Accounting for multiple testing in the analysis of spatio-temporal environmental data

The statistical analysis of environmental data from remote sensing and Earth system simulations often entails the analysis of gridded spatio-temporal data, with a hypothesis test being performed for each grid cell. When the whole image or a set of grid cells are analyzed for a global effect, the problem of multiple testing arises. When no global effect is present, we expect $$ \alpha $$% of all grid cells to be false positives, and spatially autocorrelated data can give rise to clustered spurious rejections that can be misleading in an analysis of spatial patterns. In this work, we review standard solutions for the multiple testing problem and apply them to spatio-temporal environmental data. These solutions are independent of the test statistic, and any test statistic can be used (e.g., tests for trends or change points in time series). Additionally, we introduce permutation methods and show that they have more statistical power. Real-world data are used to provide examples of the analysis, and the performance of each method is assessed in a simulation study. Unlike other simulation studies, our study compares the statistical power of the presented methods in a comprehensive simulation study. In conclusion, we present several statistically rigorous methods for analyzing spatio-temporal environmental data and controlling the false positives. These methods allow the use of any test statistic in a wide range of applications in environmental sciences and remote sensing.     

Identifying landscape scale patterns from individual scale processes

Extrapolating across scales is a critical problem in ecology. Explicit mechanistic models of ecological systems provide a bridge from measurements of processes at small and short scales to larger scales; spatial patterns at large scales can be used to test the outcomes of these models. However, it is necessary to identify patterns that are not dependent on initial conditions, because small scale initial conditions will not normally be measured at large scales. We examined one possible pattern that could meet these conditions, the relationship between mean and variance in abundance of a parasitic tick in an individual based model of a lizard tick interaction. We scaled discrepancies between the observed and simulated patterns with a transformation of the variance–covariance matrix of the observed pattern to objectively identify patterns that are “close”.     

Pattern-oriented modeling of bird foraging and pest control in coffee farms

We develop a model of how land use and habitat diversity affect migratory bird populations and their ability to suppress an insect pest on Jamaican coffee farms. Bird foraging—choosing which habitat patch and prey to use as prey abundance changes over space and time—is the key process driving this system. Following the “pattern-oriented” modeling strategy, we identified nine observed patterns that characterize the real system's dynamics. The model was designed so that these patterns could potentially emerge from it. The resulting model is individual-based, has fine spatial and temporal resolutions, represents very simply the supply of the pest insect and other arthropod food in six habitat types, and includes foraging habitat selection as the only adaptive behavior of birds. Although there is an extensive heritage of bird foraging theory in ecology, most of it addresses only the individual level and is too simple for our context. We used pattern-oriented modeling to develop and test foraging theory for this across-scale problem: rules for individual bird foraging that cause the model to reproduce a variety of patterns observed at the system level. Four alternative foraging theories were contrasted by how well they caused the model to reproduce the nine characteristic patterns. Four of these patterns were clearly reproduced with the “null” theory that birds select habitat randomly. A version of classical theory in which birds stay in a patch until food is depleted to some threshold caused the model to reproduce five patterns; this theory caused lower, not higher, use of habitat experiencing an outbreak of prey insects. Assuming that birds select the nearby patch providing highest intake rate caused the model to reproduce all but one pattern, whereas assuming birds select the highest-intake patch over a large radius produced an unrealistic distribution of movement distances. The pattern reproduced under none of the theories, a negative relation between bird density and distance to trees, appears to result from a process not in the model: birds return to trees at night to roost. We conclude that a foraging model for small insectivorous birds in diverse habitat should assume birds can sense higher food supply but over short, not long, distances.     

Fourth-corner correlation is a score test statistic in a log-linear trait–environment model that is useful in permutation testing

Ecologists wish to understand the role of traits of species in determining where each species occurs in the environment. For this, they wish to detect associations between species traits and environmental variables from three data tables, species count data from sites with associated environmental data and species trait data from data bases. These three tables leave a missing part, the fourth-corner. The fourth-corner correlations between quantitative traits and environmental variables, heuristically proposed 20 years ago, fill this corner. Generalized linear (mixed) models have been proposed more recently as a model-based alternative. This paper shows that the squared fourth-corner correlation times the total count is precisely the score test statistic for testing the linear-by-linear interaction in a Poisson log-linear model that also contains species and sites as main effects. For multiple traits and environmental variables, the score test statistic is proportional to the total inertia of a doubly constrained correspondence analysis. When the count data are over-dispersed compared to the Poisson or when there are other deviations from the model such as unobserved traits or environmental variables that interact with the observed ones, the score test statistic does not have the usual chi-square distribution. For these types of deviations, row- and column-based permutation methods (and their sequential combination) are proposed to control the type I error without undue loss of power (unless no deviation is present), as illustrated in a small simulation study. The issues for valid statistical testing are illustrated using the well-known Dutch Dune Meadow data set.     

