Developing Negative Binomial Mixed Models to Partition Variance in Fishery-Independent Survey Data

Partitioning total variability into multiple temporal and spatial sources (i.e., variance components) is a powerful approach to accommodate complex data structures.  For example, an environmental state variable may vary among repeated samples from a single site, from site-to-site within a lake, from lake-to-lake, and over time.  Models for estimating variance components have been applied to a wide variety of aquatic indices including water chemistry variables, measurements of species richness, stream habitat characteristics, metrics of fish growth, and catch-per-unit effort data.  To date, most variance-components frameworks have been based on linear models that assume normally distributed error structures.  When these models are applied to count data, the response variable is commonly transformed (usually using a logarithmic transformation) prior to fitting the model in an attempt to accommodate the normality and homogeneity of variance assumptions.  Assuming a normal distribution for observations of fish abundance is often not ideal because these counts are typically non-negative integers with high variances and low means, not to mention other issues that arise when log-transforming data such as how to treat zero observations during the analysis.  The negative binomial distribution represents an alternative to log-transformation (e.g., an alternative assumption about the mean-variance relationship) that can be applied to discrete count data; however, the partitioning of variance in this context is less straightforward than for generalized linear mixed models that assume normality.  We developed a method of estimating variance components using negative binomial mixed models.  We applied these models to count data generated by multiple fishery-independent surveys of percids from across the Great Lakes basin.  These surveys varied in overall sampling intensity, general magnitude of the catch, and in the proportion of zero catches.  Even so, negative binomial models produced reasonable approximation to the count data.  Results show that spatial and temporal partitioning of variability in survey catch, adjusted for effort, differed among the lakes, and this has an influence on what survey design will best achieve different objectives.