Jul 10, 2009 - Spatial statistics for fine-scale analysis mostly assume that .... out using algorithms and computer programs which I have described elsewhere.
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Fine-scale distributions of tropical animal mounds: a revised statistical analysis Martin Fisher Journal of Tropical Ecology / Volume 9 / Issue 03 / August 1993, pp 339 - 348 DOI: 10.1017/S0266467400007392, Published online: 10 July 2009
Link to this article: http://journals.cambridge.org/abstract_S0266467400007392 How to cite this article: Martin Fisher (1993). Fine-scale distributions of tropical animal mounds: a revised statistical analysis. Journal of Tropical Ecology, 9, pp 339-348 doi:10.1017/S0266467400007392 Request Permissions : Click here
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Journal of Tropical Ecology (1993) 9:339-348. With 4 figures
Fine-scale distributions of tropical animal mounds: a revised statistical analysis MARTIN FISHER Department of Biology, Sultan Qaboos University, PO Box 32486 Al-Khod, Muscat, Sultanate of Oman
ABSTRACT. Descriptions of the fine scale distribution of organisms have frequently been used to investigate various ecological phenomena. Unfortunately, the most widely used spatial analysis techniques are based on single index statistics, which convey only minimal information about the biological processes underlying the studied distributions. Such statistics cannot detect changes in pattern over different scales, and cannot identify some types of distribution. Additionally, both the use of such statistics on the distribution of individuals which have a non-negligible size, and the frequent failure to use an edge correction for points close to the boundaries of a sampled area, have led to the over-reporting of 'spaced out' ('regular') distributions. Using two spatial distributions recently analysed with a single index statistic (termite mounds, and earthmounds created by termites), I illustrate the benefits gained from using the spatial functions K(t), G(y) and F(x) to analyse both 'point events' and events which have a non-negligible size. These functions are considerably more informative about the nature of a spatial pattern and offer wide scope for the fitting of spatial models to biological distributions. KEY WORDS: earthmound, fine-scale spatial distribution, Isoptera, pattern, spacing, spatial analysis, termite.
INTRODUCTION
Descriptions of the fine-scale spatial arrangement of organisms have been used as a tool for investigating competition (Anderson 1971, Fowler 1986, McClure 1976), life-history phenomena (Jose et al. 1991, Sterner et al. 1986), seed dispersal (Hatton 1989), intra- and interspecific colony interactions in ants and termites (Cushman et al. 1988, Gontijo & Domingos 1991, Ryti & Case 1984, 1986, Spain et al. 1986) and many other spatially-based biological phenomena. However, the value of any insights gained from spatial analysis depends on both the validity and informativeness of the statistical methods used to describe the data. A wide variety of statistical techniques have been used for describing fine-scale spatial patterns... (Diggle. 1983, Greig-Smith 1983, Ripley 1981, Upton & Fingleton 1985). The data can be either counts of individuals within repeated quadrats, or the mapped positions of individuals. Whilst count data may rje\easier to collect and have been widely used in plant ecology (Greig-Smith 1983), the analysis of such data is especially sensitive to the choice 339
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of quadrat size (Diggle 1983). Completely mapped data may be more laborious to collect, but potentially contain more information. I am concerned here with techniques for the fine-scale spatial analysis of mapped data. Several statistics are available for examining the spatial distribution of mapped point events (Diggle 1983, Ripley 1979b), but in ecology the most widely used test is the statistic R (Clark & Evans 1954), which is based on nearest-neighbour distances. Spatial statistics for fine-scale analysis mostly assume that the sizes of individuals are negligible compared with the dimensions of the study area. Techniques also exist for describing the distribution of approximately circular individuals with a non-negligible size, such as a shrub or tree (Diggle 1981, Fisher 1990, Simberloff 1979), but they have been little used. When calculating R, allowance has to be made for nearest neighbours lying outside the study area, either by recording such events but not using them as a centre of measurement (Clark & Evans 1954) or by using an edge correction (Donnolly 1978). Failure to do so biases analyses towards the detection of 'spaced out' distributions. Many workers have not used any form of correction, and the ecological literature abounds with incorrectly described spatial patterns. Additionally, R, or any similar statistic, reduces a two dimensional map to an index. Information about spacing between events at different distances is lost, and certain patterns (such as regularly spaced clusters and regular spacing within clusters) cannot be detected. However, if all that is required is a description of a pattern as 'clustered', 'random' or 'regular', then the corrected ClarkEvans statistic may suffice, and has reasonable power (Ripley 1979b). More informative and powerful techniques for describing and testing the spatial arrangement of completely mapped patterns have been available for some time (reviewed in Diggle 1983, Ripley 1981 and Upton & Fingleton 1985), but have not been widely used in ecology (however, see Andersen 1992, Harkness & Isham 1983, Hatton 1989, Spain et al. 1986 and Sterner et al. 1986). These techniques are based on functions rather than on a single statistic. A statistician closely associated with the development of the new techniques has complained that'. . many weak, inappropriate or even misleading methods [of spatial analysis] continue to be used and can be seen in almost any issue of an ecological journal' (Ripley 1987). This complaint is still valid. Whilst the book by Diggle (1983) has encouraged use of the newer techniques, the relative complexity of the mathematics and computer programming involved, and the requirement for simulation for significance testing, have all hindered dissemination of the methods amongst ecologists. My aim here is to attempt to persuade ecologists to abandon simplistic nearest neighbour analyses for more rigorous and informative testing. To illustrate the benefits of such a move, I reanalyse two data sets which were previously analysed with the Clark-Evans' statistic. I indicate the greater depth of insight that can be gained from the use of analysis based on statistical functions and show how a second problem beguiling nearest-neighbour analyses - the analysis of biological 'events' which have a non-negligible size - can be solved.
