New derivation reduces bias and increases power of Ripley’s L index

Abstract

The development of distance-dependent growth and survival models depends on an understanding of the spatial distribution of the population in question. Ripley’s L index ( L R) has found wide application for examining the spatial dispersion of plants. L R is calculated as the square root of a weighted sum of the number of observed plant pairs that are less than a certain distance apart. The weighting used by L R inflates the pair count sum to compensate for reduced pair counts for plants near the plot boundary. Using Monte Carlo simulations, we show that the variance in the observed number of tree pairs is not stabilized by the square root transformation at low expected counts. The non-linearity of the square root transformation introduces a consistent bias in both the first and second moments of the tree pair distribution. We present a derived estimator for Ripley’s analytical L index ( L A) that provides a more accurate estimate of variance and mean. This new approach, based on a true Poisson variate, includes a modification of the previous edge correction method that incorporates a global estimate of mean pair density, rather than local values. This reduces variance caused by stochastic placement of point pairs near the boundary. Monte Carlo simulations verified the predictions of this model over a wide range of population sizes (25–1400). Simulation results showed that the L R numerical estimate of the confidence limit was overly conservative by nearly a factor of two. The improved power and accuracy provided by L A suggest that it would be fruitful to reexamine population spatial dispersion data in the literature using the analytical estimator ( L A). As an illustration, the power and accuracy of L R and L A to detect non-random spatial dispersions is compared using generated populations and six stands of mapped trees in Connecticut.