Keith M. Somers

How many principal components? stopping rules for determining the number of non-trivial axes revisited

Principal component analysis is one of the most widely applied tools in order to summarize common patterns of variation among variables. Several studies have investigated the ability of individual methods, or compared the performance of a number of methods, in determining the number of components describing common variance of simulated data sets. We identify a number of shortcomings related to these studies and conduct an extensive simulation study where we compare a larger number of rules available and develop some new methods. In total we compare 20 stopping rules and propose a two-step approach that appears to be highly effective. First, a Bartletts test is used to test the significance of the first principal component, indicating whether or not at least two variables share common variation in the entire data set. If significant, a number of different rules can be applied to estimate the number of non-trivial components to be retained. However, the relative merits of these methods depend on whether data contain strongly correlated or uncorrelated variables. We also estimate the number of non-trivial components for a number of field data sets so that we can evaluate the applicability of our conclusions based on simulated data.

GIVING MEANINGFUL INTERPRETATION TO ORDINATION AXES: ASSESSING LOADING SIGNIFICANCE IN PRINCIPAL COMPONENT ANALYSIS

Principal component analysis (PCA) is one of the most commonly used tools in the analysis of ecological data. This method reduces the effective dimensionality of a multivariate data set by producing linear combinations of the original variables (i.e., com- ponents) that summarize the predominant patterns in the data. In order to provide meaningful interpretations for principal components, it is important to determine which variables are associated with particular components. Some data analysts incorrectly test the statistical significance of the correlation between original variables and multivariate scores using standard statistical tables. Others interpret eigenvector coefficients larger than an arbitrary absolute value (e.g., 0.50). Resampling, randomization techniques, and parallel analysis have been applied in a few cases. In this study, we compared the performance of a variety of approaches for assessing the significance of eigenvector coefficients in terms of type I error rates and power. Two novel approaches based on the broken-stick model were also evaluated. We used a variety of simulated scenarios to examine the influence of the number of real dimensions in the data; unique versus complex variables; the magnitude of eigen- vector coefficients; and the number of variables associated with a particular dimension. Our results revealed that bootstrap confidence intervals and a modified bootstrap confidence interval for the broken-stick model proved to be the most reliable techniques.

Similarity Coefficients: Measures of Co-Occurrence and Association or Simply Measures of Occurrence?

Data on the presence or absence of 25 fish species in a survey of 52 lakes from the watersheds of the Black and Hollow rivers of south-central Ontario were analyzed with eight similarity coefficients. Comparisons were made of Jaccard, Ochiai, Phi, Rogers-Tanimoto, Russell and Rao, Simple Matching, Sorensen-Dice, and Yule similarity coefficients using results from R-mode cluster analysis, principal-coordinates analysis (PCoA), and nonmetric multidimensional scaling. Coefficients were grouped into those representing measures of co-occurrence and those measuring association. Coefficients of co-occurrence (i.e., Jaccard, Rogers-Tanimoto, Russell and Rao, Simple Matching, and Sorensen-Dice) incorporate information associated with the frequency of occurrence of the fish species analyzed. Dendrograms faithfully revealed this size effect. Similarly, first axes of PCoA were linear or curvilinear functions of species frequency of occurrence. Measures of association (i.e., Phi and Yule) and Ochiais coefficient were less affected by the frequency of occurrence. The first axes of PCoA, based on centered coefficients (i.e., Phi, Yule, and Ochiai), were highly correlated with the second axes from ordinations using co-occurrence coefficients. The second axes from analyses of centered coefficients were correlated with the third axes based on non-centered measures. We propose that co-occurrence coefficients reflect a general size effect similar to that commonly found in principal-components analysis. Measures of association and Ochiais coefficient incorporate implicit centering transformations that reduce the size influence associated with the frequency of occurrence. Cluster analyses using co-occurrence coefficients are most susceptible to this size effect. We believe that the interpretations of many dendrograms fail to recognize size effects that arise from employing non-centered similarity coefficients (e.g., Strauss 1982; Nemec and Brinkhurst 1987). Additionally, arguments contrasting phenetic and phylogenetic methods may unknowingly debate the utility of centered versus non-centered coefficients, since the size effect undoubtedly contributes to the apparent strength of phylogenetic approaches.

