Confidence Estimation of Ecological Aggregation Indices Based on Counts-a Robust Procedure

Abstract

SUMMARY The problem of confidence-interval estimation of indices of spatial aggregation based on counts, such as Lloyd's indices of patchiness and of mean crowding, and the variance-mean ratio, is considered. A distribution-free method based on a straightforward application of the jackknife is presented. The accuracy of the method in small samples is investigated by simulation. A transformation which should improve the small-sample performance is suggested. Application of the method to both one-sample and two-sample problems is illustrated with data previously analyzed by methods that require specific distributional assumptions. Indices of nonrandomness in the spatial distribution of organisms, based on the mean and variance of counts of individuals per sampling unit, are frequently used by ecologists. Examples are the 'variance-mean ratio', David and Moore's (1954) 'index of clumping', Morisita's (1962) 'index of dispersion' and Lloyd's (1967) 'index of mean crowding' and 'index of patchiness'. Certain theoretical objections to the use of any of these indices, when the sampling units are quadrats placed randomly over a continuum, have been pointed out. Paloheimo and Vukov (1976), for example, show that none of the indices is invariant under changes in quadrat size, for a variety of theoretical clustering processes. For this reason it has been suggested that, whenever possible, count-based indices should be eschewed in favour of distance-based measures. Although, in certain circumstances, distance-based methods are on surer theoretical ground than the count-based methods, there remain, nonetheless, situations in which count-based methods present the only alternative. This may be for theoretical or practical reasons. An example of the latter, given by Patil and Stiteler (1974), is in the study of soil arthropods. Counts of animals per given volume are relatively easy to obtain, but distance measurements are virtually impossible. A similar situation pertains when plankton are collected in a plankton net. In the first example, contiguous quadrat analysis, along the lines initially proposed by Greig-Smith (1952), would present an alternative to the use of a single count-based index or statistic, but in the second example this alternative would not be possible. Count-based methods are theoretically justified when organisms occupy discrete habitat areas (such as, for example, insects living on leaves of trees), for in this case the habitat areas form natural sampling units. Distance-based methods are not useful in such situations. However, even in situations where the use of count-based indices can be justified, there often remains a statistical problem associated with their use, namely how to assess the sampling variation associated with an index calculated from sample data. Without this, of