Spatial Autocorrelation Analysis of Migration and Selection

We test various assumptions necessary for the interpretation of spatial autocorrelation analysis of gene frequency surfaces, using simulations of Wright's isolation-by-distance model with migration or selection superimposed. Increasing neighborhood size enhances spatial autocorrelation, which is reduced again for the largest neighborhood sizes. Spatial correlograms are independent of the mean gene frequency of the surface. Migration affects surfaces and correlograms when immigrant gene frequency differentials are substantial. Multiple directions of migration are reflected in the correlograms. Selection gradients yield clinal correlograms; other selection patterns are less clearly reflected in their correlograms. Sequential migration from different directions and at different gene frequencies can be disaggregated into component migration vectors by means of principal components analysis. This encourages analysis by such methods of gene frequency surfaces in nature. The empirical results of these findings lend support to the inference structure developed earlier for spatial autocorrelation analysis.     

TESTING INFERENCES ABOUT MICRO-EVOLUTIONARY PROCESSES BY MEANS OF SPATIAL AUTOCORRELATION ANALYSIS

We generated numerous simulated gene-frequency surfaces subjected to 200 generations of isolation by distance with, in some cases, added migration or selection. From these surfaces we assembled six data sets comprising from 12 to 15 independent allele-frequency surfaces, to simulate biologically plausible population samples. The purpose of the study was to investigate whether spatial autocorrelation analysis will correctly infer the microevolutionary processes involved in each data set. The correspondence between the simulated processes and the inferences made concerning them is close for five of the six data sets. Errors in inference occurred when the effect of migration was weak, due to low gene frequency differential or low migration strength; when selection was weak and against a background with a complex pattern; and when a random process—isolation by distance—was the only one acting. Spatial correlograms proved more sensitive to detecting trends than inspection of gene-frequency surfaces by the human eye. Joint interpretation of the correlograms and their clusters proved most reliable in leading to the correct inference. The inspection and clustering of surfaces were useful for determining directional components. Because this method relies on common patterns across loci, as many gene frequencies as feasible should be used. We recommend spatial autocorrelation analysis for the detection of microevolutionary processes in natural populations.     

The consequences of spatial structure for the design and analysis of ecological field surveys

In ecological field surveys, observations are gathered at different spatial locations. The purpose may be to relate biological response variables (e.g., species abundances) to explanatory environmental variables (e.g., soil characteristics). In the absence of prior knowledge, ecologists have been taught to rely on systematic or random sampling designs. If there is prior knowledge about the spatial patterning of the explanatory variables, obtained from either previous surveys or a pilot study, can we use this information to optimize the sampling design in order to maximize our ability to detect the relationships between the response and explanatory variables?
The specific questions addressed in this paper are: a) What is the effect (type I error) of spatial autocorrelation on the statistical tests commonly used by ecologists to analyse field survey data? b) Can we eliminate, or at least minimize, the effect of spatial autocorrelation by the design of the survey? Are there designs that provide greater power for surveys, at least under certain circumstances? c) Can we eliminate or control for the effect of spatial autocorrelation during the analysis? To answer the last question, we compared regular regression analysis to a modified t‐test developed by Dutilleul for correlation coefficients in the presence of spatial autocorrelation.
Replicated surfaces (typically, 1000 of them) were simulated using different spatial parameters, and these surfaces were subjected to different sampling designs and methods of statistical analysis. The simulated surfaces may represent, for example, vegetation response to underlying environmental variation. This allowed us 1) to measure the frequency of type I error (the failure to reject the null hypothesis when in fact there is no effect of the environment on the response variable) and 2) to estimate the power of the different combinations of sampling designs and methods of statistical analysis (power is measured by the rate of rejection of the null hypothesis when an effect of the environment on the response variable has been created).
Our results indicate that: 1) Spatial autocorrelation in both the response and environmental variables affects the classical tests of significance of correlation or regression coefficients. Spatial autocorrelation in only one of the two variables does not affect the test of significance. 2) A broad‐scale spatial structure present in data has the same effect on the tests as spatial autocorrelation. When such a structure is present in one of the variables and autocorrelation is found in the other, or in both, the tests of significance have inflated rates of type I error. 3) Dutilleul's modified t‐test for the correlation coefficient, corrected for spatial autocorrelation, effectively corrects for spatial autocorrelation in the data. It also effectively corrects for the presence of deterministic structures, with or without spatial autocorrelation.
The presence of a broad‐scale deterministic structure may, in some cases, reduce the power of the modified t‐test.     

