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.     

Model selection and information theory in geographical ecology

Aim   Although parameter estimates are not as affected by spatial autocorrelation as Type I errors, the change from classical null hypothesis significance testing to model selection under an information theoretic approach does not completely avoid problems caused by spatial autocorrelation. Here we briefly review the model selection approach based on the Akaike information criterion (AIC) and present a new routine for Spatial Analysis in Macroecology (SAM) software that helps establishing minimum adequate models in the presence of spatial autocorrelation.
Innovation    We illustrate how a model selection approach based on the AIC can be used in geographical data by modelling patterns of mammal species in South America represented in a grid system ( n  = 383) with 2° of resolution, as a function of five environmental explanatory variables, performing an exhaustive search of minimum adequate models considering three regression methods: non-spatial ordinary least squares (OLS), spatial eigenvector mapping and the autoregressive (lagged-response) model. The models selected by spatial methods included a smaller number of explanatory variables than the one selected by OLS, and minimum adequate models contain different explanatory variables, although model averaging revealed a similar rank of explanatory variables.
Main conclusions    We stress that the AIC is sensitive to the presence of spatial autocorrelation, generating unstable and overfitted minimum adequate models to describe macroecological data based on non-spatial OLS regression. Alternative regression techniques provided different minimum adequate models and have different uncertainty levels. Despite this, the averaged model based on Akaike weights generates consistent and robust results across different methods and may be the best approach for understanding of macroecological patterns.     

Methods to account for spatial autocorrelation in the analysis of species distributional data: a review

Species distributional or trait data based on range map (extent‐of‐occurrence) or atlas survey data often display spatial autocorrelation, i.e. locations close to each other exhibit more similar values than those further apart. If this pattern remains present in the residuals of a statistical model based on such data, one of the key assumptions of standard statistical analyses, that residuals are independent and identically distributed (i.i.d), is violated. The violation of the assumption of i.i.d. residuals may bias parameter estimates and can increase type I error rates (falsely rejecting the null hypothesis of no effect). While this is increasingly recognised by researchers analysing species distribution data, there is, to our knowledge, no comprehensive overview of the many available spatial statistical methods to take spatial autocorrelation into account in tests of statistical significance. Here, we describe six different statistical approaches to infer correlates of species’ distributions, for both presence/absence (binary response) and species abundance data (poisson or normally distributed response), while accounting for spatial autocorrelation in model residuals: autocovariate regression; spatial eigenvector mapping; generalised least squares; (conditional and simultaneous) autoregressive models and generalised estimating equations. A comprehensive comparison of the relative merits of these methods is beyond the scope of this paper. To demonstrate each method's implementation, however, we undertook preliminary tests based on simulated data. These preliminary tests verified that most of the spatial modeling techniques we examined showed good type I error control and precise parameter estimates, at least when confronted with simplistic simulated data containing spatial autocorrelation in the errors. However, we found that for presence/absence data the results and conclusions were very variable between the different methods. This is likely due to the low information content of binary maps. Also, in contrast with previous studies, we found that autocovariate methods consistently underestimated the effects of environmental controls of species distributions. Given their widespread use, in particular for the modelling of species presence/absence data (e.g. climate envelope models), we argue that this warrants further study and caution in their use. To aid other ecologists in making use of the methods described, code to implement them in freely available software is provided in an electronic appendix.     

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.     

