A test for spatial relationships between neighbouring plants in plots of heterogeneous plant density

Abstract. Maps of plant individuals in (x, y) coordinates (i.e. point patterns) are currently analysed through statistical methods assuming a homogeneous distribution of points, and thus a constant density within the study area. Such an assumption is seldom met at the scale of a field plot whilst delineating less heterogeneous subplots is not always easy or pertinent. In this paper we advocate local tests carried out in quadrats partitioning the plot and having a size objectively determined via a trade‐off between squared bias and variance. In each quadrat, the observed pattern of points is tested against complete spatial randomness (CSR) through a classical Monte‐Carlo approach and one of the usual statistics. Local tests yield maps of p‐values that are amenable to diversified subsequent analyses, such as computation of a variogram or comparison with co‐variates. Another possibility uses the frequency distribution of p‐values to test the whole point pattern against the null hypothesis of an inhomogeneous Poisson process. The method was demonstrated by considering computer‐generated inhomoge‐neous point patterns as well as maps of woody individuals in banded vegetation (tiger bush) in semi‐arid West Africa. Local tests proved able to properly depict spatial relationships between neighbours in spite of heterogeneity/clustering at larger scales. The method is also relevant to investigate interaction between density and spatial pattern in the presence of resource gradients.     

Analyzing species distributions among temporary ponds with a permutation test approach to the join-count statistic

The Bluff Springs Sand Ponds (BSSPs) are a set of closely-spaced temporary ponds of varying hydroperiod, depth and surface area. We sampled crustacean communities of 15 ponds throughout hydroperiods in 1996 to examine species distributions among ponds. Although ponds were closely spaced (within ca. 16 ha), most species were present in subsets of the 15 ponds. We then analyzed spatial patterns of 12 crustacean species for complete spatial randomness (CSR) using join-count statistics. However, the join-count was designed for large-samples (n>50), so we further analyzed (by simulation) the join-count and a variation of the join-count (Cliff & Ord, 1981) for small-scale reliability. Simulation results revealed that neither testing distribution was reliable for n<30. We then used a permutation test to analyze species distributions and concluded that some species were distributed non-randomly. Therefore, further investigations of mechanisms causing species distributions (e.g., hydroperiod, physical/chemical conditions, biotic interactions) are clearly prescribed. The permutation test should be useful for studies of species distribution patterns among other temporary waters, and can help focus studies on causal mechanisms of distributions among small numbers of temporary aquatic habitats.     

Testing Equality of Correlated Means in the Presence of Unequal Variances and Missing Values

Several statistics are proposed for testing the hypothesis of equality of the means of bivariate normal distribution with unknown variances and correlation coefficient when observations are missing on both variatea. The null distributions of the statistics are approximated by well-known distributions. The empirical sizes and powers of the statistics are computed and compared with paired t test and some of the known statistics based on available data. The comparisons support the use of two of the statistics proposed in this paper.     

Measurement of non-randomness in spatial distributions

The measurement of departure from randomness in spatial distributions has widespread application in ecological work. Several “indices of non-randomness” are compared with regard to their dependence on sample number, sample size and density. Criteria for the best choice of index for specific situations are discussed. A new coefficient C_x is proposed for use with positively contagious distributions and tests of significance are given. When C_x and another index (S²/m−1) are used for positive and negative contagion respectively, values ranging from −1 through 0 (random) to +1 are obtained, regardless of sample number, sample size or density.     

Spatial Point Pattern Analysis by using Voronoi Diagrams and Delaunay Tessellations – A Comparative Study

Given a spatial point pattern, we use various characteristics of its Voronoi diagram and Delaunay tessellation to extract information of the dependence between points. In particular, we use the characteristics to construct statistics for testing complete spatial randomness. It is shown that the minimum angle of a typical Delaunay triangle is sensitive to both regularity and clustering alternatives, whilst the triangle's area or perimeter is more sensitive to clustering than regularity. These statistics are also sensitive to the Baddeley‐Silverman cell process.     