Complementary strengths of spatially-explicit and multi-species distribution models

Species distribution models (SDMs) project the outcome of community assembly processes – dispersal, the abiotic environment and biotic interactions – onto geographic space. Recent advances in SDMs account for these processes by simultaneously modeling the species that comprise a community in a multivariate statistical framework or by incorporating residual spatial autocorrelation in SDMs. However, the effects of combining both multivariate and spatially-explicit model structures on the ecological inferences and the predictive abilities of a model are largely unknown. We used data on eastern hemlock Tsuga canadensis and five additional co-occurring overstory tree species in 35 569 forest stands across Michigan, USA to evaluate how the choice of model structure, including spatial and non-spatial forms of univariate and multivariate models, affects ecological inference about the processes that shape community composition as well as model predictive ability. Incorporating residual spatial autocorrelation via spatial random effects did not improve out-of-sample prediction for the six tree species, although in-sample model fit was higher in the spatial models. Spatial models attributed less variation in occurrence probability to environmental covariates than the non-spatial models for all six tree species, and estimated higher (more positive) residual co-occurrence values for most species pairs. The non-spatial multivariate model was better suited for evaluating habitat suitability and hypotheses about the processes that shape community composition. Environmental correlations and residual correlations among species pairs were positively related, perhaps indicating that residual correlations were due to shared responses to unmeasured environmental covariates. This work highlights the importance of choosing a non-spatial model formulation to address research questions about the species–environment relationship or residual co-occurrence patterns, and a spatial model formulation when within-sample prediction accuracy is the main goal.     

SPATIAL GENETIC STRUCTURE OF CLONAL AND SEXUAL REPRODUCTION IN POPULATIONS OF ADENOPHORA GRANDIFLORA (CAMPANULACEAE)

The spatial distribution of clonal versus sexual reproduction in plant populations should generally have differing effects on the levels of biparental inbreeding and the apparent selfing rate, produced via mating by proximity through limited pollen dispersal. We used allozyme loci, join-count statistics, and Moran's spatial autocorrelation statistics to separate the spatial genetic structure caused by clonal reproduction from that maintained in sexually reproduced individuals in two populations of Adenophora grandiflora, a perennial herb. Join-count statistics showed that there were statistically significant clustering of clonal genotypes within distances less than 4 m. Both the entire populations and the sets of sexually reproduced individuals exhibited significant spatial autocorrelation at less than about 12 m, and the sexually reproduced individuals are substantially structured in an isolation-by-distance manner, consistent with a neighborhood size of about 50.     

Local spatial autocorrelation in biological variables

Spatial autocorrelation (SA) methods have recently been extended to include the detection of local spatial autocorrelation at individual sampling stations. We review the formulas for these statistics and report on the results of an extensive population-genetic simulation study we have published elsewhere to test the applicability of these methods in spatially distributed biological data. We find that most biological variables exhibit global SA, and that in such cases the methods proposed for testing the significance of local SA coefficients reject the null hypothesis excessively. When global SA is absent, permutational methods for testing significance yield reliable results. Although standard errors have been published for the local SA coefficients, their employment using an asymptotically normal approach leads to unreliable results; permutational methods are preferred. In addition to significance tests of suspected non-stationary localities, we can use these methods in an exploratory manner to find and identify hotspots (places with positive local SA) and coldspots (negative local SA) in a dataset. We illustrate the application of these methods in three biological examples from plant population biology, ecology and population genetics. The examples range from the study of single variables to the joint analysis of several variables and can lead to successful demographic and evolutionary inferences about the populations studied.     

