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

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) otherwise.