What spatial data do we need to develop global mammal conservation strategies?

Spatial data on species distributions are available in two main forms, point locations and distribution maps (polygon ranges and grids). The first are often temporally and spatially biased, and too discontinuous, to be useful (untransformed) in spatial analyses. A variety of modelling approaches are used to transform point locations into maps. We discuss the attributes that point location data and distribution maps must satisfy in order to be useful in conservation planning. We recommend that before point location data are used to produce and/or evaluate distribution models, the dataset should be assessed under a set of criteria, including sample size, age of data, environmental/geographical coverage, independence, accuracy, time relevance and (often forgotten) representation of areas of permanent and natural presence of the species. Distribution maps must satisfy additional attributes if used for conservation analyses and strategies, including minimizing commission and omission errors, credibility of the source/assessors and availability for public screening. We review currently available databases for mammals globally and show that they are highly variable in complying with these attributes. The heterogeneity and weakness of spatial data seriously constrain their utility to global and also sub-global scale conservation analyses.     

基于不同零模型的点格局分析

在种群空间格局研究中,定量分析格局及其形成过程已成为生态学家的主要目标。在量化分析的众多方法中,点格局分析是最常用的方法,而在选择零模型时,完全空间随机模型以外的复杂零模型很少使用,实际上,这些零模型可能有助于认识格局的内在特征。为此,我们在研究实例中,选择完全空间随机模型(complete spatial randomness)、泊松聚块模型(Poisson cluster process)和嵌套双聚块模型(nested double-cluster process)对典型草原处于不同恢复演替阶段的羊草(Leymus chinensis)种群空间格局进行了分析。结果发现:完全空间随机模型仅能检测种群在不同尺度下的格局类型;而通过泊松聚块模型和嵌套双聚块模型检验表明,在恢复演替的初期阶段,羊草种群在小尺度范围内偏离泊松聚块模型,而在整个取样范围内完全符合嵌套双聚块模型;随着恢复演替时间的推移,在恢复演替的后期,在整个取样尺度上,羊草种群与泊松聚块模型相吻合。这是很有意义的生态学现象。这一实例表明在应用点格局分析种群空间格局时,仅通过完全空间随机模型的检验来分析格局特征,或许很难论证复杂的生态过程,而选择一些完全空间随机模型以外的较复杂的零模型,可能发现一些有价值的生态学现象,对揭示格局掩盖下的内在机制有所裨益。     

On methods of spatial analysis for genotyped individuals

Spatial autocorrelation methods have commonly been applied to individual-based spatial genetic studies, although their properties and the relations among the statistics have not been carefully examined. This paper first introduces a reformulation of widely used spatial statistics using point processes. When Moran's I statistics are applied to allele frequencies within an individual, the frequencies are no longer continuous variables but have only three discrete values and specific interpretations of Moran's I statistics and the number of alleles in common (NAC) can be expressed as the weighted sum of join-count statistics. The distributions of minor genotypes are amplified in Moran's I depending on the allele frequency in the population, while NAC uses a constant weighting system. Under the point process framework, spatial analysis can be conducted on the common theoretical base, from individual locations to genetic distributions of different levels, (for example, genotype and allele). The methodology is demonstrated by application to field data for molecular ecological studies of Fagus crenata population dynamics.     

The spatial distribution of birds and carabid beetles in pine plantation forests: the role of landscape composition and structure

