Ratio Data with Random Denominators: Models and Applications

• Activity Number: 160
• Type: Invited
• Date/Time: Tuesday, August 13, 2002 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistics & the Environment*
• Author(s): Martin Liermann*+ and E. Steel
• Affiliation(s): NW Fisheries Science Center and NW Fisheries Science Center
• Keywords: ratios ; fish density ; ANCOVA ; randomization tests ; waste composition ; catch per unit effort

Abstract

Observations in which one random value is divided by another random value present unique analytical challenges. The best statistical technique for a particular situation varies depending on the unit of inference and the joint distribution of the numerator and denominator. We present three case studies of environmental research questions in which the distribution of ratios is compared between two populations. To augment these, we also construct three hypothetical statistical models which allow us to generate ratio data for which we know the distributional properties. Using data collected in the field for the case studies and generated data from the models, we compare standard approaches to drawing inference about differences between two distributions of ratios: t-tests, t-tests with transformations, randomization tests, non-parametric tests, 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 develop a few simple guidelines for choosing a test. They are based on the unit of inference and relationship between the numerator and denominator.