Novel Tools for Analyzing Proportional Size Distribution Index Data

—Proportional size distribution (PSD) indices are convenient measures for numerically summarizing size structure of fish populations. Several novel methods for analyzing PSD index data were examined as a means for potentially improving inferences of size structure: generalized linear mixed modeling, calculation of location quotients, and simultaneous construction of 100(1 a)% confidence intervals for PSD indices through multinomial modeling. The methods are demonstrated using simulated and previously published PSD index data. Generalized linear mixed modeling was used to (1) conduct a model selection analysis of PSD index data relative to a change in harvest regulation for walleyes Sander vitreus, (2) partition the variance of PSD index data among spatial, temporal, and environmental attributes, and (3) nonlinearly relate PSD data to adult density for largemouth bass Micropterus salmoides. The location quotient, which is an index for describing relative concentration of proportional data, was used to describe spatial and temporal localization of largemouth bass and walleye PSD index data. Finally, a modified method of constructing simultaneous 100(1 a)% confidence intervals for multinomial proportions was used to produce 95% confidence intervals for traditional and incremental PSD indices based on length frequency data from muskellunge Esox masquinongy. The approaches described herein can assist fishery biologists in size structure assessments of fish populations and ultimately aid in cost-effective fisheries management. Proportional size distribution (PSD) indices, which formerly were referred to separately as proportional and relative stock densities, are convenient measures for numerically summarizing fish length frequency data and facilitating comparisons of size structure among fish populations (Willis et al. 1993; Guy et al. 2007). Statistical techniques that frequently are used to analyze PSD index data include the chi-square test, Student’s t-test, and logistic regression (Neumann and Allen 2007). Although these methods are appropriate for analyzing PSD index data, the types of research questions that they can address are limited. Chi-square and Student’s t-tests examine whether PSD index data were generated from a particular statistical distribution, whereas logistic regression can be used to determine whether PSD indices are conditionally related in a linear fashion to one or more independent variables. Many other research questions pertaining to size structure may be of interest to fishery biologists in exploring PSD index data but cannot be analyzed using the aforementioned methods. For example, biologists may be interested in (1) partitioning variability in PSD index data among different habitat scales (e.g., cove, lake, and region), (2) exploring whether PSD indices are nonlinearly related to one or more independent variables, or (3) incorporating possible temporal or spatial autocorrelations in PSD index data to improve inferences regarding size structure. Given the various size structure questions that are of interest to fishery biologists, we identified the need for a general framework of PSD index data analysis that would be applicable to a variety of management situations. Such a framework ultimately would help simplify analyses of PSD index data, facilitate comparisons among different analytic techniques, and aid in determining the effects of different analytic assumptions on the recommended courses of action. The goal for our research was to describe how generalized linear mixed models (GLMMs) provide such an analytic framework for PSD index data and demonstrate how GLMMs can be used to investigate research questions that are of interest to fishery managers. Additional goals were to (1) introduce the location quotient as a method for quantifying relative concentration of PSD index data and (2) evaluate whether simultaneous 100(1 a)% confidence intervals constructed via multinomial modeling of PSD * Corresponding author: [email protected] Received April 27, 2007; accepted December 30, 2007 Published online August 25, 2008 1233 North American Journal of Fisheries Management 28:1233–1242, 2008 Copyright by the American Fisheries Society 2008 DOI: 10.1577/M07-078.1 [Article]