Aim To evaluate the joint and independent effects of spatial location, landscape composition and landscape structure on the distribution patterns of bird and carabid beetle assemblages in a mosaic landscape dominated by pine plantation forests. Location A continuous 3000‐ha landscape mosaic with native maritime pine Pinus pinaster plantations of different ages, deciduous woodlands and open habitats, located in the Landes de Gascogne forest of south‐western France. Methods We sampled breeding birds by 20‐min point counts and carabid beetles by pitfall trapping using a systematic grid sampling of 200 points every 400 m over the whole landscape. Explanatory variables were composed of three data sets derived from GIS habitat mapping: (1) spatial variables (polynomial terms of geographical coordinates of samples), (2) landscape composition as the percentage cover of the six main habitats, and (3) landscape structure metrics including indices of fragmentation and spatial heterogeneity. We used canonical correspondence analysis with variance partitioning to evaluate the joint and independent effects of the three sets of variables on the ordination of species assemblages. Moran's I correlograms and Mantel tests were used to assess for spatial structure in species distribution and relationships with separate landscape attributes. Results Landscape composition was the main factor explaining the distribution patterns of birds and carabids at the mesoscale of 400 × 400 m. Independent effects of spatial variables and landscape structure were still significant for bird assemblages once landscape composition was controlled for, but not for carabid assemblages. Spatial distributions of birds and carabids were primarily influenced by the amount of heathlands, young pine plantations, herbaceous firebreaks and deciduous woodlands. Deciduous woodland species had positive responses to edge density, while open habitat species were positively associated with mean patch area. Main conclusions Forest birds were favoured by an increase in deciduous woodland cover and landscape heterogeneity, but there was no evidence for a similar effect on carabid beetles. Fragmentation of open habitats negatively affected both early‐successional birds and carabids, specialist species being restricted to large heathlands and young plantations. Several birds of conservation concern were associated with mosaics of woodlands and grasslands, especially meadows and firebreaks. Conserving biodiversity in mosaic plantation landscapes could be achieved by the maintenance of a significant amount of early‐successional habitats and deciduous woodland patches within a conifer plantation matrix.     

Accounting for tree spatial distribution in a comparison of plot sizes and shapes in dense forest and woodland in Benin (West Africa)

The study examined simultaneously, the effect of tree spatial distribution, inventory plot size and shape on the estimation error of basal area in two contrasting environments. Twenty and fifteen square plots of 1 ha each (divided into 100 quadrats of 0.01 ha) were randomly set in dense forest and woodland, respectively. Thirteen subplots of various shapes and sizes were obtained from the association of adjacent quadrats. Estimation error was calculated using residual mean square of one‐way ANOVA, based on replications of subplot within 1 ha plots. Tree spatial distribution was measured using Green index. Weighted linear regression and mixed effect models were applied to Box & Cox transformed data. In general, the estimation error of basal area decreased with increase in subplot size. However, the effects of tree spatial distribution and plot shape varied with the vegetation type. Where trees tended to be aggregated, estimation error increased with degree of aggregation, and rectangular plots of 0.24 ha produced an acceptable precision. It was concluded that 0.24 ha rectangular plots can be used in tropical environments where the target parameters vary constantly according to one direction, while square plots of the same size are optimal for reliable analysis in case of randomness.     

基于零模型的宁夏荒漠草原优势种群点格局分析

植物种群空间分布格局是多种生态过程综合作用的结果。明确植物优势种群个体的空间分布格局与形成机制有助于认识种群生态适应对策与群落多样性维持机制。以宁夏荒漠草原优势种群蒙古冰草、短花针茅、牛枝子和牛心朴子为研究对象,采用完全空间随机零模型分析其种群空间分布格局特征,并通过异质泊松零模型与泊松聚块零模型探讨生境异质性、扩散限制等因子在其空间分布格局形成过程中的作用。结果显示:(1)完全空间随机零模型下,4个物种在4 m尺度范围内表现为聚集分布,随尺度增大,逐渐过渡到随机分布和均匀分布。(2)在排除生境异质性的异质泊松零模型下,蒙古冰草种群在整个研究尺度上表现为随机分布;牛枝子、短花针茅和牛心朴子种群仅分别在0—0.2、0.1—0.4 m与0—0.2 m尺度范围内发生偏离,表现为均匀分布与聚集分布,其他尺度均为随机分布。(3)在排除扩散限制的泊松聚块零模型下,所研究种群均表现为随机分布。综上,荒漠草原优势种群在小尺度范围内主要表现为聚集分布;生境异质性与扩散限制均是驱动其空间分布格局形成的重要因子,相对而言,小尺度空间范围内扩散限制的作用更为显著。     