A simulation study of microevolutionary inferences by spatial autocorrelation analysis

To explore the extent to which microevolutionary inference can be made using spatial autocorrelation analysis of gene frequency surfaces, we simulated sets of surfaces for nine evolutionary scenarios, and subjected spatially-based summary statistics of these to linear discriminant analysis. Scenarios varied the amounts of dispersion, selection, migration, and deme sizes, and included: panmixia, drift, intrusion, and stepping-stone models with 0–2 migrations, 0–2 selection gradients, and migration plus selection. To discover how weak evolutionary forces could be and still allow discrimination, each scenario had both a strong and a weak configuration. Discriminant rules were calculated using one collection of data (the training set) consisting of 250 sets of 15 surfaces for each of the nine scenarios. Misclassification rates were verified against a second, entirely new set of data (the test set) equal in size. Test set misclassification rates for the 20 best discriminating variables ranged from 39.3% (weak) to 3.6% (strong), far lower than the expected rate of 88.9% absent any discriminating ability. Misclassification was highest when discriminating the number of migrational events or the presence or number of selection events. Discrimination of drift and panmixia from the other scenarios was perfect. A subsequent subjective analysis of a subset of the data by one of us yielded comparable, although somewhat higher, misclassification rates. Judging by these results, spatial autocorrelation variables describing sets of gene frequency surfaces permit some microevolutionary inferences.     

Proceedings of the SMBE Tri-National Young Investigators' Workshop 2005. Accurate inference and estimation in population genomics

Both intra- and interspecific genomic comparisons have revealed local similarities in the level and frequency of mutational variation, as well as in patterns of gene expression. This autocorrelation between measurements leads to violations of assumptions of independence in many statistical methods, resulting in misleading and incorrect inferences. Here I show that autocorrelation can be due to many factors and is present across the genome. Using a one-dimensional spatial stochastic model, I further show how previous results can be employed to correct for autocorrelation along chromosomes in population and comparative genomics research. When multiple hypothesis tests are autocorrelated, I demonstrate that a simple correction can lead to increased power in statistical inference. I present a preliminary analysis of population genomic data from Drosophila simulans to show the ubiquity of autocorrelation and applicability of the methods proposed here.     

The spatial structure of the gene pool of a viviparous population of Poa alpina - environmental controls and spatial constraints

Bjømstad, O. N., Iversen, A. & Hansen, M. 1995. The spatial structure of the gene pool of a viviparous population of Poa alpina — environmental controls and spatial constraints. — Nord. J. Bot. 15: 347–354. Copenhagen. ISSN 0107–055X.
Because both the genetic make-up and the environmental conditions of a population are spatially autocorrelated, it is difficult to infer processes of selection or drift for population genetic mappings. We propose a methodology based on partial Mantel techniques and partial autocorrelation techniques to separate the action of these processes. The method is applied to data on Poa alpina to indicate that isolation-by-distance (drift) is the main process inducing positive autocorrelation at the scale of diaspore dispersal (< 100m). The pattern at larger distances is more consistent with selection.     

On the role of spatial stochastic models in understanding landscape indices in ecology

Spatial stochastic models play an important role in understanding and predicting the behaviour of complex systems. Such models may be implemented with explicit knowledge of only a limited number of parameters relating to spatial relationships among locations. Consequently, they are often used instead of deterministic‐mechanistic models, which may potentially require an unrealistically large number of parameters. Currently, in contrast to spatial stochastic models, the parameterization of the joint spatial distribution of objects in landscape models is more often implicit than explicit. Here, we investigate the similarities and differences between bona fide spatial stochastic models and landscape models by focusing mostly on the relationships between processes, their realizations (patterns), representation and measurement, and their use in exploratory as well as confirmatory data analysis. One of the most important outcomes of recognizing the importance of stochastic processes is the acknowledgement that the spatial pattern observed in a landscape is only one realization of that process. Hence, while ecologists have been using landscape pattern indices (LPIs) to characterize landscape heterogeneity and/or make inferences about processes shaping the landscape, no stochastic modelling framework has been developed for their proper statistical elucidation. Consequently, several (mis)uses of LPIs draw conclusions about landscapes which are suspect. We show that several reports about sensitivities of LPIs to measurements have common roots that can be made explicitly manageable by adopting stochastic models of spatial structure. The key parameters of these stochastic models are composition and configuration, which, in general, cannot be estimated independently from each other. We outline how to develop the stochastic framework to interpret observations and make some recommendations to practitioners about everyday usage. The conceptual linkages between patterns and processes are particularly important in light of recent efforts to bridge the static‐structural and the dynamic‐analytic traditions of ecology.     