城市住区形态时空模拟——以厦门岛为例

住区形态变迁受到人口迁移、住区满意度和低碳城市发展政策等因素的限制,常用的土地利用模型难以有效表征这一相互制约关系,使得这方面的研究仍然相对不足。通过耦合SD模型和CLUE-S模型,充分发挥了2个模型在宏观情景模拟和微观土地分配上的优势,模拟了住区、人口、住区碳足迹等制约因素的相互关系,为住区形态变迁时空模拟提供了一种有效的方法。以厦门岛为例,根据研究区历史统计数据、问卷调查数据构建了住区形态变迁SD模型,模拟了基准情景、紧凑情景和低碳情景3种不同发展情景下各类住区类型的用地需求,结合CLUE-S模型预测了3种情景下2009年—2020年各类住区类型的用地范围。结果表明,基准年住区类型Ⅰ、Ⅱ、Ⅲ三者占地面积比例为1∶1.18∶0.83,基准情景下2018年住区类型Ⅲ将成为主要的住区类型。低碳发展和紧凑发展是惯性发展的两种极端情况,体现在总住区面积、人均住宅面积和人均碳足迹大小的变化,但是对厦门岛总人口数量的影响并不大。根据目前厦门的发展趋势,低碳发展情景与紧凑发展情景相结合可能更靠近现实。在空间分布上,住区类型Ⅰ未来不再新建;住区类型Ⅱ遵循现状继续发展的惯性较大;住区类型Ⅲ分布在征地成本相对较低的区域。模型模拟结果能够为住区用地规划、住区发展对策建议提供有效的技术支撑。     

Less than eight (and a half) misconceptions of spatial analysis

Spatial analyses are indispensable analytical tools in biogeography and macroecology. In a recent Guest Editorial, Hawkins (Journal of Biogeography, 2012, 39 , 1–9) raised several issues related to spatial analyses. While we concur with some points, we here clarify those confounding (1) spatial trends and spatial autocorrelation, and (2) spatial autocorrelation in the response variable and in the residuals. We argue that recognizing spatial autocorrelation in statistical modelling is not only a crucial step in model diagnostics, but that disregarding it is essentially wrong.     

Random sampling does not exclude spatial dependence: The importance of neutral models for ecological hypothesis testing

Lájer (2007) notes that, to investigate phytosociological and ecological relationships, many authors apply traditional inferential tests to sets of relevés obtained by non-random methods. Unfortunately, this procedure does not provide reliable support for hypothesis testing because non-random sampling violates the assumptions of independence required by many parametric inferential tests. Instead, a random sampling scheme is recommended. Nonetheless, random sampling will not eliminate spatial autocorrelation. For instance, a classical law of geography holds that everything in a piece of (biotic) space is interrelated, but near objects are more related than distant ones. Because most ecological processes that shape community structure and species coexistence are spatially explicit, spatial autocorrelation is a vital part of almost all ecological data. This means that, independently from the underlying sampling design, ecological data are generally spatially autocorrelated, violating the assumption of independence that is generally required by traditional inferential tests. To overcome this drawback, randomization tests may be used. Such tests evaluate statistical significance based on empirical distributions generated from the sample and do not necessarily require data independence. However, as concerns hypothesis testing, randomization tests are not the universal remedy for ecologists, because the choice of inadequate null models can have significant effects on the ecological hypotheses tested. In this paper, I emphasize the need of developing null models for which the statistical assumptions match the underlying biological mechanisms.     

Spatial genetic structure of the southeastern North American endemic, Ceratiola ericoides (Empetraceae)

Understanding the spatial distribution of genetic diversity (i.e., spatial genetic structure [SGS]) within plant populations can elucidate mechanisms of seed dispersal and patterns of recruitment that may play an important role in shaping the demography and spatial distribution of individuals in subsequent generations. Here we investigate the SGS of allozyme diversity in 2 populations of the southeastern North American endemic shrub, Ceratiola ericoides. The data suggest that the 2 populations have similar patterns of SGS at distances of 0-45 m that likely reflect the isolation by distance (IBD) model of seed dispersal. However, at distances >or=50 m, the pattern of SGS differs substantially between the 2 populations. Whereas one population continues to reflect the classical IBD pattern, the second population shows a marked increase in autocorrelation coefficient (r) values at 50-75 m. Furthermore, r values at these distances are as much as 33% higher than at 0-5 m where the highest r value would be predicted by IBD. A likely explanation is the differing frequencies of 2 fruit morphologies in these populations and the greater role that birds play in seed dispersal in the second population.     