Distributions of Hardy-Weinberg equilibrium test statistics

It is well established that test statistics and P-values derived from discrete data, such as genetic markers, are also discrete. In most genetic applications, the null distribution for a discrete test statistic is approximated with a continuous distribution, but this approximation may not be reasonable. In some cases using the continuous approximation for the expected null distribution may cause truly null test statistics to appear nonnull. We explore the implications of using continuous distributions to approximate the discrete distributions of Hardy–Weinberg equilibrium test statistics and P-values. We derive exact P-value distributions under the null and alternative hypotheses, enabling a more accurate analysis than is possible with continuous approximations. We apply these methods to biological data and find that using continuous distribution theory with exact tests may underestimate the extent of Hardy–Weinberg disequilibrium in a sample. The implications may be most important for the widespread use of whole-genome case–control association studies and Hardy–Weinberg equilibrium (HWE) testing for data quality control.     

A likelihood ratio test for equality of deviations from randomness

The use of the sample variance-to-mean ratio as a measure of deviation from randomness in spatial pattern is reviewed. The likelihood ratio method of constructing a statistical test for the equality of several population variance-to-mean ratios is described, and details are provided for the special case where counts are modelled as arising from a negative binomial distribution. This test is illustrated by application to example data sets in ecology. Likelihood ratio tests represent a general methodology whereby relationships among several indices of aggregation can be systematically investigated, provided one is able to specify a suitable parametric form for the underlying distributions.     

基于加权Ripley's K-function的多尺度景观格局分析——以江苏盐城滨海湿地为例

以1987,1992,1997,2002,2007年的遥感影像为例,首次尝试使用加权Ripley's K-function的多尺度格局分析方法,计算了20年来景观异质性在江苏盐城滨海湿地的时间变化和空间分布趋势。通过对研究区的样带划分以及景观类型的点状化处理,建立滨海湿地样带图层和1987—2007年间各类型景观的点格局数据库,从而分析滨海湿地不同类型景观的空间聚集特征变化。基于加权Ripley's K-function的计算表明,在各级空间尺度和时间变化上,各类型湿地的斑块都呈现出空间聚集分布状态,且1987年以来,不同湿地类型的聚集空间特征尺度和空间分布强度均出现了大幅的增减变化,除互花米草滩之外的自然湿地的聚集空间特征尺度和强度都有明显下降甚至少到无法被检测到,而人工湿地却呈现聚集特征尺度和强度的双增长,且该聚集程度有逐渐增强的趋势。分析表明,既考虑样点的空间位置信息又考虑样点分布范围的加权Ripley's K-function方法能很好地表征湿地景观在多尺度上的变异,且与传统空间景观指数等分析方法的结论在一定程度上保持一致。     

Comparing the performances of Diggle's tests of spatial randomness for small samples with and without edge-effect correction: application to ecological data

Diggle's tests of spatial randomness based on empirical distributions of interpoint distances can be performed with and without edge-effect correction. We present here numerical results illustrating that tests without the edge-effect correction proposed by Diggle (1979, Biometrics 35, 87-101) have a higher power for small sample sizes than those with correction. Ignoring the correction enables detection of departure from spatial randomness with smaller samples (down to 10 points vs. 30 points for the tests with correction). These results are confirmed by an example with ecological data consisting of maps of two species of trees in a West African savanna. Tree numbers per species per map were often less than 20. For one of the species, for which maps strongly suggest an aggregated pattern, tests without edge-effect correction enabled rejection of the null hypothesis on three plots out of five vs. on only one for the tests with correction.     

A generalization of Pielou's index of non-randomness.

Pielou's index of non-randomness a measuring the distribution of point spatial patterns on a plane generalizes to an index in n-dimensional Euclidean space Eⁿ. This defines an index of non-randomness measuring the extent of regularity, randomness, or aggregation in distributions of point spatial patterns in Eⁿ.     