Statistical selection of perimeter-area models for patch mosaics in multiscale landscape analysis

This paper presents a statistical method for detecting distinct scales of pattern for mosaics of irregular patches, by means of perimeter–area relationships. Krummel et al. (1987) were the first to develop a method for detecting different scaling domains in a landscape of irregular patches, but this method requires investigator judgment and is not completely satisfying. Grossi et al. (2001) suggested a modification of Krummel's method in order to detect objectively the change points between different scaling domains. Their procedure is based on the selection of the best piecewise linear regression model using a set of statistical tests. Even though the change points were estimated, the null distributions used for testing purposes were those appropriate for known change points. The present paper investigates the effect that estimating the change points has on the underlying distribution theory. The procedure we suggest is based on the selection of the best piecewise linear regression model using a likelihood ratio (LR) test. Each segment of the piecewise linear model corresponds to a fractal domain. Breakpoints between different segments are unknown, so the piecewise linear models are non-linear. In this case, the frequency distribution of the LR statistic cannot be approximated by a chi-squared distribution. Instead, Monte Carlo simulation is used to obtain an empirical null distribution of the LR statistic. The suggested method is applied to three patch types (CORINE biotopes) located in the Val Baganza watershed of Italy.     

The maximum likelihood based intensity estimate for circular point processes

Identifying the geometrical nature of spatial point patterns plays an important role in many areas of scientific research. Common types of spatial point processes involve random, regular, and cluster patterns. However, some point patterns suggest identifiable geometrical shapes such as a circular or other conic patterns. These patterns may be recognized as either a specific clustered shape or an inhomogeneous point pattern. Less noisy conic shapes, including circular patterns, are heavily discussed in the pattern recognition literature, but the goodness-of-fit of conic-fitting algorithms is rarely discussed for very noisy data. This study addresses a parameter estimation technique for noisy circular point patterns using the maximum likelihood principle. Additionally, a spatial statistical tool known as the L-function is used to investigate whether the fitted location pattern is reasonably attributable to a circular shape. A novel quantity named ‘relative log-error’ (\(\gamma \)) is introduced to quantify the goodness-of-fit for circular model fits. An iteratively re-weighted least squares procedure is introduced and robustness is evaluated under several error structures. Computational efficiency of the current and novel circle-fitting methods is also discussed. The findings are applied to two environmental science data sets.     

Linking chronic wasting disease to mule deer movement scales: a hierarchical Bayesian approach.

Observed spatial patterns in natural systems may result from processes acting across multiple spatial and temporal scales. Although spatially explicit data on processes that generate ecological patterns, such as the distribution of disease over a landscape, are frequently unavailable, information about the scales over which processes operate can be used to understand the link between pattern and process. Our goal was to identify scales of mule deer (Odocoileus hemionus) movement and mixing that exerted the greatest influence on the spatial pattern of chronic wasting disease (CWD) in northcentral Colorado, USA. We hypothesized that three scales of mixing (individual, winter subpopulation, or summer subpopulation) might control spatial variation in disease prevalence. We developed a fully Bayesian hierarchical model to compare the strength of evidence for each mixing scale. We found strong evidence that the finest mixing scale corresponded best to the spatial distribution of CWD infection. There was also evidence that land ownership and habitat use play a role in exacerbating the disease, along with the known effects of sex and age. Our analysis demonstrates how information on the scales of spatial processes that generate observed patterns can be used to gain insight when process data are sparse or unavailable.     

