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 tmax-based 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 the use of a tmax-based estimator (M=4.899tmax−0.916, prediction error = 0.32) when possible and a growth-based method (M=4.118K0.73L∞−0.33 , prediction error = 0.6, length in cm) otherwise.
Three common cross‐sectional catch‐curve methods for estimating total mortality rate (Z) are the Chapman–Robson, regression, and Heincke estimators. There are five unresolved methodological issues: (1) which is the best estimator, (2) how one should determine the first age‐group to use in the analysis, (3) how the variance estimators perform; and, for regression estimators, (4) how the observations should be weighted, including (5) whether and how the oldest ages should be truncated. We used analytical methods and Monte Carlo simulation to evaluate the three catch‐curve methods, including unweighted and weighted versions of the regression estimator. We evaluated four criteria for specifying the first age‐class used. Regression estimators were evaluated with four different methods of right data truncation. Heincke's method performed poorly and is generally not recommended. The two‐tailed χ2 test and one‐tailed z‐test for full selectivity described by Chapman and Robson did not perform as well as simpler criteria and are not recommended. Estimates with the lowest mean square error were generally provided by (1) the Chapman–Robson estimator with the age of full recruitment being the age of maximum catch plus 1 year and (2) the weighted regression estimator with the age of full recruitment being the age of maximum catch and with no right truncation. Differences in performance between the two methods were small (<6% of Z). The Chapman–Robson estimator of the variance of had large negative bias when not corrected for overdispersion; once corrected, it performed as well as or better than all other variance estimators evaluated. The regression variance estimator is generally precise and slightly negatively biased. We recommend that the traditional Chapman–Robson approach be corrected for overdispersion and used routinely to estimate Z. Weighted linear regression may work slightly better but is completely ad hoc. Unweighted linear regression should no longer be used for analyzing catch‐curve data. Received November 30, 2011; accepted July 4, 2012
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