Grinnellian and Eltonian niches and geographic distributions of species

In the recent past, availability of large data sets of species presences has increased by orders of magnitude. This, together with developments in geographical information systems and statistical methods, has enabled scientists to calculate, for thousands of species, the environmental conditions of their distributional areas. The profiles thus obtained are obviously related to niche concepts in the Grinnell tradition, and separated from those in Elton's tradition. I argue that it is useful to define Grinnellian and Eltonian niches on the basis of the types of variables used to calculate them, the natural spatial scale at which they can be measured, and the dispersal of the individuals over the environment. I use set theory notation and analogies derived from population ecology theory to obtain formal definitions of areas of distribution and several types of niches. This brings clarity to several practical and fundamental questions in macroecology and biogeography.     

Landscape Features Fail to Explain Spatial Genetic Structure in White-Tailed Deer Across Ohio,USA

Landscape features influence wildlife movements across spatial scales and have the potential to influence the spread of disease. Chronic wasting disease (CWD) is a fatal prion disease affecting members of the family Cervidae, particularly white-tailed deer (Odocoileus virginianus), and the first positive CWD case in a wild deer in Ohio, USA, was recorded in 2020. Landscape genetics approaches are increasingly used to better understand potential pathways for CWD spread in white-tailed deer, but little is known about genetic structure of white-tailed deer in Ohio. The objectives of our study were to evaluate spatial genetic structure in white-tailed deer across Ohio and compare the support for isolation by distance (IBD) and isolation by landscape resistance (IBR) models in explaining this structure. We collected genetic data from 619 individual deer from 24 counties across Ohio during 2007–2009. We used microsatellite genotypes from 619 individuals genotyped at 11 loci and haplotypes from a 547-base pair fragment of the mitochondrial DNA control region. We used spatial and non-spatial genetic clustering tests to evaluate genetic structure in both types of genetic data and empirically optimized landscape resistance surfaces to compare IBD and IBR using microsatellite data. Non-spatial genetic clustering tests failed to detect spatial genetic structure, whereas spatial genetic clustering tests indicated subtle spatial genetic structure. The IBD model consistently outperformed IBR models that included land cover, traffic volume, and streams. Our results indicated widespread genetic connectivity of white-tailed deer across Ohio and negligible effects of landscape features. These patterns likely reflect some combination of minimal resistive effects of landscape features on white-tail deer movement in Ohio and the effects of regional recolonization or translocation. We encourage continued CWD surveillance in Ohio, particularly in the proximity of confirmed cases. © 2021 The Wildlife Society. This article has been contributed to by US Government employees and their work is in the public domain in the USA.     

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.     

A computer program to simulate multilocus genotype data with spatially autocorrelated allele frequencies

Many models for inference of population genetic parameters are based on the assumption that the data set at hand consists of groups displaying within-group Hardy-Weinberg equilibrium at individual loci and linkage equilibrium between loci. This assumption is commonly violated by the presence of within-group spatial structure arising from nonrandom mating of individuals due to isolation by distance (IBD). This paper proposes a model and simulation method implemented in a computer program to flexibly simulate data displaying such patterns. The program permits displaying of smooth spatial variations of allele frequencies due to IBD and more abrupt variations due to presence of strong barriers to gene flow. It is useful in assessing performance of various statistical inference methods and in designing spatial sampling schemes. This is shown by a simulation study aimed at assessing the extent to which IBD patterns affect accuracy of cluster inferences performed in models assuming panmixia. The program is also used to study the effects of spatial sampling scheme (e.g. sampling individuals in clumps or uniformly across the spatial domain). The accuracy of such inferences is assessed in terms of number of inferred populations, assignment of individuals to populations and location of borders between populations. The effect of spatial sampling was weak while the effect of IBD may be substantial, leading to the inference of spurious populations, especially when IBD was strong with respect to the size of the sampling domain. The model and program are new and have been embedded in the R package Geneland, for user convenience and compliance with existing data formats.     