The spectre of 'spurious' correlations

SummaryEcologists often ‘standardize’ data through the use of ratios and indices. Such measures are employed generally to remove a ‘size effect’ induced by some relatively uniteresting variable. The implications of using the resultant data in correlation and regression analyses are poorly recognized. We show that ratios and indices often provide surprising and ‘spurious’ results due to their unusual properties. As a solution, we advocate the use of randomization tests to evaluate hypotheses confounded by ‘spurious’ correlations. In addition, we emphasize that identifying the appropriate null correlation is of utmost importance when statistically evaluating ratios, although this issue is frequently ignored.

Null Models and Fish Communities: Evidence of Nonrandom Patterns

Studies of ecological communities often make implicit assumptions that the species have nonrandom patterns organized through biotic and abiotic factors. Although such assumptions are generally not tested, the analyses and conclusions derived depend on nonrandom patterns being present. Several null or neutral models have been proposed to test for these patterns. We contrast two of the more prevalent models and develop two new models, subsequently evaluating them with five sets of fish community data. Three of the null models provide similar results, from which it is concluded that fish communities from five regions of Ontario are nonrandomly structured. These three models evaluate pairs of species according to departures from null or random co-occurrence expectations. A Monte Carlo model based on the procedure proposed by E. F. Connor and D. Simberloff supports random community organization, but we attribute this discrepancy to the conservative pooling of species-pair information in that model. We recommend a hybrid model combining Monte Carlo and log-linear methods for future studies, although the log-linear model of M. E. Gilpin and J. M. Diamond provides a reasonable approximation. On the basis of species associations derived from the various models, we attribute much of the nonrandom structure to common habitat requirements among co-occurring species. A predominance of positively associated species generally involves pairs of species with similar ecological characteristics. Strong negative associations typically involve predatorprey species. Although competition is often identified as a significant feature in community ecology, we do not believe that competition is a major force structuring these fish communities.

Putting Things in Order: The Ups and Downs of Detrended Correspondence Analysis

Two of the fundamental questions in community ecology are: (1) How do species respond to environmental gradients? and (2) What are the interspecific associations in a community? Such data typically involve sampling many sites and variables (e.g., species, environmental conditions), which are often so numerous as to make simple bivariate analyses unrealistic. As a result, ecologists have adopted and adapted multivariate methods in order to best organize samples on the basis of the site attributes. Methods such as principal components analysis (PCA; see, e.g., Crome and Richards 1988; Schlesinger et al. 1989), correspondence analysis (CA; Blondel et al. 1988; Jackson and Harvey 1989), and nonmetric multidimensional scaling (NMDS; Kenkel and Orloci 1986) are commonly employed. Owing to the heterogeneous nature of ecological data (i.e., high species turnover along a gradient or beta diversity), these multivariate methods frequently produce a horseshoe or an arch configuration in the first two axes of the resulting ordination (Kendall 1971; Hill and Gauch 1980). Wartenberg et al. state that this arch occurs because sites are considered similar due to the corresponding lack of individuals of most species, rather than the presence of members of the same species. This similarity leads to involution, the closeness (in species space) of dissimilar extremes of an environmental gradient (1987, p. 441). Some researchers regard the arch as undesirable, and several approaches are available to remove the arch from ordination analyses (e.g., Phillips 1978; Williamson 1978; Hill and Gauch 1980; Bradfield and Kenkel 1987; ter Braak 1987). Other researchers propose that the arch reflects attributes of the data rather than its being a mathematical artifact (e.g., Noy-Meir and Austin 1970; Swan 1970; Pielou 1984; Beals 1985; Wartenberg et al. 1987). The debate about the origin and implications of the arch and horseshoe remains unresolved. In fact, Peet et al. (1988, p. 926) state that although the CA arch is certainly an inherent property of the method, it is not the same phenomenon as the related PCA horseshoe, as suggested by others (e.g., Pielou 1984; Wartenberg et al. 1987). The most popular method of removing the arch effect from ordinations is detrended correspondence analysis (DCA; Hill and Gauch 1980), a modification of traditional CA. The process of detrending (or removal of the arch) is implemented through the division of the first axis from CA into a number of segments. Within each of these segments, the site scores for the second axis are adjusted by subtracting the within-segment mean on the second axis from the score for each site. As a result, each segment has a mean value of zero for scores on the second axis. This process is repeated several times, and the results are averaged to determine

The Temporal Coherence of Zooplankton Population Abundances in Neighboring North‐Temperate Lakes