Quantitative approaches in climate change ecology

Contemporary impacts of anthropogenic climate change on ecosystems are increasingly being recognized. Documenting the extent of these impacts requires quantitative tools for analyses of ecological observations to distinguish climate impacts in noisy data and to understand interactions between climate variability and other drivers of change. To assist the development of reliable statistical approaches, we review the marine climate change literature and provide suggestions for quantitative approaches in climate change ecology. We compiled 267 peer‐reviewed articles that examined relationships between climate change and marine ecological variables. Of the articles with time series data (n = 186), 75% used statistics to test for a dependency of ecological variables on climate variables. We identified several common weaknesses in statistical approaches, including marginalizing other important non‐climate drivers of change, ignoring temporal and spatial autocorrelation, averaging across spatial patterns and not reporting key metrics. We provide a list of issues that need to be addressed to make inferences more defensible, including the consideration of (i) data limitations and the comparability of data sets; (ii) alternative mechanisms for change; (iii) appropriate response variables; (iv) a suitable model for the process under study; (v) temporal autocorrelation; (vi) spatial autocorrelation and patterns; and (vii) the reporting of rates of change. While the focus of our review was marine studies, these suggestions are equally applicable to terrestrial studies. Consideration of these suggestions will help advance global knowledge of climate impacts and understanding of the processes driving ecological change.     

Genealogical inference from microsatellite data.

Ease and accuracy of typing, together with high levels of polymorphism and widespread distribution in the genome, make microsatellite (or short tandem repeat) loci an attractive potential source of information about both population histories and evolutionary processes. However, microsatellite data are difficult to interpret, in particular because of the frequency of back-mutations. Stochastic models for the underlying genetic processes can be specified, but in the past they have been too complicated for direct analysis. Recent developments in stochastic simulation methodology now allow direct inference about both historical events, such as genealogical coalescence times, and evolutionary parameters, such as mutation rates. A feature of the Markov chain Monte Carlo (MCMC) algorithm that we propose here is that the likelihood computations are simplified by treating the (unknown) ancestral allelic states as auxiliary parameters. We illustrate the algorithm by analyzing microsatellite samples simulated under the model. Our results suggest that a single microsatellite usually does not provide enough information for useful inferences, but that several completely linked microsatellites can be informative about some aspects of genealogical history and evolutionary processes. We also reanalyze data from a previously published human Y chromosome microsatellite study, finding evidence for an effective population size for human Y chromosomes in the low thousands and a recent time since their most recent common ancestor: the 95% interval runs from approximately 15, 000 to 130,000 years, with most likely values around 30,000 years.     

Isolation by resistance across a complex coral reef seascape

A detailed understanding of the genetic structure of populations and an accurate interpretation of processes driving contemporary patterns of gene flow are fundamental to successful spatial conservation management. The field of seascape genetics seeks to incorporate environmental variables and processes into analyses of population genetic data to improve our understanding of forces driving genetic divergence in the marine environment. Information about barriers to gene flow (such as ocean currents) is used to define a resistance surface to predict the spatial genetic structure of populations and explain deviations from the widely applied isolation-by-distance model. The majority of seascape approaches to date have been applied to linear coastal systems or at large spatial scales (more than 250 km), with very few applied to complex systems at regional spatial scales (less than 100 km). Here, we apply a seascape genetics approach to a peripheral population of the broadcast-spawning coral Acropora spicifera across the Houtman Abrolhos Islands, a high-latitude complex coral reef system off the central coast of Western Australia. We coupled population genetic data from a panel of microsatellite DNA markers with a biophysical dispersal model to test whether oceanographic processes could explain patterns of genetic divergence. We identified significant variation in allele frequencies over distances of less than 10 km, with significant differentiation occurring between adjacent sites but not between the most geographically distant ones. Recruitment probabilities between sites based on simulated larval dispersal were projected into a measure of resistance to connectivity that was significantly correlated with patterns of genetic divergence, demonstrating that patterns of spatial genetic structure are a function of restrictions to gene flow imposed by oceanographic currents. This study advances our understanding of the role of larval dispersal on the fine-scale genetic structure of coral populations across a complex island system and applies a methodological framework that can be tailored to suit a variety of marine organisms with a range of life-history characteristics.     

