NOTE Kimura: Using the empirical Bayes method to estimate and evaluate bycatch rates of seabirds 



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Table 1 



Simulation results from the Poisson-gamma empirical Bayes (EB) model with the assumption of 50 vessels (strata), and with 

 random sample sizes (observed number of 1000 hooks) distributed as [/(O, r^^^) and replicated 1000 times. The gamma distri- 

 bution prior was assumed to have parameters a=0.603, /3=0.030 , as were estimated from the seabird bycatch analysis, and 

 the average a and p were calculated from the 1000 replications. The seabird bycatch rate for vessel /, A,, was estimated by the 

 maximum likelihood (ML) estimator A,, the empirical Bayes (EB) estimator A,, and the global unstratified (GU) estimator A. Per- 

 formance of these estimators was measured by the total mean squared error (TMSE) averaged over the 1000 replications, and R 

 was defined as the ratio ofTMSE values calculated for ML and EB estimates. 



Sample size t^^^ 



TMSEi A.) Mh 



TMSEU.) EB 



TMSE{A,)GV 



TMSEiX, 1 



Average a Average ^ 



200 



500 



1000 



2000 



5000 



0.001390 

 0.001440 

 0.001000 

 0.000935 

 0.000906 



0.000189 

 0.000103 

 0.000063 

 0.000038 

 0.000018 



0.000542 

 0.000531 

 0.000541 

 0.000532 

 0.000537 



7.4 



14.0 



15.9 



24.6 



50.3 



0.712 

 0.655 

 0.638 

 0.621 

 0.473 



0.029 

 0.029 

 0.030 

 0.031 

 0.041 



Results from fitting the Poisson-gamma model 



The bycatch per thousand hooks in 1997-98, when bird 

 avoidance gear was not as common, was 0.085 birds, 

 compared with 0.013 birds in 2002-03 when seabird 

 avoidance gear was voluntarily employed. Thus, the 

 bycatch rate for all seabirds was reduced in 2002-03 to 

 15% of the 1997-98 value. 



For the 2002 bycatch data, initial parameter esti- 

 mates for the gamma distribution were made with the 

 moment estimators described earlier. These initial es- 

 timates were refined by using the maximum of the 

 marginal likelihood also described earlier. The final 

 MML estimates were ci= 0.603 and ^=0.030. 



For 2002, the resulting EB bycatch rate estimates per 

 vessel, Aj (Table 2), differed little from conventional ML 

 estimates per vesseL A similar result occurred in the 

 1997-98 analysis. However, vessel 28 (Table 2), showed 

 a large adjustment between the ML and EB estimates. 

 It is apparent from Table 2 that this vessel had unusu- 

 ally low effort (T28=34) and a relatively large seabird 

 bycatch {y2g=8). This adjustment towards the aggregate 

 mean is a predictable EB adjustment for situations 

 where individual stratum data are weak. For vessel 28, 

 the predicted EB estimate of seabird bycatch rate per 

 thousand hooks was 4.4, whereas the actual observed 

 bycatch rate was 8 (Table 2). 



When the 2002 seabird bycatch rates were used to 

 predict the 2003 seabird bycatch rates for individual 

 vessels, neither the ML or EB estimates provided a sig- 

 nificant correlation (p=0.036, /i = 38). In contrast, when 

 the 1997 bycatch rates were used to predict the 1998 

 bycatch rates, there was a significant correlation for the 

 one-tailed test p=0.324, 7i = 33, P=0.033). 



Discussion and conclusion 



Empirical Bayes estimators are superior to Bayes esti- 

 mators in the sense that prior distributions can be 

 estimated rather than assumed. If one prefers the Bayes 



method, one would counter that noninformative priors 

 make the assumption of priors relatively benign, whereas 

 for the empirical Bayes model, the assumption of the 

 family of priors may be quite critical. 



The empirical Bayes method can be applied even when 

 the marginal distribution is analytically intractable, by 

 substituting numerical integration for analytical inte- 

 grals. However, the computational intensity required by 

 using numerical integration can appear daunting even 

 with the current speed of desktop computers (Laslett 

 et al., 2002). 



Nevertheless, if the prior family is properly selected, 

 the empirical Bayes method can provide very precise 

 estimates. For our Poisson-gamma simulation, the 

 empirical Bayes method provided uniformly superior 

 estimates of the Poisson A, for a wide range of t^^^^^ 

 values. Although the ratio values in Table 1 indicate 

 that the EB estimator is most useful when t^^^^ is large, 

 the greatest benefit of the EB method is probably on 

 the opposite end of the scale when individual stratum 

 sampling is relatively weak. Note that the bias in (a,/3) 

 when T^jj^ = 5000 (Table 1) may be simply bias in mar- 

 ginal maximum likelihood estimates because maximum 

 likelihood estimators are not generally unbiased. Anoth- 

 er possibility is that bias was caused by computational 

 error in calculating the marginal likelihood when the 

 T,'s were large, even though the marginal likelihood was 

 calculated on the log-scale. 



In the seabird bycatch analysis, results show that 

 in almost all cases estimates of bycatch rates at the 

 individual vessel level were not significantly affected 

 by using the EB method. These results may indicate 

 that individual vessel sampling levels (i.e., t,) are at a 

 sufficiently high level that ML estimates are already 

 precise estimates of seabird bycatch rates. For the pre- 

 diction of the 2003 bycatch rate of seabirds from the 

 2002 analysis, the TMSE of the ML estimator of A, was 

 reduced a minor amount from 0.0007904 to 0.0007339 

 by using the EB estimator of A,. However, the important 

 issue is that neither the ML nor EB estimates for 2002 

 significantly correlated with the observed 2003 bycatch 