安徽黄山青冈种群遗传结构的空间自相关分析

以黄山－青冈（Ｃyclobalanopsis glauca)种群为例,研究了种群内等位基因的空间格局,在种群内,大多数等位基因的Moran'sI数大于期望值,但只有两个等位基因存在显著的正空间自相关;如果考虑不同的无性系分株时,大多数等位基因在短距离内存在显著的空间自相关。相关图表明不同距离间隔,Moran'sI指数变化无规律,表明没有哪个进化因子起决定作用,但无性系繁殖在空间自相关中起重要作用,尤其是在近距离。     

兴安落叶松种群格局的分形特征——关联维数

种群格局关联维数揭示出个体空间关联的尺度变化规律,表明种群个体的空间相关程度。采用关联维数对大兴安岭兴安落叶松种群格局的研究表明,各类兴安落叶松林中兴安落叶松种群格局具有较高的关联维数（接近２）,个体空间关联程度较高,其次序为杜鹃－兴安落叶松林（１．７４６）＞草类－兴安落叶松林（１．７４０）＞越桔－兴安落叶松林（１．５５０）＞杜香－兴安落叶松林（１．４６８）。兴安落叶松－白桦林中兴安落叶松种群格局的关联维数较小（＜１．５１２,远离２）,个体间关联较弱,处于劣势伴生地位。通过将各天然森林类型与兴安落叶松人工林比较显示,个体空间相关程度由高至低的次序为兴安落叶松人工林（１．７６２）＞兴安落叶松天然林（１．６２６）＞兴安落叶松－白桦林（１．４３４）,揭示出兴安落叶松种群在不同森林类型中个体空间关联的尺度变化的差异。文中还对综合运用各种分形维数揭示种群格局尺度变化特征的问题进行了讨论。     

Generating and testing predictions about community structure: Which theory is relevant and can it be tested with observational data?

Three issues are discussed relevant to the controversy over using null models and observational data on guild structure to test community-level predictions based on limiting similarity theory. First, I argue that most limiting similarity theory is not based on reasonable assumptions for plants and that the theory that is relevant does not generate any predictions about expected guild proportionality on a small spatial scale. Therefore, regardless of adequacy of the statistical methods, the predictions being tested by the body of literature using null models to test for niche limitation are unlikely to be relevant in most plant comunities. Second, assuming that the predictions are after all worth being tested, I argue that most tests using the guild approach do not provide adequate explanations of how the defined guilds could lead to greater competition within vs. between guilds. If this is not true for the particular guilds used, the predictions of guild proportionality or size constancy will not be valid. Third, I address the controversy over whether field experiments can provide more solid evidence than observational data about the role of competition in determining community structure by (1) suggesting methods of dealing with potential drawbacks of field experiments, and (2) suggesting alternative experimental approaches for directly addressing issues about community structure.     

Spatial autocorrelation and red herrings in geographical ecology

Aim Spatial autocorrelation in ecological data can inflate Type I errors in statistical analyses. There has also been a recent claim that spatial autocorrelation generates ‘red herrings’, such that virtually all past analyses are flawed. We consider the origins of this phenomenon, the implications of spatial autocorrelation for macro‐scale patterns of species diversity and set out a clarification of the statistical problems generated by its presence. Location To illustrate the issues involved, we analyse the species richness of the birds of western/central Europe, north Africa and the Middle East. Methods Spatial correlograms for richness and five environmental variables were generated using Moran's I coefficients. Multiple regression, using both ordinary least‐squares (OLS) and generalized least squares (GLS) assuming a spatial structure in the residuals, were used to identify the strongest predictors of richness. Autocorrelation analyses of the residuals obtained after stepwise OLS regression were undertaken, and the ranks of variables in the full OLS and GLS models were compared. Results Bird richness is characterized by a quadratic north–south gradient. Spatial correlograms usually had positive autocorrelation up to c. 1600 km. Including the environmental variables successively in the OLS model reduced spatial autocorrelation in the residuals to non‐detectable levels, indicating that the variables explained all spatial structure in the data. In principle, if residuals are not autocorrelated then OLS is a special case of GLS. However, our comparison between OLS and GLS models including all environmental variables revealed that GLS de‐emphasized predictors with strong autocorrelation and long‐distance clinal structures, giving more importance to variables acting at smaller geographical scales. Conclusion Although spatial autocorrelation should always be investigated, it does not necessarily generate bias. Rather, it can be a useful tool to investigate mechanisms operating on richness at different spatial scales. Claims that analyses that do not take into account spatial autocorrelation are flawed are without foundation.     

