper emphasizes that choice of the underlying tree distribution to be used as a reference in testing significance of a dendrogram comparison measure should be consistent with the descriptor incorporated by that measure.