Hypothesis-testing methods for multivariate data are needed to make rigorous probability statements about the effects of factors and their interactions in experiments. Analysis of variance is particularly powerful for the analysis of univariate data. The traditional multivariate analogues, however, are too stringent in their assumptions for most ecological multivariate data sets. Non-parametric methods, based on permutation tests, are preferable. This paper describes a new non-parametric method for multivariate analysis of variance, after McArdle and Anderson (in press). It is given here, with several applications in ecology, to provide an alternative and perhaps more intuitive formulation for ANOVA (based on sums of squared distances) to complement the description pro- vided by McArdle and Anderson (in press) for the analysis of any linear model. It is an improvement on previous non-parametric methods because it allows a direct additive partitioning of variation for complex models. It does this while maintaining the flexibility and lack of formal assumptions of other non-parametric methods. The test- statistic is a multivariate analogue to Fisher's F-ratio and is calculated directly from any symmetric distance or dissimilarity matrix. P-values are then obtained using permutations. Some examples of the method are given for tests involving several factors, including factorial and hierarchical (nested) designs and tests of interactions.

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A new method for non-parametric multivariate analysis of variance

We propose a method (the ‘gap statistic’) for estimating the number of clusters (groups) in a set of data. The technique uses the output of any clustering algorithm (e.g. K-means or hierarchical), comparing the change in within-cluster dispersion with that expected under an appropriate reference null distribution. Some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature.

Estimating the number of clusters in a data set via the gap statistic

Estimating the number of clusters in a dataset via the gap statistic

A Monte Carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2, 3, 4, or 5 distinct nonoverlapping clusters. To provide a variety of clustering solutions, the data sets were analyzed by four hierarchical clustering methods. External criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level. Thus, the clustering present in the data was quite strong. The simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data. Several procedures worked fairly well, whereas others performed rather poorly. Thus, the latter group of rules would appear to have little validity, particularly for data sets containing distinct clusters. Applied researchers are urged to select one or more of the better criteria. However, users are cautioned that the performance of some of the criteria may be data dependent.

An examination of procedures for determining the number of clusters in a data set

SYSTAT provides a variety of cluster analysis methods on rectangular or symmetric data matrices. Cluster analysis is a multivariate procedure for detecting natural groupings in data. It resembles discriminant analysis in one respect—the researcher seeks to classify a set of objects into subgroups although neither the number nor members of the subgroups are known. CLUSTER provides three procedures for clustering: Hierarchical Clustering, K-Clustering, and Additive Trees. The Hierarchical Clustering procedure comprises hierarchical linkage methods. The K-Clustering procedure splits a set of objects into a selected number of groups by maximizing between-cluster variation and minimizing within-cluster variation. The Additive Trees Clustering procedure produces a Sattath-Tversky additive tree clustering. Hierarchical Clustering clusters cases, variables, or both cases and variables simultaneously; K-Clustering clusters cases only; and Additive Trees clusters a similarity or dissimilarity matrix. Several distance metrics are available with Hierarchical Clustering and K-Clustering including metrics for binary, quantitative and frequency count data. Hierarchical Clustering has ten methods for linking clusters and displays the results as a tree (dendrogram) or a polar dendrogram. When the MATRIX option is used to cluster cases and variables, SYSTAT uses a gray-scale or color spectrum to represent the values. SYSTAT further provides five indices, viz., statistical criteria by which an appropriate number of clusters can be chosen from the Hierarchical Tree. Options for cutting (or pruning) and coloring the hierarchical tree are also provided. In the K-Clustering procedure SYSTAT offers two algorithms, KMEANS and KMEDIANS, for partitioning. Further, SYSTAT provides nine methods for selecting initial seeds for both KMEANS and KMEDIANS. Cluster analysis is a multivariate procedure for detecting groupings in data. The objects in these groups may be: Cases (observations or rows of a rectangular data file). For example, suppose health indicators (numbers of doctors, nurses, hospital beds, life expectancy, etc.) are recorded for countries (cases), then developed nations may form a subgroup or cluster separate from developing countries. Variables (characteristics or columns of the data). For example, suppose causes of death (cancer, cardiovascular, lung disease, diabetes, accidents, etc.) are recorded for each U.S. state (case); the results show that accidents are relatively independent of the illnesses. Cases and variables (individual entries in the data matrix). For example, certain wines are associated with good years of production. Other wines have other years that are better. Clusters may be of two sorts: overlapping or exclusive. Overlapping clusters allow the same object to appear in more than one …

Cluster analysis

A method for identifying clusters of points in a multidimensional Euclidean space is described and its application to taxonomy considered. It reconciles, in a sense, two different approaches to the investigation of the spatial relationships between the points, viz., the agglomerative and the divisive methods. A graph, the shortest dendrite of Florek etal. (1951a), is constructed on a nearest neighbour basis and then divided into clusters by applying the criterion of minimum within cluster sum of squares. This procedure ensures an effective reduction of the number of possible splits. The method may be applied to a dichotomous division, but is perfectly suitable also for a global division into any number of clusters. An informal indicator of the "best number" of clusters is suggested. It is a"variance ratio criterion" giving some insight into the structure of the points. The method is illustrated by three examples, one of which is original. The results obtained by the dendrite method are compared with those...

Harabasz J

Papers

A dendrite method for cluster analysis