Incorporating animal behavior into seed dispersal models: implications for seed shadows

Seed dispersal fundamentally influences plant population and community dynamics but is difficult to quantify directly. Consequently, models are frequently used to describe the seed shadow (the seed deposition pattern of a plant population). For vertebrate-dispersed plants, animal behavior is known to influence seed shadows but is poorly integrated in seed dispersal models. Here, we illustrate a modeling approach that incorporates animal behavior and develop a stochastic, spatially explicit simulation model that predicts the seed shadow for a primate-dispersed tree species (Virola calophylla, Myristicaceae) at the forest stand scale. The model was parameterized from field-collected data on fruit production and seed dispersal, behaviors and movement patterns of the key disperser, the spider monkey (Ateles paniscus), densities of dispersed and non-dispersed seeds, and direct estimates of seed dispersal distances. Our model demonstrated that the spatial scale of dispersal for this V. calophylla population was large, as spider monkeys routinely dispersed seeds >100 m, a commonly used threshold for long-distance dispersal. The simulated seed shadow was heterogeneous, with high spatial variance in seed density resulting largely from behaviors and movement patterns of spider monkeys that aggregated seeds (dispersal at their sleeping sites) and that scattered seeds (dispersal during diurnal foraging and resting). The single-distribution dispersal kernels frequently used to model dispersal substantially underestimated this variance and poorly fit the simulated seed-dispersal curve, primarily because of its multimodality, and a mixture distribution always fit the simulated dispersal curve better. Both seed shadow heterogeneity and dispersal curve multimodality arose directly from these different dispersal processes generated by spider monkeys. Compared to models that did not account for disperser behavior, our modeling approach improved prediction of the seed shadow of this V. calophylla population. An important function of seed dispersal models is to use the seed shadows they predict to estimate components of plant demography, particularly seedling population dynamics and distributions. Our model demonstrated that improved seed shadow prediction for animal-dispersed plants can be accomplished by incorporating spatially explicit information on disperser behavior and movements, using scales large enough to capture routine long-distance dispersal, and using dispersal kernels, such as mixture distributions, that account for spatially aggregated dispersal.     

Multiscale codependence analysis: an integrated approach to analyze relationships across scales

The spatial and temporal organization of ecological processes and features and the scales at which they occur are central topics to landscape ecology and metapopulation dynamics, and increasingly regarded as a cornerstone paradigm for understanding ecological processes. Hence, there is need for computational approaches which allow the identification of the proper spatial or temporal scales of ecological processes and the explicit integration of that information in models. For that purpose, we propose a new method (multiscale codependence analysis, MCA) to test the statistical significance of the correlations between two variables at particular spatial or temporal scales. Validation of the method (using Monte Carlo simulations) included the study of type I error rate, under five statistical significance thresholds, and of type II error rate and statistical power. The method was found to be valid, in terms of type I error rate, and to have sufficient statistical power to be useful in practice. MCA has assumptions that are met in a wide range of circumstances. When applied to model the river habitat of juvenile Atlantic salmon, MCA revealed that variables describing substrate composition of the river bed were the most influential predictors of parr abundance at 0.4-4.1 km scales whereas mean channel depth was more influential at 200-300 m scales. When properly assessed, the spatial structuring observed in nature may be used purposefully to refine our understanding of natural processes and enhance model representativeness.     

Development of an individual-based model to evaluate elk (Cervus elaphus nelsoni) movement and distribution patterns following the Cerro Grande Fire in north central New Mexico, USA

Though studies have modeled the effects of fires on elk, no studies have related the effects of post-fire landscape succession on ungulate movements and distribution using dynamic modeling techniques. The purpose of this study was to develop and test a spatially-explicit, stochastic, individual-based model (IBM) to evaluate potential movement and distribution patterns of elk (Cervus elaphus nelsoni) in relation to spatial and temporal aspects of the Cerro Grande Fire that burned north central New Mexico in May of 2000. Following extensive literature review, the SAVANNA Ecosystem Model was selected to simulate the underlying post-fire successional processes driving elk movement and distribution. Standard logisitic regression was used to analyze habitat-use patterns of ten elk from data collected using global positioning system radio collars while an additional five animals were used as an independent test set during model validation. Static variables in the form of roads, buildings, fences, and habitual use/memory were used to modify a map of impedance values based on the logistic regression of slope, aspect, and elevation. Integration with SAVANNA came through the application of a habitat suitability index (HSI), which combined movement rules written for the IBM and variables modified and produced by the dynamic ecological processes run in SAVANNA. Overall pattern analysis indicated that realistic migrational processes and habitat-use patterns emerged from movement rules incorporated into the IBM in response to advancing and receding snow when compared to the independent test set. Primary and secondary movement pathways emerged from the collective responses of simulated individuals. Using regression analyses, no significant differences between simulated animals and animals used in either model development or an independent test set revealed any differences in response to snow patterns. These considerations suggest the model was adequately corroborated based on existing data and outlined objectives.     

