Ortiz et al : Estimates of bycatch from tfie shrimp trawl fisfiery in the Gulf of Mexico 



591 



Evaluation of the delta lognormal model 



Before comparing the annual bycatch estimates of total 

 finfish, Atlantic croaker, red snapper, and Spanish mack- 

 erel from the general linear model (Nichols"'! and the delta 

 lognormal model, the delta lognormal model was evalu- 

 ated and assessed. Because there is not yet a formal strat- 

 egy for model verification, acceptance of a particular model 

 should not be based exclusively on "goodness of fit" scores 

 (McCullagh and Nelder, 1989). 



In general, model assessments can be classified into two 

 main groups. The first group checks for systematic departure 

 from the underlying model, testing for additional factors, 

 factor interactions, or covariates that could explain a signifi- 

 cant proportion of the residual model variation. The second 

 group involves evaluation of particular or isolated points 

 in the data. McCullagh and Nelder (1989) and O'Brien and 

 Keli'" have described six specific tests for evaluating general- 

 ized linear models: 1 ) assessment of the scale of independent 

 variables; 2) assessment of the link function adequacy; 3) 

 assessment of variance function adequacy; 4) investigation 

 of systematic departure from the assumed model; 5) investi- 

 gation of outliers; and. 6) investigation of omitted predictor 

 variables. Most of these analyses are based on the behavior 

 of the model residuals, either as graphical or informal tests, 

 rather than an exact statistical test. For the delta lognormal 

 model, only tests evaluating systematic departure from the 

 assumed model were performed on each of the model com- 

 ponents (i.e. an estimation of the proportion of positive tows 

 and the estimation of bycatch rates) separately. 



With the delta lognormal model, an evaluation of the 

 proportion of positive tows was restricted to a graphical 

 analysis of the frequency distribution of positive tows of 

 observed and predicted data. This restriction was warranted 

 because most of the tests suggested for assessing model 

 adequacy are uninformative for binomial data (McCullagh 

 and Nelder, 1989: O'Brien and Kell'^). Figure 4 shows the 

 standardized frequency distributions of proportion of posi- 

 tive tows per cell for the combined finfish category, Atlan- 

 tic croaker, red snapper, and Spanish mackerel. Each plot 

 shows the obsei-ved and the predicted proportions estimated 

 by the binomial distribution of the delta lognormal model. 

 The predicted frequencies fitted closely those observed in 

 all four cases. The assumed binomial distribution is able to 

 predict appropriately the proportion of positive tows in a 

 broad range (from the combined finfish category case where 

 almost all tows were positive [97%] to the case of Spanish 

 mackerel where only d^c of the tows were positive). 



The suitability of the delta lognormal general linear 

 model component for the positive tows was evaluated by 

 the following gi-aphical tests: 1) adequacy of the link func- 

 tion, 2) adequacy of the variance function, and 3) system- 

 atic departure from the assumed model. 



Nichols, S. 1996. Estimates of annual shrimp fleet bycatch 



in the offshore waters of the Gulf of Mexico. Personal comniun. 



NMFS Pascagoula laboratory. 3209 Frederic St. Pascagoula, MS 



:39567. 



O'Brien, C. M.. and L. T. Kell. 1996. The use of generalized 



linear models for the modelling of catch-effort series. I. 



Theory. ICCAT Collect. Vol. Sci. Pap. 46(41:476-482. 



By plotting the adjusted dependent variable Hog CPUE) 

 we were able to assess the link function against the esti- 

 mated linear predictor ( f] ). A linear configuration is expected 

 for normal, assumed Poisson or gamma error distributions. 

 In our case, the delta lognormal model assumed a normal 

 error distribution for log CPUE of positive catch. Figure 

 5 A shows the plots of the linear predictor (Ip-logcp) against 

 the adjusted dependent variable (log CPUE) for red snap- 

 per In the case of high density of points as in Figure 5A, 

 locally weighted regression smoothing procedures ( i.e LOESS 

 smoothing) have been suggested for showing the trend of the 

 response variable (McCullagh and Nelder, 1989). 



Adequacy of the variance or assumed error distribution 

 function was evaluated by using a plot of residuals against 

 fitted values. The spread of residuals is expected to be 

 approximately constant and independent of the fitted values, 

 confirming the adequacy of the assumed error distribution in 

 the model. Figure 5B shows the plots of residuals (R-logcpu ) 

 against the fitted values (P-logcpu) for red snapper The 

 residuals are evenly distributed about the zero line and are 

 without any apparent trend with respect to the fitted values. 

 Likewise, a plot of residuals versus the normalized cumu- 

 lative residuals (QQ plot) can be used to assess the vari- 

 ance function adequacy. A linear relationship is expected for 

 residuals from a normal error distribution. 



A plot of standardized residuals (rs-logcp) against fitted 

 values ( log CPLTE ) was used to identify possible trends or cur- 

 vatures that would suggest a departure from the assumed 

 model (Fig. 5C ). The null pattern of this plot is a linear config- 

 uration of the standardized residuals (O'Brien and Kell''). In 

 conclusion, assessments of each of the delta lognormal model 

 components did confirm the model choices and assumptions 

 for the finfish group and the fish species examined (similar 

 plots were created for finfish, Spanish mackerel, and Atlantic 

 croaker but are not presented here for briefiiess). 



As shown before, bycatch estimates from the current 

 general linear model depend upon the standard time unit 

 chosen to convert catches in numbers to CPUE values. 

 Similarly, the same tow time evaluation with the delta log- 

 normal model was performed as with the general linear 

 model. CPLIE values were calculated by using 10-, 30- and 

 240-min tow times, and concurrently, shrimping effort unit, 

 given in hours, were multiplied by a scale factor to make 

 the time unit compatible with the modified CPUE values. 

 With the delta lognormal model, the annual bycatch esti- 

 mates were exactly the same, independent of the time unit 

 used to calculate the CPUE values, further demonstrating 

 the benefits of using a model that separates the zero catch 

 observations from the positive catch. In addition, delta 

 models do not require adding a constant value to loga- 

 rithmic transformed values because the estimated density 

 component is restricted to positive catch only, thus avoid- 

 ing the uncertainty in selecting a c value to log transform 

 CPUE values as required in the general linear model. 



Results and discussion 



Because the bycatch database complied with the delta log- 

 normal model specifications, a stepwise analysis of devi- 