Spatial point pattern analysis of available and exploited resources

A patchy spatial distribution of resources underpins many models of population regulation and species coexistence, so ecologists require methods to analyse spatially‐explicit data of resource distribution and use. We describe a method for analysing maps of resources and testing hypotheses about species' distributions and selectivity. The method uses point pattern analysis based on the L‐function, the linearised form of Ripley's K‐function. Monte Carlo permutations are used for statistical tests. We estimate the difference between observed and expected values of L(t), an approach with several advantages: 1) The results are easy to interpret ecologically. 2) It obviates the need for edge correction, which has largely precluded the use of L‐functions where plot boundaries are “real”. Including edge corrections may lead to erroneous conclusions if the underlying assumptions are invalid. 3) The null expectation can take many forms, we illustrate two models: complete spatial randomness (to describe the spatial pattern of resources in the landscape) and the underlying pattern of resource patches in the landscape (akin to a neutral landscape approach). The second null is particularly useful to test whether spatial patterns of exploited resource points simply reflect the spatial patterns of all resource points. We tested this method using over 100 simulated point patterns encompassing a range of patterns that might occur in ecological systems, and some very extreme patterns. The approach is generally robust, but Type II decision errors might arise where spatial patterns are weak and when trying to detect a clumped pattern of exploited points against a clumped pattern of all points. An empirical example of an intertidal lichen growing on barnacle shells illustrates how this technique might be used to test hypotheses about dispersal mechanisms. This approach can increase the value of survey data, by permitting quantification of natural resource patch distribution in the landscape as well as patterns of resource use by species.     

Spatial pattern analysis in ecology based on Ripley's K-function: Introduction and methods of edge correction

Abstract. Spatial pattern analysis based on Ripley's K-function is a second-order analysis of point patterns in a twodimensional space. The method is increasingly used in studies of spatial distribution patterns of plant communities, but the statistical methods involved are sometimes poorly understood or have been modified without evaluating the effects on results. The procedures of field data acquisition, statistical analysis, and the test for the null hypothesis of complete spatial randomness are described and the presentation of results is discussed. Different methods of edge correction were tested on a computer-generated random pattern and a mapped distribution of a Mediterranean shrubland. The inclusion of buffer zones around mapped plots describes the spatial pattern most accurately, but may not warrant the additional labour involved. Three variations of the weighted edge correction yielded comparable results for the distribution patterns tested. The toroidal edge correction may give biased results for non-random patterns. Recommendations for standardisation of the statistical procedures and data presentation are given.     

Mapping the spatial variability of the field distribution of sterile Anastrepha ludens over a release area

The sterile insect technique (SIT) is an environmental-friendly method used against Anastrepha ludens Loew (Diptera: Tephritidae) populations. This study aimed to perform an analysis of the spatial variability of the field distribution of sterile A. ludens using a geostatistical approach along with Geographic Information Systems (GIS). Field data on captures of sterile A. ludens during a Valencia orange season over a release area were analysed using spherical, exponential and Gaussian variograms. Such variograms were evaluated by criteria such as the mean absolute error, average standard error, root mean square error and the coefficient of determination. Results revealed a spatially structured distribution of sterile A. ludens across the release area. Interpolated models by Ordinary Kriging technique exhibited continuous surfaces evidencing spatial heterogeneity of the distribution of flies. Such a result evidenced that the spatial dynamics of flies significantly varied despite the planned uniform coverage of the release. The GIS led to integrating spatial information of the spatial dynamics through one single model. The release activity should be improved westward of the studied area, as the final model suggested that the ratio sterile: wild is lower than that in the east. This study provides insights into the spatial analysis of the distribution of sterile flies further than one single geographical point. Moreover, it highlights geostatistical techniques and GIS as useful tools for the assessment of the impact and quality of the release activity over fruit-growing areas subjected to an area-wide integrated pest management approach.