Correct in theory but wrong in practice: Bias caused by using a lognormal distribution to penalize annual recruitments in fish stock assessment models

• We evaluated estimation performance of two equally justified recruitment penalties.
• A normally distributed penalty had far less bias than a lognormal penalty.
• A lognormal penalty should not be used for recruitment.
• More research is needed on the performance of penalties in stock assessment.

Penalties are widely used for a range of parameters while fitting fish stock assessment models. Penalizing annual recruitments for deviating from an underlying mean recruitment is probably the most common. Assuming that recruits are log-normally distributed for the purposes of this penalty is theoretically justifiable. In practice, however, bias may be induced because this distributional assumption includes a term equal to the summation of the log observed data, which in the case of recruitment equals the summation of the log recruitment parameters that are not data. Using simulation, the potential for bias caused by assuming that recruits were log-normally distributed was explored, and results were contrasted with the assumption that log-recruitment was normally distributed, an alternative that avoids the potentially troublesome summation term. Spawning stock biomass (SSB) and recruitment were negatively biased, while fishing mortality (F) was positively biased under the assumption of log-normally distributed recruitments, and the bias worsened closer to the terminal year. The bias also worsened when the true underlying F was low relative to natural mortality, and with domed fishery selectivity. Bias in SSB, recruitment, and F was nonexistent or relatively small under the assumption that log-recruitment was normally distributed. Distributional assumptions for penalties used in assessment models should be reviewed to reduce the potential for biased estimation. These results also provide further support for simulation testing to evaluate statistical behavior of assessment models.