Evaluating the predictive performance of empirical estimators of natural mortality rate using information on over 200 fish species

ICES Journal of Marine Science ICES Journal of Marine Science (2015), 72(1), 82– 92. doi:10.1093/icesjms/fsu136 Original Article Evaluating the predictive performance of empirical estimators of natural mortality rate using information on over 200 fish species Amy Y. Then 1,2 *, John M. Hoenig 1, Norman G. Hall 3,4, and David A. Hewitt 5 1 Virginia Institute of Marine Science, College of William & Mary, PO Box 1346, Gloucester Point, VA 23062, USA Institute of Biological Sciences, Faculty of Science, University of Malaya, Lembah Pantai, 50603 Kuala Lumpur, Malaysia 3 Centre for Fish and Fisheries Research, Murdoch University, South Street, Murdoch, Western Australia 6150, Australia 4 Department of Fisheries, Western Australian Fisheries and Marine Research Laboratories, PO Box 20, North Beach, Perth, Western Australia 6920, Australia 5 US Geological Survey, Western Fisheries Research Center, Klamath Falls, OR, USA 2 *Corresponding author: tel: +60128977292; fax: +60379674178; e-mail: , Then, A. Y., Hoenig, J. M., Hall, N. G., and Hewitt, D. A. Evaluating the predictive performance of empirical estimators of natural mortality rate using information on over 200 fish species. – ICES Journal of Marine Science, 72: 82– 92. Received 17 September 2013; revised 17 July 2014; accepted 18 July 2014; advance access publication 20 August 2014. Many methods have been developed in the last 70 years to predict the natural mortality rate, M, of a stock based on empirical evidence from comparative life history studies. These indirect or empirical methods are used in most stock assessments to (i) obtain estimates of M in the absence of direct information, (ii) check on the reasonableness of a direct estimate of M, (iii) examine the range of plausible M estimates for the stock under consideration, and (iv) define prior distributions for Bayesian analyses. The two most cited empirical methods have appeared in the literature over 2500 times to date. Despite the importance of these methods, there is no consensus in the literature on how well these methods work in terms of prediction error or how their performance may be ranked. We evaluate estimators based on various combinations of maximum age (tmax), growth parameters, and water temperature by seeing how well they reproduce .200 independent, direct estimates of M. We use tenfold cross-validation to estimate the prediction error of the estimators and to rank their performance. With updated and carefully reviewed data, we conclude that a tmaxbased estimator performs the best among all estimators evaluated. The tmax-based estimators in turn perform better than the Alverson –Carney method based on tmax and the von Bertalanffy K coefficient, Pauly’s method based on growth parameters and water temperature and methods based just on K. It is possible to combine two independent methods by computing a weighted mean but the improvement over the tmax-based methods is slight. Based on cross-validation prediction error, model residual patterns, model parsimony, and biological considerations, we recommend 0.73 −0.33 the use of a tmax-based estimator (M = 4.899t−0.916 L1 , max , prediction error ¼ 0.32) when possible and a growth-based method (M = 4.118K prediction error ¼ 0.6, length in cm) otherwise. Keywords: Alverson and Carney, data limited, data poor situations, fish mortality, Hoenig, indirect estimators of M, Jensen, natural mortality, Pauly, prior distribution. Introduction One of the most influential stock assessment parameters, natural mortality rate (M), is generally believed to be difficult to estimate reliably and directly. By direct, we refer to estimation of M using information strictly pertaining to the species or stock of interest. Five examples are (i) measuring total mortality in an unexploited stock, (ii) relating total mortality to the amount of fishing and extrapolating to zero fishing effort, (iii) measuring both total mortality and exploitation rates and solving for components of mortality (e.g. Hewitt et al., 2007), (iv) mark-recapture and telemetry studies (e.g. Hoenig et al., 1998; Knip et al., 2012), and (v) estimating M internally in an integrated stock assessment model (see Maunder and Punt, 2013). Direct estimation methods of M are often data intensive, thus limiting their application to relatively data-rich stocks. A host of methods have been developed in the last 70 years to estimate M from surrogate life history information. These # International Council for the Exploration of the Sea 2014. All rights reserved. For Permissions, please email: Performance of empirical estimators of natural mortality life history correlates include maximum age tmax (Tanaka, 1960; Bayliff, 1967; Ohsumi, 1979; Hoenig, 1983), von Bertalanffy growth coefficient K (Beverton and Holt, 1959; Ralston, 1987; Charnov, 1993; Jensen, 1996) as well as composites of these variables—for example, the von Bertalanffy asymptotic size L1 or W1, growth coefficient K, and water temperature T (Pauly, 1980), and both tmax and K (Alverson and Carney, 1975). Ecological theory and empirical evidence provides strong basis for prediction of M from surrogate information not only for fish stocks but also for other animals and even plants (Hoenig, 1983; McCoy and Gillooly, 2008). We use the term indirect or empirical to categorize this suite of methods since their derivation relies on comparative life history studies to borrow strength from many species (e.g. Pauly, 1980; Hoenig, 1983). Some models for estimating M have been derived based on theoretical ecological considerations; these models constitute empirical methods by our definition if they rely on data to estimate one or more unknown parameters (e.g. Alverson and Carney, 1975; Gunderson and Dygert, 1988). Although these empirical methods are often perceived as being less reliable than their data-rich counterparts, a consensus is that empirical methods are useful and very important particularly in a data-poor setting (e.g. Brodziak et al., 2011). Empirical methods are routinely applied in stock assessments, both for data-poor and data-rich stocks, in the following ways: (i) obtain point estimates of M in the absence of direct information, (ii) examine the reasonableness of a directly estimated value of M, (iii) obtain a range of plausible values for M for the stock by applying a suite of empirical methods, and (iv) define prior distributions of M in Bayesian analyses. A large body of evidence suggests that M varies over age and size (e.g. Peterson and Wroblewski, 1984; McGurk, 1986; Lorenzen, 1996; Gislason et al., 2010). Nonetheless, most fisheries scientists would agree that a single value for M can provide a useful representation of mortality over much of the exploitable lifespan of a species. Simulation studies have indicated that the assumption of a constant M in stock assessments is still very useful even when the simulated populations are subject to age- and time-varying M dynamics (Deroba and Schu (...truncated)