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Figure 1. (A) Point map of nests and mounds of a wood feeding termite guild in an area of 50 X 50 m, digitized from Figure 2C of Gontijo (1991), and (B) map of earthmounds in an area of 227 X 177 m, digitized from the large earthmounds of Figure 3 of Oliveira-Filho (1992).
METHODS
The data
I electronically digitized two recently published sets of spatial data (Figure 1): the mounds of a wood feeding termite guild in a 0.25 ha area of savanna vegetation in south-east Brazil (Figure 2C in Gontijo & Domingos 1991), and earthmounds ('murundus') in a 5 ha area of savanna in central Brazil (the large mounds/sets type B in Figure 3 of Oliveira-Filho 1992). I also measured the maximum and minimum diameters of the earthmounds, which were not all exactly circular, and used the average diameter as the size of the mounds. The earthmounds were in an irregularly shaped area, and whilst algorithms could be devised to calculate edge effects for such an area, for simplicity I used a subset of the data. I imposed the largest rectangular area that would fit within Figure 3 of Oliveira-Filho (1992), and digitized the earthmounds within this area. The rectangle encompassed 52 of the original 71 earthmounds. To check the accuracy of the digitization technique, I digitized the point data in Figure 3 of Diggle (1983) and calculated the corrected Clark-Evans statistic for both this digitized data and for the actual data (Diggle 1983, Appendix A.3). The results were more or less identical (R = 1.561 for the actual data and 1.566 for the digitized data). Statistical methods and analysis
The Clark-Evans' statistic is given by R = fE/ fA, where fE is mean expected distance to the nearest neighbour and fA is mean actual distance to the nearest neighbour. Without an edge correction (Clark & Evans 1954) fE = 1I2\/ p, where p = density of events, whereas with an edge correction (Donnelly 1978) fE = 1/2V p + (0.051+0.041/Vn)(L/n), where L = boundary length of study
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area and n = number of events. Values of R close to 1 indicate a 'random' spatial pattern, values close to 0 a 'clustered' distribution, and values approaching the maximum of 2.149 a 'regular' distribution. The significance of any departure of R from 1 is tested using the standard deviate of the normal distribution (Clark & Evans 1954). Recently developed techniques for spatial analysis are based on the statistical functions K(t), G(y) and F(x). Different authors have used various notations for these functions; I adhere to the notation of Diggle (1983). The function K(t) was introduced by Ripley (1976, 1977). pK(t) is the expected number of points within a distance / of a randomly chosen point, where p = the density of points. Since the area that lies within distance t of a randomly chosen point is nf, the expected number of points lying within the area is /OT/2, and therefore for a simple Poisson (i.e. 'random') distribution of points, K(t) = ni2. G(y) is the empirical distribution function (EDF) of the distance of each event to its nearest neighbour, and F(x) is the EDF of the distances of random points to the nearest biological event. Diggle (1979) has referred to the use of these two functions as 'refined nearest neighbour analysis'. G(y) is particularly sensitive to 'regular' alternatives to complete spatial randomness (CSR), and F(x) to 'clustered' spatial processes (Diggle 1979, 1983). In most studies, whether the positions of individuals are mapped by ground survey, aerial photography or some other technique, a boundary (often rectangular) is used for convenience. If the mapped area is relatively homogeneous and the artificial boundary does not coincide with any environmental discontinuity, then individuals outside the area can be assumed to have the same type of distribution as those immediately within the mapped area, and a boundary correction can be used. This avoids discarding mapped events close to the boundary. Full descriptions of the three functions, including details of edge corrections, are given in Diggle (1983), Fisher (1990), Ripley (1981) and Upton & Fingleton (1985). Diggle's (1983) recommendations for the spatial analysis of completely mapped patterns were followed: (1) Calculate estimates of the functions, £(t), G(y) and F(x), for the data. (2) Test the null hypothesis (Ho) that the events were generated from a random biological process by using Monte Carlo analysis; i.e. generate points from a simple Poisson process conditioned on the density in the study area, calculate the three functions, and repeat the process 19 or 99 times, forP^ 0.05 andPsS 0.01 respectively. (3) Select the minimum and maximum values of the functions at each interval of distance to give confidence envelopes. The functions calculated for the data, and for the envelopes from Monte Carlo simulations are then plotted. A plot of y/{£(t)/n} against t is more useful than fc(t)t since under CSR, y/{£(t)/n} = t. G(y) and F(x) can be plotted either against distance (y or x respectively) or, more usefully, against the EDF for the distribution being tested (denoted Gx(y) and Fx{x)), if known. For a simple Poisson process, Gt(y) = l-exp(-7tXf) and Ft(x) = l-exp(-7tXx2).
