Evaluation of using random-at-length observations and an equilibrium approximation of the population age structure in fitting the von Bertalanffy growth function

• Paired age–length observations used for estimating growth and can be treated in multiple ways.
• Lengths as random for age results in good estimates of parameters with random sampling.
• Age as random for length results in good estimates with random and length-based sampling.
• Estimating growth using age conditioned on length requires an estimate of the population age structure.
• Simulation shows that an equilibrium approximation of that age structure may be appropriate.

When paired age–length samples are collected from fishery data, estimation of the growth function parameters has typically assumed that each length observation is a random sample of fish for a given age (traditional method). An alternative methodology (length-conditional) assumes each age sample taken is random with respect to that length. The length-conditional method uses the underlying population age structure to derive predictions and therefore has been limited to implementation inside the stock assessment model. This paper used simulation methods to evaluate the estimates of the von Bertalanffy growth parameters outside the assessment model using the traditional method and a length-conditional method based on an equilibrium approximation to the population age structure derived from an estimate of the total mortality rate. Both random and length-stratified sampling designs were evaluated along with a range of sample sizes. With a random sampling design, both the traditional and approximate length-conditional methods produced unbiased estimates of Linf and K. However, only the approximate length-conditional approach produced unbiased estimates when samples were length-stratified. Variability in the length-at-age relationship was better estimated by the approximate length-conditional approach for both sampling designs. The approximate length-conditional method was robust to small errors in total mortality rate, but biased with increasing levels of mortality misspecification. With a reasonable estimate of total mortality, the approximate length-conditional approach may be a viable alternative to the traditional method when estimating growth parameters, especially the variability in growth.