Accounting for spatial pattern when modeling organism-environment interactions

Statistical models of environment-abundance relationships may be influenced by spatial autocorrelation in abundance, environmental variables, or both. Failure to account for spatial autocorrelation can lead to incorrect conclusions regarding both the absolute and relative importance of environmental variables as determinants of abundance. We consider several classes of statistical models that are appropriate for modeling environment-abundance relationships in the presence of spatial autocorrelation, and apply these to three case studies: 1) abundance of voles in relation to habitat characteristics; 2) a plant competition experiment; and 3) abundance of Orbatid mites along environmental gradients. We find that when spatial pattern is accounted for in the modeling process, conclusions about environmental control over abundance can change dramatically. We conclude with five lessons: 1) spatial models are easy to calculate with several of the most common statistical packages; 2) results from spatially-structured models may point to conclusions radically different from those suggested by a spatially independent model; 3) not all spatial autocorrelation in abundances results from spatial population dynamics; it may also result from abundance associations with environmental variables not included in the model; 4) the different spatial models do have different mechanistic interpretations in terms of ecological processes – thus ecological model selection should take primacy over statistical model selection; 5) the conclusions of the different spatial models are typically fairly similar – making any correction is more important than quibbling about which correction to make.     

Isolation by distance within a river system: genetic population structuring of Atlantic salmon, Salmo salar, in tributaries of the Varzuga River in northwest Russia

An important issue for designing any conservation programme aimed at preserving genetic diversity is estimation of the scale at which genetic structuring occurs. Additional relevant factors include distinguishing whether or not population structuring is expected to be stable as predicted by the member-vagrant hypothesis, or alternatively, whether populations are more prone to local extinction-recolonization processes, as predicted by the metapopulation evolutionary model. In this study, the population genetic structure of Atlantic salmon from 11 locations within or nearby the Varzuga River tributary system was assessed using 17 microsatellites. Mantel tests and spatial autocorrelation analyses revealed a significant isolation-by-distance signal within the tributary system as well as a negative association between the level of genetic diversity and waterway distance from the river mouth, indicating that dispersal is less likely to occur to populations deep in the tributary system. Individual-level spatial autocorrelation analyses indicated that the majority of migration occurred between populations situated within 20 km of each other. The relatively high level of genetic structuring and significant isolation-by-distance signal observed in the Varzuga tributaries are concordant with the predictions of the member-vagrant evolutionary model. However, one subpopulation in particular revealed signs of instability which may be due to its location in the tidal zone, or due to the fact that it is more affected by human impacts. The results suggest that preservation of a number of spawning sites spaced throughout the tributary system is recommendable for ensuring sustainable fishing tourism in the river.     

Limiting relationships between selection and recombination

It is difficult to directly observe processes like natural selection at the genetic level, but relatively easy to estimate genetic frequencies in populations. As a result, genetic frequency data are widely used to make inferences about the underlying evolutionary processes. However, multiple processes can generate the same patterns of frequency data, making such inferences weak. By studying the limits to the underlying processes, one can make inferences from frequency data by asking how strong selection (or some other process of interest) would have to be to generate the observed pattern. Here we present results of a study of the limits to the relationship between selection and recombination in two-locus, two-allele systems in which we found the limiting relationships for over 30 000 sets of parameters, effectively covering the range of two-locus, two-allele problems. Our analysis relates T _min—the minimum time for a population to evolve from the initial to the final conditions—to the strengths of selection and recombination, the amount of linkage disequilibrium, and the Nei distance between the initial and final conditions. T _min can be large with either large disequilibrium and small Nei distance, or the reverse. The behavior of T _min provides information about the limiting relationships between selection and recombination. Our methods allow evolutionary inferences from frequency data when deterministic processes like selection and recombination are operating; in this sense they complement methods based entirely on drift.     

Testing mutual independence between two discrete-valued spatial processes: a correction to pearson chi-squared

A common feature of data collected in environmental and earth sciences is that they typically exhibit spatial autocorrelation. Violating the assumption of independent observations can have dramatic effects on inferences derived from standard statistical methods. In this article, we examine the consequences of spatial autocorrelation on Pearson's chi-squared test of mutual independence between two categorical responses with a general number of classes. Correspondingly, we suggest a simple modification to the standard test statistic that allows for spatial autocorrelation. Our modified statistic is based on a first-order correction factor and thus provides only an approximate test. However, we show by Monte Carlo simulation that this approximation results in satisfactory inferences in several situations of practical interest. The usefulness of the method is displayed through an application to categorical data arising in the study of the relationship between the distribution pattern of plant species and woodland age in a forest in northern Belgium.     