If any of the three functions calculated for the data lie outside the envelopes, Ho is rejected, otherwise it is accepted. (4) If Ho is rejected, the position of the
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functions with respect to the envelopes can be used to indicate the type of spatial point process that might have generated the data. (5) Other spatial processes can then be tested, again using Monte Carlo analysis, until a satisfactory fit is obtained. If an Ho of CSR is rejected, there are many spatial processes that can be considered as possible models for the phenomena that have generated the finescale distribution under study (Diggle 1983, Ripley 1981, Upton & Fingleton 1985). Biological data is often clustered to some degree, and a useful model is the Poisson cluster process, in which a number of parent seeds are generated and then the required number of points are distributed around the seeds according to a radially symmetric normal distribution (Diggle 1983). The greater the standard deviation (a) of the distribution of the points around the seeds, the 'looser' is the clustering. The seeds do not normally form part of the final pattern. The distribution of individuals which have a non-negligible size cannot be initially tested against an Ho of CSR. SimberlofF(1979) derived a modified form of R for analysing the spatial distribution of non-overlapping circles, but this still suffers from the drawbacks of simple nearest neighbour analysis. If the data are non-overlapping and approximately circular, a simple, pragmatic solution is to use Monte Carlo analysis to test an Ho of 'size-conditioned' CSR: the spacing of points from a simple Poisson process is conditioned on a random permutation of the radii of the events in the data set, such that points are not allowed to lie closer to each other than the combined distance of their radii (Fisher 1990). Exact probabilities can be calculated for the goodness-of fit of a particular spatial process (Diggle 1983), but for exploratory work the graphical analyses described here suffice. Analysis can halt at step (4), above, if only an indication of the type and scale of pattern is required. All statistical analyses were carried out using algorithms and computer programs which I have described elsewhere (Fisher 1990). These are available upon request. For each data set I calculated both the corrected and uncorrected ClarkEvans statistic, for comparison, and the three spatial functions. Where CSR was rejected I attempted to determine what type of underlying process might have generated the data. RESULTS
For the termite guild, R = 0.969 (corrected) and R = 1.000 (uncorrected), and for the earthmounds, R = 1.418 (corrected) and R = 1.508 (uncorrected). Referring the values to the standard normal deviate indicates that the wood feeding termite guild is randomly distributed, whilst the earthmounds lie on the 'regular' side of random. The uncorrected values are, as expected, greater than the corrected values, always overestimating the spacing between points, although here the use of the correction did not alter the conclusions. My calcula-
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tions of R for the wood feeding termite guild differs from that of Gontijo & Domingos (1991), who determined the wood feeding guild to be clustered. However, since Gontijo & Domingos (1991) did not clearly state whether they used conspecifics or guild members as nearest neighbours for their analysis of the guild, whether nests of different species within the same mound had the same spatial coordinates, and since their Figure 2 from which I digitized the data is somewhat confused, it is not possible to determine the reason for the difference in our calculations. Nevertheless, this does not detract from further analysis of the data. My calculation of R for a subset of the large earthmounds compares well with R = 1.317 calculated for the whole data set (Oliveira-Filho 1992). Using the spatial functions R(t), 0(y) and P(x) an Ho of CSR was clearly rejected for the wood feeding termite guild (Figure 2A-G), in contrast to the conclusion based on R. All three functions lay at least in part outside the Monte Carlo envelopes. "\/{R.(t)/7t}, which is useful for indicating the type of spatial process that could have generated the data, lies beneath the minimum envelope at distances
