Review and evaluation of likelihood functions for composition data in stock-assessment models: Estimating the effective sample size

Abstract Catch-at-age (or catch-at-length) data are one of the major components of most modern statistical stock assessment methods. Catch-at-age data provide, among other things, information about gear selectivity and recruitment strength. Catch-at-age data can also have a large influence on the estimates of fishing mortality, absolute abundance, and trends in abundance. The multinomial distribution describes the theoretical sampling process that is used to collect catch-at-age data, but only under the assumption of random sampling. Sampling designs generally employed to collect fishery-related data lead to age–composition estimates that depart from the strict theoretical multinomial probability distribution. Lack of independence can, for example, be due to size- or age-specific schooling or aggregating, causing positive correlations among individuals, and overdispersion. An additional cause of inadequacy of the multinomial assumption is model misspecification. Therefore, the effective sample size that should be used in an assessment model can be much smaller than the actual sample size. This can cause inappropriate weighting among data sets and negatively biased estimates of uncertainty. I use simulation analysis to evaluate five methods to estimate the effective sample size for catch-at-age data: (1) iterative multinomial likelihood; (2) normal approximation, using binomial variance; (3) lognormal likelihood with variance proportional to the inverse of the proportion; (4) Dirichlet likelihood; and (5) a multivariate normal approximation. The results show that all five methods perform similarly, but do not reduce estimation error relative to using the actual sample size unless the effective sample size is about one fifth of the actual sample size. All but (4) produced positively biased estimates of the effective sample size. If the effective sample size is not known within half an order of magnitude, I recommend using the lognormal likelihood with variance proportional to the inverse of the proportion and a regression against the actual sample size. This method is less computationally intense than (1), more robust than (2) and (4), and produces the least biased estimates of effective sample size, except for (4). Unlike (1), it can be included in Bayesian analysis.

[1]  Mark N. Maunder,et al.  A general framework for integrating the standardization of catch per unit of effort into stock assessment models , 2001 .

[2]  John R. Skalski,et al.  Variance estimation in integrated assessment models and its importance for hypothesis testing , 2007 .

[3]  Donald A. Jackson,et al.  Robust Regression Approach to Analyzing Fisheries Data , 1994 .

[4]  Kazuhiko Hiramatsu,et al.  Estimating mortality rates from tag recoveries: incorporating over-dispersion, correlation, and change points , 1994 .

[5]  André E. Punt,et al.  POPULATION MODELLING OF TASMANIAN ROCK LOBSTER, JASUS EDWARDSII, RESOURCES , 1997 .

[6]  Paul J. Starr,et al.  Bayesian assessment of the SNA1 snapper (Pagrus auratus) stock on the north‐east coast of New Zealand , 2001 .

[7]  David A. Fournier,et al.  MULTIFAN a Likelihood-Based Method for Estimating Growth Parameters and Age Composition from Multiple Length Frequency Data Sets Illustrated using Data for Southern Bluefin Tuna (Thunnus maccoyii) , 1990 .

[8]  Eric R. Ziegel,et al.  Generalized Linear Models , 2002, Technometrics.

[9]  Mark N. Maunder,et al.  Estimation of recruitment in catch-at-age models , 2003 .

[10]  Birgir Hrafnkelsson,et al.  A model for categorical length data from groundfish surveys , 2004 .

[11]  Jon T. Schnute,et al.  The influence of error on population estimates from catch-age models , 1995 .

[12]  Yong Chen,et al.  Impacts of outliers and mis-specification of priors on Bayesian fisheries-stock assessment , 2000 .

[13]  Kristin Helle,et al.  Survey design considerations for estimating the length composition of the commercial catch of some deep-water species in the northeast Atlantic , 2004 .

[14]  Rowan Haigh,et al.  Compositional analysis of catch curve data, with an application to Sebastes maliger , 2007 .

[15]  A. Maccall,et al.  STATUS OF BOCACCIO OFF CALIFORNIA IN 2003 , 2003 .

[16]  Terese H. Kendrick,et al.  Empirical weighting of multiple stock-abundance indices for parameter estimation and stock assessment in a multi-zone or multi-species fishery , 2001 .

[17]  David B. Sampson,et al.  Evaluation of Assumed Error Structure in Stock Assessment Models That Use Sample Estimates of Age Composition , 1998 .

[18]  James N. Ianelli,et al.  Bayesian stock assessment using catch-age data and the sampling - importance resampling algorithm , 1997 .

[19]  John R. Skalski,et al.  Integrating design- and model-based inference to estimate length and age composition in North Pacific longline catches , 2006 .

[20]  James N. Ianelli,et al.  A Statistical Model for Stock Assessment of Southern Bluefin Tuna with Temporal Changes in Selectivity , 2003 .

[21]  David A. Fournier,et al.  Impacts of atypical data on Bayesian inference and robust Bayesian approach in fisheries , 1999 .

[22]  Daniel K. Kimura,et al.  Approaches to Age-Structured Separable Sequential Population Analysis , 1990 .

[23]  P. R. Neal,et al.  Catch-Age Analysis with Auxiliary Information , 1985 .

[24]  M. Pennington,et al.  On estimating the age composition of the commercial catch of Northeast Arctic cod from a sample of clusters , 2003 .

[25]  John Sibert,et al.  AD Model Builder: using automatic differentiation for statistical inference of highly parameterized complex nonlinear models , 2012, Optim. Methods Softw..

[26]  J. Renwick,et al.  Quantifying annual variation in catchability for commercial and research fishing , 2003 .

[27]  Bruce C. Lubow,et al.  Fitting population models to multiple sources of observed data , 2002 .

[28]  A Gamma/Dirichlet model for estimating uncertainty in age-specific abundance of Norwegian spring-spawning herring , 2002 .

[29]  David A. Fournier,et al.  A General Theory for Analyzing Catch at Age Data , 1982 .

[30]  Mark N. Maunder,et al.  Integrating the standardization of catch-per-unit-of-effort into stock assessment models: testing a population dynamics model and using multiple data types , 2004 .