A mathematical proof comparing the statistical properties between two common approaches for parameterizing sex-composition likelihoods in fishery stock assessments

Abstract

Two primary methods for parameterizing sex-specific age and length composition likelihoods in fishery stock assessments exist, which we refer to as the ‘Joint and ‘Split’ approaches. When using the ‘Joint’ approach, sex-composition data are assumed to arise from a single statistical model that describes the probability of sampling across all ages and sexes in a given year. By contrast, the ‘Split’ approach assumes that sex-composition data arises from several statistical models: sex-specific models that describe the probability of sampling ages within each sex, and an additional model that describes the sex-ratio information from composition data. In this mathematical proof, we derive the statistical properties of both approaches under multinomial and Dirichlet-multinomial sampling and show that they produce equivalent model expectations. However, we illustrate that the ‘Split’ approach leads to smaller assumed variances when sampling follows a Dirichlet-multinomial distribution, because overdispersion acts independently within each sex rather than jointly across sexes. Given that both approaches yield equivalent model expectations, we generally recommend using the ‘Joint’ approach for parameterizing sex-composition likelihoods. The ‘Joint’ approach is simpler to implement, aligns with most fisheries sampling designs, and is able to jointly account for overdispersion and sampling correlations across sexes. However, we acknowledge that in some cases, the ‘Split’ approach may be more appropriate.