A distribution-free model of length-frequency distribution to inform fishery stratification for integrated assessments

Length frequency distributions (LFD) of catch collected as a random sub-sample from hauls are an important input to integrated fish stock assessments. Statistical tests for variables that significantly influence the LFD helps in evaluating how to account for important differences in availability and/or gear specific selectivity. A distribution-free model of LFD is described which is fitted to the set of length quantiles L for a predetermined set of corresponding probabilities P (in this instance 0.05, 0.1 to 0.9 in 0.1 increments, and 0.95), constructed from individual length measurements of individual hauls as sampling units or after pooling hauls to sampling units defined by combinations of discrete-valued covariates of gear type, spatial block, depth strata, and optionally the sex of sampled fish. The Ls were modelled as a Gaussian response variable using a Generalised Additive Mixed Model (GAMM) with cubic smoothing splines fitted to P for each combination of the covariates (i.e. gear type, depth strata and sex). Within-sampling unit error structure incorporated a statistical weight for sample size and a continuous autoregressive component due to the cumulative nature of the quantiles. Additional error terms included a random spatial block term and interactions of spatial blocks with other covariates. Graphical presentation of fitted splines along with standard error of difference bounds were used to investigate where differences were significant. 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 cubic splines allows modelling of standard sigmoidally-shaped cumulative LFDs as well as other more complex shapes, for example, due to multi-modality. The model and model testing using analysis of residuals and random effect estimates are demonstrated using LFD from commercial fishing for Patagonian toothfish at Heard Island (Division 58.5.2). The model supports fitting separate selectivity functions by gear, depth and sex in Division 58.5.2.