586 



Fishery Bulletin 98(3) 



Finfish 



72 74 76 78 80 82 84 86 



90 92 94 



Red snapper 



72 74 76 78 



90 92 94 



Year 



Atlantic croaker 



6.0 



72 74 76 78 80 82 84 86 88 90 92 94 



Spanish mackerel 



72 74 76 78 80 82 84 86 88 90 92 94 



Figure 1 



Estimated total bycatch with the general linear model and modified models of the general linear model for the finfish group, Atlan- 

 tic croaker, red snapper, and Spanish mackerel. (See Table 1 for descriptions of the modified models.) 



in the matrix, we found that this value increased from 39'^ 

 in the base scenario of the current general linear model to 

 81'^'f . in what was defined as the "minimum model" where 

 only the factors year and dataset were included. 



Overall, the results showed that total bycatch estimates 

 did not vary substantially, although the assumed model was 

 radically modified ( Fig. 1 ). These results suggest that season, 

 area, and depth zone are factors that do not significantly 

 contribute to the explanation of the observed variability in 

 the data. Although the F-values from the ANOVA tables 

 were highly significant (P<0.05) for each factor in all gen- 

 eral linear model matrix scenarios, this significance may be 

 a response to the large number of degrees of freedom. Alter- 

 natively, it is possible that the structure of the general linear 

 model does not reflect all the main factors that account for 

 bycatch variability among years, except for dataset source. 

 Indeed, interactions between the main factors may also be 

 important. Given the limited data coverage, however, the 

 inclusion of other factors or interactions among factors in 

 the general linear model is clearly not advisable. 



In summary, the simple model with year and dataset as 

 factors produced similar estimates of bycatch in relation 

 to the complex model, including season, area, and depth 

 zone factors. In particular, for species that are not common 

 as shrimp bycatch, a simple model avoids empty cells and 

 highly unbalanced input matrix designs. 



Use of logarithms in the general linear model 



One of the assumptions in the linear regi"ession model 

 is that the error within the matrix cells should follow a 

 normal distribution and have a constant equal variance. 

 In the bycatch dataset, the CPUE variance increases as 

 the mean CPUE increases, indicating a constant coefficient 

 of variation. This condition suggests a logarithmic trans- 

 formation of mean CPUE values. To avoid the problem of 

 undefined logarithms for zero catches, a constant value c 

 of 1 was added to all observed CPUE (Eq. 1) in the model. 

 Then the linearization procedure was carried out on the log 

 base 10 of the modified CPUE. This c value was then sub- 

 tracted in the back transformation of the predicted means 

 (Eq. 2). No particular explanation for the choice of 1 in the 

 current general linear model has been given. 



Thus, we considered the effects of using different c values 

 in the general linear model. Three different c values where 

 used: 10. 0.5, and the smallest positive CPUE-value for each 

 species (i.e. 0.0178 for finfish, 0.0779 for Atlantic croaker, 

 0.0685 for red snapper, and 0.0685 for Spanish mackerel). 

 The results showed that annual bycatch estimates vary dra- 

 matically depending upon the c value used in the algorithm 

 (Fig. 2). Although the magnitudes varied with changes in 

 the c value, the trends were the same for each species. How- 

 even the direction of change was not the same among spe- 



