A method for calculating a meta-analytical prior for the natural mortality rate using multiple life history correlates

ICES Journal of Marine Science ICES Journal of Marine Science (2015), 72(1), 62– 69. doi:10.1093/icesjms/fsu131 Original Article A method for calculating a meta-analytical prior for the natural mortality rate using multiple life history correlates Owen S. Hamel 1* 1 Northwest Fisheries Science Center, National Oceanographic and Atmospheric Administration, 2725 Montlake Boulevard East, Seattle, WA 98112, USA *Corresponding author: tel: +1 206 860 3481; fax: +1 206 860 6792; e-mail: Hamel, O. S. A method for calculating a meta-analytical prior for the natural mortality rate using multiple life history correlates. – ICES Journal of Marine Science, 72: 62 –69. Received 30 October 2013; revised 1 July 2014; accepted 7 July 2014; advance access publication 2 August 2014. The natural mortality rate M is an important parameter for understanding population dynamics, and is extraordinarily difficult to estimate for many fish species. The uncertainty associated with M translates into increased uncertainty in fishery stock assessments. Estimation of M within a stock assessment model is complicated by its confounding with other life history and fishery parameters which are also uncertain, some of which are typically estimated within the model. Ageing error and variation in growth, which may not be fully modelled, can also affect estimation of M, as can various assumptions, including the form of the stock – recruitment function (e.g. Beverton– Holt, Ricker) and the level of compensation (or steepness), which may be fixed (or limited by a prior) in the model. To avoid these difficulties, stock assessors often assume point estimates for M derived from meta-analytical relationships between M and more easily measured life history characteristics, such as growth rate or longevity. However, these relationships depend on estimates of M for a great number of species, and those estimates are also subject to errors and biases (as are, to a lesser extent, the other life history parameters). Therefore, at the very least, some measure of uncertainty in M should be calculated and used for evaluating uncertainty in stock assessments and management strategy evaluations. Given error-free data on M and the covariate(s) for a metaanalysis, prediction intervals would provide the appropriate measure of uncertainty in M. In contrast, if the relationship between the covariate(s) and M is exact and the only error is in the estimates of M used for the meta-analysis, confidence intervals would appropriate. Using multiple published meta-analyses of M’s relationship with various life history correlates, and beginning with the uncertainty interval calculations, I develop a method for creating combined priors for M for use in stock assessment. Keywords: fish, Gunderson, Hoenig, Jensen, McCoy and Gillooly, meta-analysis, natural mortality rate, Pauly, prediction interval, prior distribution, stock assessment. Introduction The majority of models commonly used for stock assessment, except surplus production models, require or produce an estimate of the instantaneous natural mortality rate (M ). However, M is a particularly difficult parameter to estimate for many commercially important fish species, whether estimated independently or within a population model. The high degree of uncertainty in the value of this parameter is a problem that can greatly affect the accuracy and precision of the stock assessment results. Yet most stock assessments use a point estimate for M and, at best, include sensitivity analyses across a range of values which are considered plausible for M. Management strategy evaluations, which are used to evaluate how well management strategies work given uncertainty in and estimation errors for population and fisheries parameters within a stock assessment framework, rarely consider misspecification of M (e.g. Punt, 2003; Kraak et al., 2008; Punt et al., 2008). The single value of M used in many stock assessment models must be taken to represent an averaging of the natural mortality rates experienced by subsets of a stock (defined by, e.g. size, age, sex, period, and/or location) above a minimum age. The reality that M varies with time, age, sex, and cohort, as well as due to environmental conditions, predation, and inter- and intraspecific competition, makes estimation even more difficult. However, the data rarely exist to estimate the variation in even some of these directions (Vetter, 1988). Thus, most statistical catch-at-age stock assessments Published by Oxford University Press on behalf of International Council for the Exploration of the Sea 2014. This work is written by (a) US Government employee(s) and is in the public domain in the US. The natural mortality rate assume a constant M, or, at most, include some variability across age or length and/or differences between the sexes. While Lorenzen (2000) and Gislason et al. (2010) provide compelling arguments for having M vary with age, the data available to estimate even a single, time-, and age-invariant M are subject to a number of errors and uncertainties. These include ageing error and bias, which can lead to quite poor estimates of M (e.g. Beamish, 1979), uncertainty in fishery and survey selectivity (particularly for dome-shaped selectivity, for which the estimate of the drop-off in selectivity with age or size can be highly correlated with the estimate of M ), and uncertainty in catch levels and the fishing mortality rate (F ) over time. Variance and bias in otolith-based ageing tends to increase with fish age, especially for those fish that essentially stop growing past a certain age or size, resulting in annuli which are extremely narrow and difficult to identify (Chilton and Beamish, 1982; Campana, 2001). Bias in ageing can lead to poor understanding of fish population dynamics (Yule et al., 2008). Fishery and survey selectivity can peak at an intermediate size or age, decreasing thereafter due to ontogenetic movement into areas that are more difficult to fish, or due to physiological changes allowing larger fish to escape capture. However, estimating the degree of reduced (dome-shaped) selectivity is also difficult due to correlation of the rate of decline in selectivity with age with M and other parameters (including steepness (h), catchability (q), and F), and the fact that selectivity itself is often time varying. Finally, it can be difficult to decompose the total mortality rate into its natural and fishing-related components (Aenes et al., 2007). Estimating M within stock assessment models is difficult and often ill advised due to the issues raised above. Simulation analyses using a statistical catch-at-age model found that estimation of M was relatively accurate when age data and index data were available from the beginning of significant fishing, h and selectivity parameters were known, and the stock was fished down to a relatively low level during the period of the data (Magnusson and Hilborn, 2007). However, these ar (...truncated)