Power evaluation of focused cluster tests

Many statistical tests have been developed to assess the significance of clusters of disease located around known sources of environmental contaminants, also known as focused disease clusters. The majority of focused-cluster tests were designed to detect a particular spatial pattern of clustering, one in which the disease cluster centers around the pollution source and declines in a radial fashion with distance. However, other spatial patterns of environmentally related disease clusters are likely given that the spatial dispersion patterns of environmental contaminants, and thus human exposure, depend on a number of factors (i.e., meteorology and topography). For this study, data were simulated with five different spatial patterns of disease clusters, reflecting potential pollutant dispersion scenarios: (1) a radial effect decreasing with increasing distance, (2) a radial effect with a defined peak and decreasing with distance, (3) a simple angular effect, (4) an angular effect decreasing with increasing distance and (5) an angular effect with a defined peak and decreasing with distance. The power to detect each type of spatially distributed disease cluster was evaluated using Stone’s Maximum Likelihood Ratio Test, Tango’s Focused Test, Bithell’s Linear Risk Score Test, and variations of the Lawson–Waller Score Test. Study findings underscore the importance of considering environmental contaminant dispersion patterns, particularly directional effects, with respect to focused-cluster test selection in cluster investigations. The effect of extra variation in risk also is considered, although its effect is not substantial in terms of the power of tests.     

Measurement error causes scale-dependent threshold erosion of biological signals in animal movement data.

Recent advances in telemetry technology have created a wealth of tracking data available for many animal species moving over spatial scales from tens of meters to tens of thousands of kilometers. Increasingly, such data sets are being used for quantitative movement analyses aimed at extracting fundamental biological signals such as optimal searching behavior and scale-dependent foraging decisions. We show here that the location error inherent in various tracking technologies reduces the ability to detect patterns of behavior within movements. Our analyses endeavored to set out a series of initial ground rules for ecologists to help ensure that sampling noise is not misinterpreted as a real biological signal. We simulated animal movement tracks using specialized random walks known as Lévy flights at three spatial scales of investigation: 100-km, 10-km, and 1-km maximum daily step lengths. The locations generated in the simulations were then blurred using known error distributions associated with commonly applied tracking methods: the Global Positioning System (GPS), Argos polar-orbiting satellites, and light-level geolocation. Deviations from the idealized Lévy flight pattern were assessed for each track after incrementing levels of location error were applied at each spatial scale, with additional assessments of the effect of error on scale-dependent movement patterns measured using fractal mean dimension and first-passage time (FPT) analyses. The accuracy of parameter estimation (Lévy mu, fractal mean D, and variance in FPT) declined precipitously at threshold errors relative to each spatial scale. At 100-km maximum daily step lengths, error standard deviations of > or = 10 km seriously eroded the biological patterns evident in the simulated tracks, with analogous thresholds at the 10-km and 1-km scales (error SD > or = 1.3 km and 0.07 km, respectively). Temporal subsampling of the simulated tracks maintained some elements of the biological signals depending on error level and spatial scale. Failure to account for large errors relative to the scale of movement can produce substantial biases in the interpretation of movement patterns. This study provides researchers with a framework for understanding the limitations of their data and identifies how temporal subsampling can help to reduce the influence of spatial error on their conclusions.     

Wildfire, landscape diversity and the Drossel-Schwabl model

How simple can a model be that still captures essential aspects of wildfire ecosystems at large spatial and temporal scales? The Drossel-Schwabl model (DSM) is a metaphorical forest-fire model developed to reproduce only one pattern of real systems: a frequency distribution of fire sizes resembling a power law. Consequently, and because it appears oversimplified, it remains unclear what bearings the DSM has in reality. Here, we test whether the DSM is capable of reproducing a pattern that was not considered in its design, the hump-shaped relation between the diversity of succession stages and average annual area burnt. We found that the model, once reformulated to represent succession, produces realistic landscape diversity patterns. We investigated four succession scenarios of forest-fire ecosystems in the USA and Canada. In all scenarios, landscape diversity is highest at an intermediate average annual area burnt as predicted by the intermediate disturbance hypothesis. These results show that a model based solely on the dynamics of the fuel mosaic has surprisingly high predictive power with regard to observed statistical properties of wildfire systems at large spatial scales. Parsimonious models, such as the DSM can be used as starting points for systematic development of more structurally realistic but tractable wildfire models. Due to their simplicity they allow analytical approaches that further our understanding under increasing complexity.     

