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



95 



model parameters. There is very little bias in the estimate 

 of the slope of the relationship between recruitment and 

 the environmental time series. 



For the environmental model with process error, when 

 there is little or no information in the data to estimate the 

 recruitment for that year, the penalty on the annual re- 

 cruitment anomalies causes recruitment to be estimated 

 close to the recruitment predicted by the relationship 

 between recruitment and the environmental time series. 

 Therefore, if there is a relationship between recruitment 

 and the environmental time series, this model should pro- 

 vide better estimates because additional information is 

 included in the estimation procedure. This model has the 

 favorable property that if there is no relationship between 

 recruitment and the environmental time series, the model 

 estimates ji to be small, eliminating any influence of the 

 relationship between recruitment and the environmental 

 time series, and still estimates the annual recruitment 

 anomalies to represent the variation in annual recruit- 

 ment. The likelihood ratio test can be used to detect a rela- 

 tionship between recruitment and the environmental time 

 series, and if a relationship does not exist, the results with 

 P fixed at zero can be used. However, including [i in the es- 

 timation procedure, even when there was no relationship 

 between recruitment and the environmental time series, 

 did not increase the error in the parameter estimates in 

 relation to the model with j3 fixed at zero (see the results 

 for the traditional model. Table ID). 



The method we describe can be used to integrate en- 

 vironmental time series for parameters of the stock as- 

 sessment model other than recruitment. The influence of 

 the environment on catchability of the fish would be an 

 obvious choice because there are numerous publications on 

 the topic. For example. Green (1967) suggested that ther- 

 mocline data would improve estimation of tuna abundance 

 from catch and effort data, by allowing for the differentia- 

 tion between changes in tuna abundance and catchability 

 due to vertical distribution of tunas influenced by tempera- 

 ture. We have used a method similar to the method that is 

 presented in the present study to incorporate SST into the 

 purse-seine catchability parameters for yellowfin and big- 

 eye tuna (Maunder and Watters, 2001; Watters and Maun- 

 der, 2001). Maunder (2001a) presented a general method 

 to integrate the standardization of CPUE data into stock 

 assessment models, including the integration of environ- 

 mental variables. Growth rates have been observed to have 

 temporal variation, and this variation has been correlated 

 with environmental factors. Several authors have pre- 

 sented growth curves that include temperature data (e.g. 

 Mallet et al., 1999). Movement is another process that may 

 be influenced by the environment. Lehodey et al. (1997) 

 showed that spatial shifts in the western Pacific skipjack 

 tuna population are linked to the movement of a large pool 

 of warm water and that the movements of this large pool 

 are related to El Nino-Southern Oscillation events. 



Once a correlation between the environmental time se- 

 ries and the population process has been determined, this 

 relationship can be used to improve the predictive abil- 

 ity of the model. For example, if a relationship between 

 SST at the time of spawning and recruitment has been 



determined, and the age at recruitment to the fishery is 3 

 years, recruitment to the fishery can be estimated 3 years 

 in advance. One should be cautious about assuming that 

 these relationships are valid and will continue to hold into 

 the future, however Hilborn and Walters ( 1992) cautioned 

 about using environmental data because there are many 

 environmental indices that one can try, and if the data set 

 has a few large and a few small observations, it is likely 

 that one of the environmental data sets will correlate with 

 the data. Myers (1998) reviewed a number of published cor- 

 relations between recruitment and environmental factors 

 and found that few of the correlations held when retested 

 at later dates. Maunder and Starr ( 1998) also advised cau- 

 tion because they found that a strong cohort may not enter 

 the fishery when expected because of variations in growth 

 rates. We have found that, when applying this method to 

 the bigeye tuna data, there is an inconsistency in the pre- 

 1997 data and the data for 1997 and 1998 caused by much 

 stronger than expected year classes entering the fishery in 

 1997 and 1998. There is also difficulty in deciding on the 

 management strategy if environmental regime shifts are 

 influencing the productivity of the stock (Maunder, 1998b). 

 An advantage of the integrated approach, particularly the 

 environmental model with process error, is that it more 

 fully describes the uncertainty in the relationship be- 

 tween the population process and the environmental time 

 series, and therefore this uncertainty can be included in 

 any management advice based on the relationship. 



Conclusions 



Integrating environmental relationships in a statistical 

 stock assessment model is an improvement over the tra- 

 ditional statistical model when there are large gaps in the 

 data. However, it is important to include process error to 

 avoid the high probability of detecting spurious correla- 

 tions seen in the environmental model when using the like- 

 lihood ratio test. Therefore, the environmental model with 

 process error is the model of choice because 1 ) there is no 

 bias in the estimates, 2) when there is no relationship with 

 the environmental series, it is equivalent to the traditional 

 model, 3) when such a relationship exists, the recruitment 

 estimates are improved, particularly if there are important 

 gaps in the data, 4) it may be used for prediction, and 5) 

 uncertainty about the relationship can be modeled. 



Acknowledgments 



We thank Dave Fournier for advice on using AD Model 

 Builder and related software, and Bill Bayliff, Rick Deriso, 

 Shelton Harley, and Ransom Myers for commenting on the 

 manuscript. 



Literature cited 



Bertignac, M., P. Lehodey, and J. Hampton. 



1998. A spatial population dynamics simulation model of 



