Modelling catch and effort data using generalised linear models, the Tweedie distribution, random vessel effects and random stratum-by-year effects
The current standard method for modelling catch and effort data for Patagonian toothfish (Dissostichus eleginoides) for CCAMLR areas is to model the haul-by-haul ratios of catch to effort as the response variable in a generalised linear model (GLM) with a square-root link function and a unit variance function. A time series of standardised CPUE estimates and their precision can be obtained from the ‘fishing year’ parameter estimates together with ‘baseline’ parameter estimates, their variance–covariance matrix, and the inverse-link function. An alternative GLM with a more rigorous theoretical basis is introduced here. Catch is modelled as the response variable using a GLM with a power variance function, with the power parameter (λ) estimated using a profile extended quasi-likelihood, and a log link function with log of effort as an offset. For 1 < λ < 2 this model is equivalent to assuming a compound Poisson-gamma distribution (i.e. Tweedie distribution) for catch that, unlike lognormal or gamma distributions, admits zero values.
In addition, random vessel effects are introduced into the GLM, as specified by a generalised linear mixed model (GLMM), in order to provide more efficient estimates of the standardised CPUE time series and more realistic estimates of their precision. Extra efficiency is gained by recovery of inter-vessel information as a result of the imbalance in the number of hauls in the year-by-vessel cross-classification. Further, the inclusion of an area stratum by fishing year interaction as an additional random effect in the GLMM is investigated. Fitting the stratum-by-year interaction as a fixed effect is problematic since it requires weighting of the individual stratum estimates by the areal extent of the stratum in order to obtain overall yearly standardised catch-per-unit-effort (CPUE) estimates. Without stratified random sampling, the determination of stratum areas that will give unbiased standardised CPUE estimates may be difficult. Fitting the stratum-by-year interaction as a random effect avoids this difficulty, and diagnostic methods to evaluate the validity of considering this interaction as random are described.
The methods are demonstrated using catch and effort data from two commercial D. eleginoides fisheries: the longline fishery around South Georgia (Subarea 48.3) and the trawl fishery around Heard Island and MacDonald Islands (Division 58.5.2).