Accounting for rate instability and spatial patterns in the boundary analysis of cancer mortality maps

Boundary analysis of cancer maps may highlight areas where causative exposures change through geographic space, the presence of local populations with distinct cancer incidences, or the impact of different cancer control methods. Too often, such analysis ignores the spatial pattern of incidence or mortality rates and overlooks the fact that rates computed from sparsely populated geographic entities can be very unreliable. This paper proposes a new methodology that accounts for the uncertainty and spatial correlation of rate data in the detection of significant edges between adjacent entities or polygons. Poisson kriging is first used to estimate the risk value and the associated standard error within each polygon, accounting for the population size and the risk semivariogram computed from raw rates. The boundary statistic is then defined as half the absolute difference between kriged risks. Its reference distribution, under the null hypothesis of no boundary, is derived through the generation of multiple realizations of the spatial distribution of cancer risk values. This paper presents three types of neutral models generated using methods of increasing complexity: the common random shuffle of estimated risk values, a spatial re-ordering of these risks, or p-field simulation that accounts for the population size within each polygon. The approach is illustrated using age-adjusted pancreatic cancer mortality rates for white females in 295 US counties of the Northeast (1970–1994). Simulation studies demonstrate that Poisson kriging yields more accurate estimates of the cancer risk and how its value changes between polygons (i.e., boundary statistic), relatively to the use of raw rates or local empirical Bayes smoother. When used in conjunction with spatial neutral models generated by p-field simulation, the boundary analysis based on Poisson kriging estimates minimizes the proportion of type I errors (i.e., edges wrongly declared significant) while the frequency of these errors is predicted well by the p-value of the statistical test.

A new hypothesis concerning the nature of small pelagic fish clusters: An individual-based modelling study of Sardinella aurita dynamics off West Africa

Coastal populations of small pelagic fish display nested aggregation levels. Above the level of the school structure, clusters are observed the nature of which has not been definitively determined. We hypothesized that these clusters corresponded to a materialisation of the microcohorts originating from successive spawnings of fish populations in their vital domain.A candidate individual-based model was developed to investigate this hypothesis. This model is based on pattern-oriented modelling of a concrete documented case: the dynamics of the round sardinella (Sardinella aurita) population living off the West African coasts and subject to environmental fluctuations caused by seasonal upwelling. The simulated agents were round sardinella microcohorts situated and moving in a discretised physical environment. The combined effects of environmental forcing (temperature, wind, retention) and inner biological dynamics (reproduction, growth and mortality, competition) condition the dynamics of this population.The modelled behaviour generated realistic dynamic patterns (population distribution, spawning zones, periods and plasticity, biomass fluctuations), which were obtained simultaneously and successfully compared with observations. The steady-state number of microcohorts obtained after simulation convergence was similar to the number of clusters observed in situ in this area for this population.The realism and diversity of the patterns simultaneously simulated suggested the cluster-microcohort equivalence hypothesis as a candidate framework accounting for the origin of the clusters observed in situ. Within this preliminary exploration, we discuss the consistency of the hypothesis and the accuracy of the model. If the correspondence between clusters and microcohorts proves to be real, it may be transient and progressively modified by other environmental factors. If stable over time, as simulated in the model, the number of observed clusters should be related to the number of spawning events in the species’ lifetime.     

On the use of nearest neighbor contingency tables for testing spatial segregation

For two or more classes (or types) of points, nearest neighbor contingency tables (NNCTs) are constructed using nearest neighbor (NN) frequencies and are used in testing spatial segregation of the classes. Pielou’s test of independence, Dixon’s cell-specific, class-specific, and overall tests are the tests based on NNCTs (i.e., they are NNCT-tests). These tests are designed and intended for use under the null pattern of random labeling (RL) of completely mapped data. However, it has been shown that Pielou’s test is not appropriate for testing segregation against the RL pattern while Dixon’s tests are. In this article, we compare Pielou’s and Dixon’s NNCT-tests; introduce the one-sided versions of Pielou’s test; extend the use of NNCT-tests for testing complete spatial randomness (CSR) of points from two or more classes (which is called CSR independence, henceforth). We assess the finite sample performance of the tests by an extensive Monte Carlo simulation study and demonstrate that Dixon’s tests are also appropriate for testing CSR independence; but Pielou’s test and the corresponding one-sided versions are liberal for testing CSR independence or RL. Furthermore, we show that Pielou’s tests are only appropriate when the NNCT is based on a random sample of (base, NN) pairs. We also prove the consistency of the tests under their appropriate null hypotheses. Moreover, we investigate the edge (or boundary) effects on the NNCT-tests and compare the buffer zone and toroidal edge correction methods for these tests. We illustrate the tests on a real life and an artificial data set.     

