592 



Fishery Bulletin 98(3) 



ance was performed to assess the importance of the factors 

 selected in the delta model. Table 2 gives the percent change 

 in deviance as each factor is added to the binomial fitted 

 proportion of the zero versus positive tows component of the 

 delta lognormal model. The deviance explained by the model 

 is equivalent to the r-' concept in linear models (McCullagh 

 and Nelder, 1989; Stefansson, 1996). Tests of significance 

 were based on the x^ statistic for the binomial distribution 

 of the proportion of positive tows (McCullagh and Nelder 

 1989). Overall, the delta lognormal model, with all factors, 

 explained between 557^ and TS'/f of the total deviance for 

 the finfish group and the three fish species. However, as 

 expected, the percentage of deviance explained by each 

 factor differed for each species. For example, the dataset 

 factor appeared to be unimportant in estimating the propor- 

 tion of positive tows for red snapper and Atlantic croaker 

 Instead, area and season factors were more important for 

 red snapper, and area and depth zone for Atlantic croaker 

 Table 3 shows the lognormal component of the delta 

 model r- values, sum of squares error or residual deviance. 



residual degrees of freedom, and the P values. Similarly to 

 the proportion of positive tows, a stepwise analysis of the r-' 

 shows that dataset, year, season, area, and depth zone are 

 significant factors in explaining the overall variability of 

 the model. An exception is the depth zone factor in estimat- 

 ing bycatch CPUE rates for red snapper The delta lognor- 

 mal estimated density model explained from \1'7( (Atlantic 

 croaker) to SS'/f (Spanish mackerel) of the total variation, 

 indicating that a significant portion of the bycatch CPUE 

 variability is still unexplained by the model. 



The annual shrimp bycatch estimates for the four spe- 

 cies groups in the U.S. Gulf of Mexico differed in several 

 aspects between the delta lognormal model and the current 

 general linear model. Results varied for the finfish group 

 and the fish species analyzed. Differences were found both 

 in the absolute magnitude of bycatch estimates and in the 

 trend over the time series 1972-95. For the total finfish 

 bycatch, the delta lognormal model estimated an average 

 of 795 million lbs. for the period 1972-95, or 147^ lower 

 than the equivalent general linear model estimate of 916 



Finfish 



] Binomial Deita model 

 -Observed 



0.30 

 0.25 

 0.20 

 015 

 0.10 

 0.05 

 0.00 



^ 



18 28 38 46 58 68 78 68 98 



Bin mid point 



Red snapper 



1 Binomial Delta model 

 -Observed 



008 018 028 038 48 58 68 78 88 9e 



0.16 



0.12 



008 



0.04 



0.00 



0.80 

 0.70 

 0.60 

 0.50 

 0.40 

 0.30 

 0.20 

 0.10 

 0.00 



Atlantic croaker 



] Binomial delta model 

 - Observed 



008 Die 028 038 48 58 68 78 68 96 



Spanish mackerel 



] Binomial Delta model 

 - Observed 



II I * I — I — I t » 



0.08 0.18 0.28 0.38 0.48 58 0.68 0.78 



Bin midpoint 



Figure 4 



.Standardized frequency distribution of the proportion of positive tows from the bycatch data 1972-9.5 and the proportions esti- 

 mated by the binomial-based delta lognormal model component. 



