A nonparametric model of empirical length distributions to inform stratification of fishing effort for integrated assessments

Length frequency data (LFD) are an important input to integrated stock assessments, and statistical tests for variables that significantly influence the length distribution of fish can assist in the definition of effort strata, typically denoted as fisheries or sub-fisheries, in order to account for important systematic differences due to availability and/or gear-specific selectivity of size classes. Here, a nonparametric model of the probability density function of lengths is described which, instead of fitting to LFD directly, is fitted to the set of length quantiles for a pre-determined set of corresponding probabilities p (in this instance 0.05, 0.1-0.9 in 0.1 increments, and 0.95). These length quantile data (LQD) can be constructed with individual hauls as sampling units or after pooling hauls to sampling units defined by combinations of covariates such as gear type, spatial block, depth strata, or the sex of sampled fish. The length quantiles are modelled as a Gaussian response variable using a Generalised Additive Mixed Model (GAMM) with smoothing splines fitted for each combination of the covariates (i.e. gear type, depth strata and sex). Graphical presentation of the fitted splines along with standard error of difference bounds were used to investigate where differences were significant in order to assist in the optimal definition of sub-fisheries. The model has the advantage of greater generality and sensitivity in detecting differences compared to modelling a single quantile such as the median. In addition, fitting splines allows flexible and parsimonious modelling of length distributions of any shape. The model is demonstrated using LQD from commercial fishing for Patagonian toothfish at Heard Island.