Geographic patterns: how to identify them and why

Geographic patterns of genetic diversity allow us to make inferences about population histories and the evolution of inherited disease. The statistical methods describing genetic variation in space, such as estimation of genetic variances, mapping of allele frequencies, and principal components analysis, have opened up the possibility to reconstruct demographic processes whose effects have been tested by a variety of approaches, including spatial autocorrelation, cladistic analyses, and simulations. These studies have significantly contributed to our understanding of human genetic variation; however, the molecular data that have accumulated since the mid-1980s have also created new complications. Reasons include the generally limited sample sizes, but, more generally, it is the nature of molecular variation itself that makes it necessary to develop and apply specific models and methods for the treatment of DNA data. The foreseeable diffusion of laboratory techniques for the rapid typing of many DNA markers will force us to change our approach to the study of human variation anyway, moving from the gene level toward the genome level. Because extensive variation among loci is the rule rather than the exception, an important practical tip is to be skeptical of inferences based on single-locus diversity.     

Prediction of soil property distribution in paddy soil landscapes using terrain data and satellite information as indicators

Sustainable land management and land use planning require reliable information about the spatial distribution of the physical and chemical soil properties affecting both landscape processes and services. Although many studies have been conducted to identify the spatial patterns of soil property distribution on various scales and in various landscapes, only little is known about the relationships underlying the spatial distribution of soil properties in intensively used, finely structured paddy soil landscapes in the southeastern part of China. In order to provide adequate soil information for the modelling of landscape processes, such as soil water movement, nutrient leaching, soil erosion and plant growth, this study investigates to what extent cheap and readily available ancillary information derived from digital elevation models and remote sensing data can be used to support soil mapping and to indicate soil characteristics on the landscape scale. This article focuses on the spatial prediction of the total carbon and nitrogen content and of physical soil properties such as topsoil silt, sand and clay content, topsoil depth and plough pan thickness. Correlation analyses indicate that, on the one side, the distribution of C, N and silt contents is quite closely related to the NDVI of vegetated surfaces and that, on the other side, it corresponds significantly to terrain attributes such as relative elevation, elevation above nearest drainage channel and topographical wetness index. Geostatistical analyses furthermore reflect a moderately structured spatial correlation of these soil variables. The combined use of the above mentioned terrain variables and the NDVI in a multiple linear regression accounted for 29% (silt) to 41% (total C) of the variance of these soil properties. In order to select the best prediction method to accurately map soil property distribution, we compared the performance of different regionalization techniques, such as multi-linear regression, simple kriging, inverse distance to a power, ordinary kriging and regression kriging. Except for the prediction of topsoil clay content, in all cases regression kriging model “C” performed best. Compared to simple kriging, the spatial prediction was improved by up to 14% (total C), 13% (total N) and 10% (silt). Since the autocorrelation lengths of these spatially well correlated soil variables range between three and five times the soil sampling density, we consider regression kriging model “C” to be a suitable method for reducing the soil sampling density. It should help to save time and costs when doing soil mapping on the landscape scale, even in intensively used paddy soil landscapes.     

Variogram analysis of the spatial genetic structure of continuous populations using multilocus microsatellite data

A geostatistical perspective on spatial genetic structure may explain methodological issues of quantifying spatial genetic structure and suggest new approaches to addressing them. We use a variogram approach to (i) derive a spatial partitioning of molecular variance, gene diversity, and genotypic diversity for microsatellite data under the infinite allele model (IAM) and the stepwise mutation model (SMM), (ii) develop a weighting of sampling units to reflect ploidy levels or multiple sampling of genets, and (iii) show how variograms summarize the spatial genetic structure within a population under isolation-by-distance. The methods are illustrated with data from a population of the epiphytic lichen Lobaria pulmonaria, using six microsatellite markers. Variogram-based analysis not only avoids bias due to the underestimation of population variance in the presence of spatial autocorrelation, but also provides estimates of population genetic diversity and the degree and extent of spatial genetic structure accounting for autocorrelation.