Ortiz et al,: Estimates of bycatcii from the shrimp trawl fishery in the Gulf of Mexico 



589 



years ( 1990-95) by 20'7f . In contrast, the estimated bycatch 

 of Spanish mackerel was reduced by 449; on average. 



The estimated CPUE should be independent of the time 

 unit used (because it is a constant factor for all observa- 

 tions). However, the differences seen in our study in esti- 

 mated bycatch were due to the presence of zero CPUE 

 values. By dividing by different time units, the relative 

 distance between the groups of zero CPUE values and the 

 positive CPUE values is changed; as a result, estimators of 

 the central tendency for these data will vary. Although the 

 end results of the time-unit and c value choices are similar 

 (biased estimates), their mathematical origin is different. 

 The time-unit choice is a multiplier of the positive catch 

 data (zero catch /any time unit=zero CPUE), whereas the 

 c value choice adds the c value to all CPUE data. Although 

 a change in time unit could be exactly matched by the 

 appropriate change on the c value, addition of more data, 

 with the same time unit, would require recalculating the 

 appropriate c value. 



Procedure for estimating bycatch with 

 the delta lognormal model 



Delta models have been used to analyze fisheries data, in 

 particular when there is a predominant group of zero obser- 

 vations. These models have been used to obtain estimates 

 of abundance for highly aggregated organisms, such as 

 planktonic samples (Pennington, 1983), in the analysis of 

 catch-per-unit-of-effort data for the development of CPUE 

 indices (Lo et al., 1992; Cooke and Lankester'*), as well as 

 in the analysis of ground trawl surveys to estimate total or 

 relative abundance (Pennington, 1996; Stefansson, 1996). 

 The main advantage of delta models is that they allow for 

 an explicit and finite probability of zero catch. In a delta 

 model, the estimated values are the product of two inde- 

 pendent components: the probability of nonzero observa- 

 tions, and the probability of effective density if there is 

 a positive observation. In the case of fishery surveys, the 

 nonzero probability can be analogous to the probability of 

 encountering a fish aggregation, whereas the probability 

 within the positive observations would correspond to the 

 estimated density of a given fish aggregation (Cooke and 

 Lankester''). 



Delta models are multivariate distributions with a non- 

 zero probability mass at the origin (Shimizu, 1988). Ste- 

 fansson (1996) presented a mathematical model based on 

 a generalized delta lognormal model for analyzing ground- 

 fish survey data. This model defines the cumulative den- 

 sity function of abundance at a given sampling station as 



F,( w) = P[Y<co] = ( 1 - p, ) 4- p^G,( CO), 



where G, = a continuous cumulative density function 

 describing the distribution of positive values 

 in a station /; and 



^ Cooke.J. G.andK. Lankester. 1995. Consideration of statistical 

 models for catch-effort indices for use in tunning VPA's. ICCAT 

 Collect. Vol. Sci. Pap. 4.5(2):125-131. 



p, = the probability of finding fish in that station. 



If p, is constant and G, is a lognormal distribution within 

 a stratum, the function is the delta lognormal model. If 

 p, is set to one (i.e. excluding zero values), and G, is set 

 to a gamma or other exponential function with a para- 

 meterized mean, this model becomes a generalized linear 

 model (GliM, Stefansson, 1996). The advantage of this 

 formulation is that each component in the delta model can 

 be expressed in terms of a GLiM (McCullagh and Nelder, 

 1989). Thus, the choice of a particular density function 

 in each of the delta model components can be related 

 to other measured variables, such as tow times, location 

 effects, and seasonal or year effects, through assumptions 

 on distribution. 



Bycatch data derived from observers in the Gulf of 

 Mexico shrimp trawl fishery typically have a high propor- 

 tion of zero bycatches and a skewed distribution of the 

 positive bycatch CPUE rates, with a large number of low 

 bycatches and very few large bycatches. The large catches 

 most likely reflect the spatial-temporal distribution char- 

 acteristics of fish stocks rather than are outliers of the 

 data. This type of distribution is far from normal, and 

 commonly used transformations are unable to make the 

 data comply with the normal assumptions with the clas- 

 sical regression models. Furthermore, in the case of so- 

 called "non-frequent bycatch species," the proportion of 

 zero observations is markedly increased (above 959^ ); this 

 significantly biases and reduces the efficiency of statisti- 

 cal estimators of central tendency and overestimates the 

 variance (Pennington, 1996). 



The delta lognormal model was used in our study to gen- 

 erate annual bycatch estimates for all finfish combined, as 

 well as for three specific finfish species: Atlantic croaker, 

 red snapper, and Spanish mackerel in the U.S. Gulf of 

 Mexico shrimp trawl fishery. Briefly, bycatch CPUE rates 

 of a given fish species in a given cell were estimated as 

 the product of two components: 1) the proportion of tows 

 with positive catch and 2 ) the mean catch rate if at least 

 one fish was caught. Bycatch per cell is then the product 

 of the estimated CPUE and the corresponding shrimping 

 effort for that particular cell. Total annual bycatch is then 

 the sum over all strata within a year for the commercial 

 component, as in the general linear model (see Eq. 3). 



Each component of the delta lognormal model, the pro- 

 portion of positive tows and the mean bycatch rate, was 

 estimated by following a general linear model approach 

 with the procedure GENMOD in the SAS statistical soft- 

 ware package (SAS Institute Inc., 1993). General linear 

 models consist of three elements: 1) the random compo- 

 nent which defines the error structure of the model, 2) the 

 sysfemafic component which defines a set of explanatory 



variables .V J, .v., .v ,and 3) the link function which defines 



the relation between the random and the systematic com- 

 ponents (McCullagh and Nelder, 1989). We described the 

 delta lognormal model for estimating shrimp bycatch on 

 the basis of the assumptions entailed with each compo- 

 nent of the model. To compare models, the same explana- 

 tory variables used in the current general linear model 

 were used with the delta lognormal model. 



