Approaches to determining the number of components to interpret from principal components analysis were compared. Heuristic procedures included: retaining components with eigenvalues (λ) > 1 (i.e., Kaiser—Guttman criterion); components with bootstrapped λ > 1 (bootstrapped Kaiser—Guttman); the scree plot; the broken—stick model; and components with λ totalling to a fixed amount of the total variance. Statistical approaches included: Bartlett's test of sphericity; Bartlett's test of homogeneity of the correlation matrix, Lawley's test of the second λ bootstrapped confidence limits on successive λ (i.e., significant differences between λ); and bootstrapped confidence limits on eigenvector coefficients (i.e., coefficients that differ significantly from zero). All methods were compared using simulated data matrices of uniform correlation structure, patterned matrices of varying correlation structure and data sets of lake morphometry, water chemistry, and benthic invertebrate abundance. The most consistent results were obtained from the broken—stick model and a combined measure using bootstrapped λ and associated eigenvector coefficients. The traditional and bootstrapped Kaiser—Guttman approaches over—estimated the number of nontrivial dimensions as did the fixed—amount—of—variance model. The scree plot consistently estimated one dimension more than the number of simulated dimensions. Barlett's test of sphericity showed inconsistent results. Both Bartlett's test of homogeneity of the correlation matrix and Lawley's test are limited to testing for only one and two dimensions, respectively.

We examine evidence for the structuring of fish communities from stream and lake systems and the roles of biotic, abiotic, and spatial factors in determining the species composition. Piscivory by fish is a dominant factor in both stream and lake systems whereas evidence for the importance of competition appears less convincing. Within small streams or lakes, the impact of predation may exclude other species, thereby leading to mutually exclusive distributions and strong differences in community composition. Within a geographic region, abiotic effects frequently dictate the relative importance of piscivory, thereby indirectly influencing the composition of prey species present. The spatial scale of studies influences our perceived importance of biotic versus abiotic factors, with small-scale studies indicating a greater importance of competition and large-scale studies emphasizing abiotic controls. The scale of the individual sites considered is critical because smaller systems have higher variability and wider extremes of conditions than larger lakes and rivers. The stability of physical systems and degree of spatial connectivity contribute to increased diversity in both larger stream and larger lake systems. We identify challenges and needs that must be addressed both to advance the field of fish community ecology and to face the problems associated with human-induced changes.

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The Mantel test provides a means to test the association between distance matrices and has been widely used in ecological and evolutionary studies. Recently, another permutation test based on a Procrustes statistic (PROTEST) was developed to compare multivariate data sets. Our study contrasts the effectiveness, in terms of power and type I error rates, of the Mantel test and PROTEST. We illustrate the application of Procrustes superimposition to visually examine the concordance of observations for each dimension separately and how to conduct hypothesis testing in which the association between two data sets is tested while controlling for the variation related to other sources of data. Our simulation results show that PROTEST is as powerful or more powerful than the Mantel test for detecting matrix association under a variety of possible scenarios. As a result of the increased power of PROTEST and the ability to assess the match for individual observations (not available with the Mantel test), biologists now have an additional and powerful analytical tool to study ecological and evolutionary relationships.

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