Perkins and Edwards A mixture model for estimating discarded bycatch 



335 



E[Yj] = (nj + /nj)£y k + /n + , 



(7) 



where n + and n are the number of positive observa- 

 tions and the total number of observations in areaj, 

 the Vi, + are the positive observations in all areas, and 

 n* is the total number of positive observations in all 

 areas. Similarly, when only the NB mean p depends 

 on area, the MLE for mean discard per set in areaj 

 reduces to 



E[y r -] = (n + /n)]T 



+ / 4 



(8) 



where n + and n are the number of positive observa- 

 tions and the total number of observations in all ar- 

 eas, they k + are the positive observations in area j, 

 and n + is the total number of positive observations 

 in area j. Again, Equation 5 is used to compute 

 "pooled" estimates in these latter two cases. Note that 

 the estimates for different areas are not independent, 

 because both Equations 7 and 8 involve observations 

 from all areas. In particular, the first term in Equa- 

 tion 7 is an area-specific estimate of the probability 

 of a positive observation, whereas the second term is 

 a "pooled" estimate of the mean for positive observa- 

 tions. This is consistent with the areal dependence 

 on which Equation 7 is based, and provides more 

 precise estimates of E[Y] than simply taking the 

 sample mean in each area. A similar interpretation 

 holds for Equation 8. 



As a consequence of Equations 6, 7, and 8, the es- 

 timate of mean discard per set ( 1-p )p can be much 

 more precise than the estimates of the individual 

 parameters involved in it, because it does not depend 

 solely on either the positive observations or the pro- 

 portion of zeros. 



While variance estimators for Equation 6 are 

 straightforward, there is no simple analytic result 

 for estimating the variance of Equations 7 or 8 (see 

 Discussion). Thus, for consistency, we used bootstrap 

 methods in all cases. Our bootstrap resampling pro- 

 cedure varied slightly for each set type, depending 

 on the particular areal dependence chosen for the model 

 parameters. When no dependence was appropriate, 

 data were resampled across all areas. When dependence 

 was important, data were resampled by area in the 

 same proportions as the original observations. 



Results 



Modelling discard per set 



Based on the results of likelihood-ratio tests, geo- 

 graphic area was a statistically significant predictor 



of discard per set for only two of the three set types 

 (log and school sets). 



Nonzero observations of discard from the third set 

 type (dolphin sets) were reported very infrequently 

 ( 19 out of 2,110 sets, Table 1). The data provided little 

 statistical information from which to distinguish 

 patterns in discard between geographic areas, and 

 area failed to produce a significant improvement in 

 the fit when included as a covariate for dolphin sets. 

 Therefore, we selected the model with no areal de- 

 pendence for any of the parameters so that the esti- 

 mates for p, a, and p for dolphin sets are fishery- 

 wide values (Table 2). The standard error of the mix- 

 ing parameter p for the dolphin model is small 

 (CV=1.53% ), reflecting the high estimate forp dictated 

 by the extremely large number of zero observations of 

 discard. The standard errors of the parameters for the 

 NB portion of the probability distribution (a and p ) are 

 quite large ( CVs > 907^ , Fig. 3 ), reflecting the few posi- 

 tive data available for their determination. 



