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Fishery Bulletin 101(1) 



little information associated with them and therefore did 

 not add anything to the estimation procedure. However, 

 they do add additional parameters, which reduce the pos- 

 sibility of accepting the model when using the likelihood 

 ratio test. If the recruitment anomalies were estimated for 

 only a limited number of years, it is likely that the envi- 

 ronmental model with process error would be a statistical- 

 ly significant improvement over the environmental model. 

 Statistical tests could be carried out to determine which 

 annual recruitment anomalies should be estimated, but 

 this would be very time consuming. Reducing the number 

 of annual recruitment anomalies may also cause an un- 

 derestimation of the confidence intervals. For the snapper 

 example, removing the anomalies for the initial conditions 

 may be a good compromise. 



Discussion 



We have developed a general framework for integrating 

 environmental time series into stock assessment models 

 that appears to perform better than traditional methods. 

 The method is flexible and it can be used to model many dif- 

 ferent functional relationships between population or fish- 

 ing processes and environmental time series and to include 

 multiple environmental time series for any population 

 model parameter (see Appendix II). Furthermore, it can 

 be used with any statistical stock assessment model. The 

 method can be used to test whether an environmental time 

 series describes temporal variation in model parameters. 



The traditional model, which estimates annual recruit- 

 ment within the stock assessment model and subsequently 

 correlates the recruitment with the environmental series 

 outside the stock assessment model, performs poorly. It 

 has a reasonable probability of detecting a relationship be- 

 tween recruitment and the environmental series, but this 

 probability decreases rapidly as the number of years with 

 missing catch-at-age data sets increases. The probability of 

 incorrectly detecting a relationship when one is not pres- 

 ent is low. This method has reasonable confidence-interval 

 coverage for average recruitment and little bias or variance 

 in the estimates of model parameters. The factor causing 

 the poor performance of the traditional model is the large 

 bias in the estimate of the slope of the relationship between 

 recruitment and the environmental time series, which in- 

 creases as the number of years with missing catch-at-age 

 data increases. The bias occurs because the traditional 

 model has a penalty on the absolute size of the annual 

 recruitment deviations. This penalty constrains an annual 

 recruitment anomaly to be close to the mean recruitment 

 when there is little or no information about the recruit- 

 ment in that year. Therefore, when the logarithm of the 

 annual recruitment is correlated with the environmental 

 time series, the estimated slope of the relationship is biased 

 downward. Even in situations for which there is sufficient 

 information for every recruitment anomaly, there will be a 

 small tradeoflin the size of the anomaly, which reduces the 

 contribution of the penalty to the objective function and the 

 likelihood from the catch-at-age data. Unfortunately, if the 

 penalty on the annual recruitment anomalies is removed. 



the estimation process can become unstable, particularly 

 in data-poor situations for which the bias is greater. The 

 amount of time that is required by the estimation algorithm 

 also increases if the penalty is removed. When the penalty 

 on the size of the recruitment anomalies is removed, the 

 bias in estimates of the slope of the relationship between 

 recruitment and the environmental time series is reduced 

 when using all the catch-at-age data, but the variance in the 

 estimates is greatly increased. In addition, when removing 

 the penalty there was a large positive bias when using only 

 the last 10 years of catch-at-age data and a large negative 

 bias when using only the first 10 years of catch-at-age data. 

 It is not known what results would be obtained if cohort 

 analysis, which does not use a constraint on the annual 

 recruitment anomalies, is used instead of the statistical 

 catch-at-age analysis. It should be remembered that cohort 

 analysis cannot be used or assumptions that are unlikely 

 to be satisfied will have to be made when catch-at-age data 

 are missing for some years. 



The environmental model, which has a deterministic 

 relationship between recruitment and the environmental 

 time series that is integrated into the stock assessment 

 model, also performs poorly. This method has poor confi- 

 dence interval coverage for average recruitment because 

 the size of the confidence intervals are gi'eatly underes- 

 timated. The method has larger bias and variance in the 

 estimates of model parameters compared to the other two 

 methods. There is a small negative bias in the estimate of 

 the slope of the relationship between recruitment and the 

 environmental time series. The environmental model has a 

 very high probability of detecting a relationship between re- 

 cruitment and the environmental series, and this probabil- 

 ity only decreases slightly as the number of missing years of 

 catch-at-age data sets increases. However, this model has a 

 very large probability of incorrectly detecting a relationship 

 when one is not present. Therefore, when using the environ- 

 mental model, the likelihood ratio test should not be used 

 to determine if there is a significant relationship between 

 recruitment and an environmental time series. The value 

 used to compare to the x- statistic in the likelihood ratio 

 test for the environmental model is highly correlated with 

 the catch-at-age sample size; therefore simulation analysis 

 is needed to find the appropriate /- statistic for the given 

 sample size (see Appendix III). This is also important for 

 calculating confidence intervals that are also based on the 

 X~ statistic and is the reason for the poor coverage for i?^. 



The environmental model with process error, which has 

 a relationship between recruitment and the environmen- 

 tal time series that is integrated into the stock assessment 

 model with additional process error, performs well. This 

 model has a reasonable probability of detecting a relation- 

 ship between recruitment and the environmental series, 

 but this probability is lower than those of the other two 

 models, and decreases as the amount of data is reduced. 

 It has a low probability of incorrectly detecting a relation- 

 ship when one is not iircsent. These probabilities could be 

 im[)r()ved by using simulation analysis to find the appro- 

 priate ;f- statistic (see Appendix III). This method has rea- 

 sonable confidence interval coverage for average recruit- 

 ment and has little bias or variance in the estimates of 



