Estimating the relationship between abundance and distribution

Numerous studies investigate the relationship between abundance and distribution using indices reflecting one of the three aspects of distribution: proportion of area occupied, aggregation, and geographical range. Using simulations and analytical derivations, we examine whether these indices provide unbiased estimates of the relationship when estimated from count data. The indices investigated include the proportion of empty samples, the proportion of structurally empty samples, Lloyds index of patchiness, measures derived from Lorenz curves (such as D-95 and the Gini index), and measures based on Euclidean distance to the centre of gravity of the spatial distribution. Only the proportion of structurally empty areas, Lloyds index, and indices of the distance to the centre of gravity of the spatial distribution are unbiased at all levels of abundance. The remaining indices generate relationships between abundance and distribution even in cases where no underlying relationships exists, although the problem decreases for measures derived from Lorenz curves when samples contain more than four individuals on average. To illustrate the problem, the indices are applied to juvenile North Sea cod, Gadus morhua.