Testing Pairwise Association between Spatially Autocorrelated Variables: A New Approach Using Surrogate Lattice Data

Background

Independence between observations is a standard prerequisite of traditional statistical tests of association. This condition is, however, violated when autocorrelation is present within the data. In the case of variables that are regularly sampled in space (i.e. lattice data or images), such as those provided by remote-sensing or geographical databases, this problem is particularly acute. Because analytic derivation of the null probability distribution of the test statistic (e.g. Pearson''s r) is not always possible when autocorrelation is present, we propose instead the use of a Monte Carlo simulation with surrogate data.

Methodology/Principal Findings

The null hypothesis that two observed mapped variables are the result of independent pattern generating processes is tested here by generating sets of random image data while preserving the autocorrelation function of the original images. Surrogates are generated by matching the dual-tree complex wavelet spectra (and hence the autocorrelation functions) of white noise images with the spectra of the original images. The generated images can then be used to build the probability distribution function of any statistic of association under the null hypothesis. We demonstrate the validity of a statistical test of association based on these surrogates with both actual and synthetic data and compare it with a corrected parametric test and three existing methods that generate surrogates (randomization, random rotations and shifts, and iterative amplitude adjusted Fourier transform). Type I error control was excellent, even with strong and long-range autocorrelation, which is not the case for alternative methods.

Conclusions/Significance

The wavelet-based surrogates are particularly appropriate in cases where autocorrelation appears at all scales or is direction-dependent (anisotropy). We explore the potential of the method for association tests involving a lattice of binary data and discuss its potential for validation of species distribution models. An implementation of the method in Java for the generation of wavelet-based surrogates is available online as supporting material.     

Spatial autocorrelation invalidates the Dow-Cheverud test

Dow and Cheverud (Am. J. Phys. Anthropol. 68:367–373, 1985) have published a statistical test for comparing any three similarity matrices. Using both simulations and analytical arguments, I establish that the presence of spatial autocorrelation, a common feature of geographically based anthropological and biological data, causes this test to reject too often. Increasing the spatial autocorrelation increases the spurious rejection rate. About 20% of the papers that reference Dow and Cheverud's paper have used their test with spatially autocorrelated data. Mantel's (Cancer Res. 27:209–220, 1967) method, when used as a test of spatial autocorrelation, is unaffected by these considerations, since its null hypothesis is that the data are uncorrelated. © 1992 Wiley-Liss, Inc.     

Effects of incorporating spatial autocorrelation into the analysis of species distribution data

Aim  Spatial autocorrelation (SAC) in data, i.e. the higher similarity of closer samples, is a common phenomenon in ecology. SAC is starting to be considered in the analysis of species distribution data, and over the last 10 years several studies have incorporated SAC into statistical models (here termed 'spatial models'). Here, I address the question of whether incorporating SAC affects estimates of model coefficients and inference from statistical models.
Methods  I review ecological studies that compare spatial and non-spatial models.
Results  In all cases coefficient estimates for environmental correlates of species distributions were affected by SAC, leading to a mis-estimation of on average c . 25%. Model fit was also improved by incorporating SAC.
Main conclusions  These biased estimates and incorrect model specifications have implications for predicting species occurrences under changing environmental conditions. Spatial models are therefore required to estimate correctly the effects of environmental drivers on species present distributions, for a statistically unbiased identification of the drivers of distribution, and hence for more accurate forecasts of future distributions.