We investigated the temporal coherence (i.e., the correlation or synchrony between time series) of annual abundances among populations of freshwater zooplankton in eight lakes in Ontario, Canada, from 1980 to 1992. We estimated temporal coherence using the intraclass correlation coefficient (ri). While values of ri were relatively low among comparisons of all eight lakes, they were statistically significant for three of the seven common cladoceran and copepod taxa (Bosmina longirostris, Leptodiaptomus minutus, and Mesocyclops edax). These significant positive correlations imply that a portion of the interannual variation in abundance was produced by factors operating on a scale larger than the individual lake catchments. Because the eight‐lake analysis might obscure strong, but conflicting, patterns among lakes in the region, we identified homogeneous and temporally coherent subsets of lakes for each species using an exploratory stepwise deletion procedure. The resultant homogeneous subsets exhibited much greater temporal coherence, accounting for 47% (Eubosmina) to 84% (Leptodiaptomus) of the interannual variation in abundance. Our results suggest that the factors affecting annual variation in zooplankton abundance must be sought both within lakes and beyond their watersheds.

Rapid Biological Assessments: How Many Animals Are Enough?

Rapid bioassessment data using counts of benthic macroinvertebrates from the littoral zones of 5 lakes in south-central Ontario were examined to determine if subsamples of 100 animals provided sufficient statistical power to distinguish these lakes. One-way analysis of variance and power analysis were completed using 17 biological indices based on counts of 100, 200, and 300 animals. Despite the common perception that more is better, the doubled or tripled effort required to sort, identify, and enumerate more animals resulted in very little improvement in our ability to distinguish lakes. Eight of the 17 indices were of limited value for separating the 5 lakes. High correlations, indicating redundancies among the best indices, suggested that 1 or 2 metrics would be sufficient to characterize the lakes. Three indices including % amphipods, % insects, and a multivariate metric representing the 1st axis from a correspondence analysis ordination were the best metrics for separating the 5 lakes. A variant of the commonly used EPT index, composed of the total number of individuals of Ephemeroptera, Plecoptera, and Trichoptera, was also useful for characterizing the lakes. From this comparative analysis, subsamples based on counts of 100 animals are sufficient to distinguish the littoral benthic communities of small inland lakes in south-central Ontario.

Multispecies crayfish declines in lakes: implications for species distributions and richness

Abstract Aquatic communities are highly threatened by anthropogenic and climate change. However, despite their importance in these communities, information regarding temporal changes in populations and assemblages of North American crayfish is scarce. Long-term monitoring of crayfish populations in south-central Ontario, Canada, indicates that the populations are in a significant state of decline. We sought to determine whether these population declines are spatially and taxonomically broad, and if so, what factors might be associated with the declines. We sampled crayfish abundance (catch per unit effort) in 100 lakes, and compared current abundances to survey results from the early 1990s. Abundances of all species (natives and nonnatives) declined significantly during this interval. Declines were both severe (63–96% loss of abundance) and geographically widespread for all species. Previous studies have documented native species declines caused by the invasive crayfish Orconectes rusticus, but this species was absent from almost all lakes and was not a factor in the declines. We hypothesize that the introduction of predatory smallmouth bass (Micropterus dolomieui), increases in Al concentrations, and reduced Ca concentrations in these lakes are negatively affecting crayfish populations.

Evaluating the effect of lake calcium concentration on the acquisition of carapace calcium by freshwater crayfish

Aquatic systems in many parts of the world, including those on the Canadian Shield in central Ontario, Canada, are facing severe declines in aquatic calcium (Ca). The relationship between the Ca content of the carapace of freshwater crayfish, Orconectes virilis, O. rusticus and Cambarus bartonii, and lake Ca concentration was statistically evaluated across a broad [Ca] range in south-central Ontario. Carapace Ca content was significantly positively related to lake [Ca] in O. virilis but not C. bartonii or O. rusticus, probably because the latter two were only sampled over a comparatively small range of lake [Ca]. We evaluated additional variation in carapace Ca content potentially explained by the characteristics of O. virilis individuals, including size (carapace length), sex, and hardness (rigidity), using multiple linear regression. The carapace Ca content of O. virilis was significantly positively related to the hardness of the carapace, and to a lesser extent the size of the individual. For O. virilis, lake [Ca] appears to limit carapace Ca acquisition below approximately 8xa0mgxa0l−1. O. virilis is therefore experiencing stress due to low [Ca] in the majority of soft-water boreal lakes where it occurs because they are undergoing [Ca] decline.