Nonparametric spatial covariance functions: Estimation and testing

Spatial autocorrelation techniques are commonly used to describe genetic and ecological patterns. To improve statistical inference about spatial covariance, we propose a continuous nonparametric estimator of the covariance function in place of the spatial correlogram. The spline correlogram is an adaptation of a recent development in spatial statistics and is a generalization of the commonly used correlogram. We propose a bootstrap algorithm to erect a confidence envelope around the entire covariance function. The meaning of this envelope is discussed. Not all functions that can be drawn inside the envelope are candidate covariance functions, as they may not be positive semidefinite. However, covariance functions that do not fit, are not supported by the data. A direct estimate of the L₀ spatial correlation length with associated confidence interval is offered and its interpretation is discussed. The spline correlogram is found to have high precision when applied to synthetic data. For illustration, the method is applied to electrophoretic data of an alpine grass (Poa alpina).     

The effects of plant distribution and frugivore density on the scale and shape of dispersal kernels

For many plant species, seed dispersal is one of the most important spatial demographic processes. We used a diffusion approximation and a spatially explicit simulation model to explore the mechanisms generating seed dispersal kernels for plants dispersed by frugivores. The simulation model combined simple movement and foraging rules with seed gut passage time, plant distribution, and fruit production. A simulation experiment using plant spatial aggregation and frugivore density as factors showed that seed dispersal scale was largely determined by the degree of plant aggregation, whereas kernel shape was mostly dominated by frugivore density. Kernel shapes ranged from fat tailed to thin tailed, but most shapes were between an exponential and that of the solution of a diffusion equation. The proportion of dispersal kernels with fat tails was highest for landscapes with clumped plant distributions and increased with increasing number of dispersers. The diffusion model provides a basis for models including more behavioral details but can also be used to approximate dispersal kernels once a diffusion rate is estimated from animal movement data. Our results suggest that important characteristics of dispersal kernels will depend on the spatial pattern of plant distribution and on disperser density when frugivores mediate seed dispersal.     

Community structure of the benthos in Scottish Sea-lochs. IV. Multivariate spatial pattern

Most studies of spatial patterns of invertebrates in soft sediments have concentrated on populations of individual species. Those that examined patterns in communities have tended to employ categorical analytical techniques. Using macrofaunal abundance data from van Veen grab samples collected 20 to 100 m apart in known spatial arrangements from Scottish sea-lochs, the relationships between patterns in macrobenthic species composition and distances between samples were explored using matrix correlations in a non-parametric framework. Using a simple definition of spatial structure, i.e. that intersample distances are monotonically correlated with intersample species-similarity, spatial structure at each of seven stations was assessed using non-parametric Mantel tests based on rank-correlations. Changes in community structure were positively correlated with distance at all sites in Loch Etive, on both current-swept muddy sands and soft deep muds. Different components of the macrobenthos contributed to spatial pattern at each site. Simple spatial structure was also detectable at a muddy-sand site in Loch Creran, but neither on soft mud, nor at the soft mud site in the Firth of Lorne. The concept of rank-correlograms was introduced. These were used to examine the extent and form of spatial structure in different components of the macrobenthos at each site. Relationships between similarity and distance were neither simple nor consistent. Results were compared to previous studies which used the same data, and it was concluded that studies carried out at a particular scale, or on a particular component of the benthos, are unlikely to be successful in predicting spatial relationships at other scales or for other components of the benthos. Correlational rather than categorical analyses are recommended for exploratory studies of spatial relationships in the benthos. Analyses of the spatial structure at these seven sea-loch sites suggests that by ensuring that samples are at least 40 m apart an investigator is unlikely to underestimate variability or otherwise invalidate statistical analyses based on the use of the samples as replicates. Spacing samples up to 100 m apart may increase variability estimates, further reducing the chance of concluding that a difference exists when one does not. Received: 4 May 1999 / Accepted: 8 March 2000