Random denominators and the analysis of ratio data

Ratio data, observations in which one random value is divided by another random value, present unique analytical challenges. The best statistical technique varies depending on the unit on which the inference is based. We present three environmental case studies where ratios are used to compare two groups, and we provide three parametric models from which to simulate ratio data. The models describe situations in which (1) the numerator variance and mean are proportional to the denominator, (2) the numerator mean is proportional to the denominator but its variance is proportional to a quadratic function of the denominator and (3) the numerator and denominator are independent. We compared standard approaches for drawing inference about differences between two distributions of ratios: t-tests, t-tests with transformations, permutation tests, the Wilcoxon rank test, and ANCOVA-based tests. Comparisons between tests were based both on achieving the specified alpha-level and on statistical power. The tests performed comparably with a few notable exceptions. We developed simple guidelines for choosing a test based on the unit of inference and relationship between the numerator and denominator.     

Maximizing Species Representation under Limited Resources: A New and Efficient Heuristic

The maximal covering species problem (MCSP) is becoming an accepted method for selecting networks of reserves for the preservation of species and their habitats. Drawing on experience with the MC location P (MCLP), we expect that the objective function in the region of the optimum is quite flat. This implies many equal-optimal and near-optimal solutions for any particular problem. We propose a new type of heuristic for the MCSP: an interchange heuristic which frequently provides a better solution than greedy adding heuristics now commonly used. The proposed interchange heuristic, with multiple random starts, gives one a range of alternative sets of sites rather than a single mathematically selected solution, from which a final satisfactory solution can be chosen.