Estimating multinomial effective sample size in catch-at-age and catch-at-size models

Catch-at-age or catch-at-size stock assessment models require specification of an effective sample size (ESS) as a weighting component for multinomial composition data. ESS weights these data relative to other data that are fit, and is not an estimable parameter within a model that uses a multinomial likelihood. The ESS is typically less than the actual sample size (the number of fish) because of factors such as sampling groups of fish (clusters) that are caught together. A common approach for specifying ESS is to iteratively re-fit the model, estimating ESS after each fit so that the standardized residual variance is “correct,” until ESS converges. We survey iterative methods for determining ESS for a multinomial likelihood and apply them to two Great Lakes whitefish stocks. We also propose an extension of such methods: (the Generalized Mean Approach - GMA) for the case where ESS is based on mean age (or length) to account for correlation structures among proportions. Our extension allows for greater flexibility in the relationship between ESS and sampling intensity. Our results show that the choice of ESS estimation method can impact assessment model results. Simulations (in the absence of correlation structures) showed that all the approaches to calculating effective sample size could provide reasonable results on average, however methods that estimated annual ESS independently across years were highly imprecise. In our simulations and application, methods: that did account for correlation structure in annual proportions produced lower ESS than those that did not and suggested that these methods are adjusting for a deviation from the multinomial correlation structure. We recommend using methods that adjust for correlation structures in the proportions, and either assuming a constant ESS or, when there is substantial inter-annual variation in sampling levels, assuming ESS is related to sampling intensity and using the GMA or a similar approach to estimate that relationship.