Maunder and Watters: Integrating environmental time series into stock assessment models 



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2 Environmental model 



/3 is estimated, each value in the parameter set f, is 

 fixed at zero, and a significant relationship is deter- 

 mined by testing if /i = 0, using a likelihood ratio test. 



3 Environmental model with process error 



Both p and e, are estimated and a significant relation- 

 ship is determined by testing if /3 = 0, with a likelihood 

 ratio test. 



Simulation testing 



Simulation analysis was carried out to test the perfor- 

 mance of the integi'ated approach and to compare this 

 approach to the traditional model. A simple age-structured 

 model (Appendix I) was set up to simulate a population for 

 20 years, starting from an unexploited population and gen- 

 erating catch, effort, and catch-at-age data. The simulated 

 recruitment was generated as having a component based 

 on an environmental time series and a random compo- 

 nent. Each component was given the same variance (0.6- ). 

 The environmental time series was randomly generated 

 with /3 = 1 for each simulation. The standard deviation of 

 the observation error in the CPUE index, o^^p^,^, was set 

 at 0.6, and the sample size of the catch-at-age data was 

 set to 50. The same age-structured model was then fitted 

 to the data to estimate the model parameters. The three 

 models defined in the previous section (traditional model, 

 environmental model, and environmental model with pro- 

 cess error) were tested with the simulated data. In addi- 

 tion to the parameters outlined in the description of the 

 three models, average recruitment, the catchability coef- 

 ficient, and the standard deviation of the fit to the CPUE 

 data were also estimated. We also used a model that had 

 constant recruitment to provide likelihood values to use in 

 testing the significance of the environmental model. 



The simulation analysis was repeated 500 times for four 

 scenarios: 1) using catch-at-age data for all years, 2) using 

 catch-at-age data for the first 10 years, 3) using catch-at- 

 age data for the last 10 years, 4) using catch-at-age data 

 for all years, but using j3 = when generating the simulat- 

 ed data. Scenario 4 was used to investigate the probability 

 of type-I error of the models when used in combination 

 with the statistical tests. For each set of simulated data 

 and each model, we determine how often a significant rela- 

 tionship between the logarithm of annual recruitment and 

 the environmental time series is detected, the estimate of 

 the slope of the relationship between the logarithm of an- 

 nual recruitment and the environmental time series, the 

 estimates of average recruitment, and the depletion level 

 (ratio of current to unexploited biomass). We also calcu- 

 late minimum-width 95% confidence intervals for average 

 recruitment, using the likelihood profile method for the 

 simulated data sets with catch-at-age data for all years. 



Application: relating recruitment in the 



Hauraki Gulf-Bay of Plenty snapper stock to SST 



Recruitment to the Hauraki Gulf snapper iPagrua auratus I 

 stock is correlated with temperature (Paul, 1976). The 

 abundance of l-i- snapper in the Hauraki Gulf estimated 



by trawl surveys has been shown to have a positive corre- 

 lation with SST (Francis, 1993) and air temperature (Gil- 

 bert, 1994) around or just after the time of spawning in the 

 previous year This relationship has also been shown with 

 catch-at-age analysis to continue to hold as snapper enter 

 the fishery at ages 4 and older (Maunder and Starr, 1998). 

 We applied the integrated approach described in this 

 study in combination with the age-structured statistical 

 catch-at-age model described in Maunder and Starr (2001 ) 

 to the Hauraki Gulf-Bay of Plenty snapper stock. The model 

 was fitted to catch-at-age data and biomass estimates. The 

 biomass estimates were available for 1985 and 1994 and 

 were obtained from analysis of tagging data. The majority 

 of the catch-at-age data were available from 1990 to 1997, 

 but there were some catch-at-age data of dubious quality, 

 small sample size, and high variability for 1970 to 1973. The 

 annual recruitment at age 1 was estimated for the time pe- 

 riod of the model ( 1970-98) and also for 18 age classes (ages 

 2 to 19) that comprised the initial conditions in 1970. 



Results 



Simulation analysis 



For all four sets of simulated data the environmental 

 model had the highest probability of detecting a relation- 

 ship between recruitment and the environmental time 

 series (Table 1). This model had a very high probability of 

 detecting a relationship even when there was no relation- 

 ship in the simulated data (Table ID). This indicates that 

 the likelihood ratio test is not appropriate for the environ- 

 mental model (see Appendix III). For all data sets, except 

 that with only catch-at-age data in the first 10 years, the 

 traditional model had a higher probability of detecting a 

 relationship between recruitment and the environmental 

 time series than did the environmental model with process 

 error The environmental model with process error had a 

 lower probability of detecting a true relationship than the 

 traditional model, but also had a slightly lower probability 

 of type-I error (the probability of incorrectly accepting a 

 nonexistent relationship) than the traditional model. The 

 probability of detecting a relationship was reduced as the 

 number of catch-at-age data sets was reduced. 



The environmental model with process error did not 

 show any bias in the estimate of the slope of the relation- 

 ship between the logarithm of annual recruitment and the 

 environmental time series, /3 (Table 1 ). For this model, the 

 variation in the estimates of /3 increased when fewer years 

 with catch-at-age data were available. The environmental 

 model showed a small negative bias and slightly more er- 

 ror in the estimates of p. The traditional model showed 

 a large negative bias in the estimate of the slope of the 

 relationship between the logarithm of annual recruitment 

 and the environmental time series, and this bias increased 

 as the amount of catch-at-age data was reduced. The tra- 

 ditional method also had larger error, which increased as 

 less catch-at-age data were available. 



The errors in the estimates of average recruitment and 

 ^ajr^B,, increased slightly with less catch-at-age data 



