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Fishery Bulletin 94(2). 1996 



the system. The total number of sets observed in each 

 area, including sets for which discard was not recorded, 

 represents the actual areal distribution of fishing ef- 

 fort during the study period. However, observation of 

 discard was not proportional to this distribution of to- 

 tal effort (Table 1). In the analysis that follows, it is 

 important to distinguish between the total number of 

 sets, denoted by N , and the number of sets for which 

 discard was recorded, denoted by n . The former de- 

 fine the actual distribution of fishing effort, whereas 

 the latter simply reflect the sample taken. Because our 

 sample of sets with discard recorded was not propor- 

 tional to the total effort, ignoring area in the analysis 

 could lead to biased estimates if the mean discard per 

 set differs from area to area for a given set type. 



Because there were clear differences between the 

 the three set types in per-set discard, we included 

 set type as a covariate for all three model param- 

 eters. Thus, with set type and geographic area as 

 the only covariates, our analysis reduced to fitting 

 the model (Eq. 1) independently for each set type, 

 with p, p, and a having possibly different values in 

 each area. To determine an appropriate dependence 

 upon area, we used stepwise likelihood-ratio tests to 

 select the simplest model that could not be signifi- 

 cantly improved by adding additional terms. We first 

 made initial fits for each set type using no areal de- 

 pendence, then progressively added dependence for 

 more of the model parameters. At each step, we used 

 a quasi-Newton numerical optimization algorithm to 

 maximize the likelihood and estimate parameters. 

 It should be noted that because this is not a linear 

 model, significance levels (i.e. p-values) from these 

 likelihood-ratio tests are approximate. We used the 

 large-sample normal approximation for MLE's to 

 compute standard errors for p, p, and a. For com- 

 parison, we also computed bootstrap standard errors. 



It can be shown from the likelihood equations for 

 the NBAZ that estimates for the parameters a and p. 

 depend solely on the positive observations in the data. 

 The estimate for the parameter/; depends on all the 

 data, but most strongly upon the proportion of zero 

 observations. Thus, the precision of the estimates for 

 a and p can be very poor if the data contain few posi- 

 tive observations, even though the precision of the 

 estimate forp may still be very good. 



Estimating mean discard per set 



We used Equation 2 and the maximum likelihood 

 estimates forp and u from the best-fit models to es- 

 timate mean discard per set for each set type in each 

 area. We also calculated a "pooled" estimate for each 

 set type as the weighted average of the area-specific 

 estimates, where weightings were proportional to 



total effort in each area. For example, mean discard 

 per set of type i in area j is estimated as 



E[V,,| = (l-p, >,,, 



(4) 



whereas the "pooled" estimate for all areas combined 

 is estimated as 



EIVJ^^IX^I^I/IX,. O) 



where iV is the total effort (in number of sets) of 

 type /' occurring in area,/. Note that this "pooled" cal- 

 culation is based on the proportion of total sets (in- 

 cluding those for which discard was not recorded) 

 observed in each area. This is an estimate of the mean 

 discard per set over the entire fishery during the 

 study period. However, it is also valid as a predic- 

 tion of future discard if the proportion of effort ( sets ) 

 in each area remains constant as the actual number 

 of sets varies, assuming that other factors in the fish- 

 ery, such as size and species composition of discard 

 and style of fishing, remain the same. 



While Equation 4 provides a straightforward way 

 to compute the MLE for the product ( 1-p )p, the vari- 

 ance of that product can be difficult to estimate ac- 

 curately. However, we were able to use the likelihood 

 equations for the NBAZ to derive explicit forms for 

 the MLE of mean discard per set. Specifically, only 

 the product il-p)p need be estimated, and we de- 

 rived, through algebraic manipulation of the likeli- 

 hood equations, simple closed-form expressions that 

 do not involve the individual parameter estimates. 

 By the invariance properties of maximum likelihood 

 estimates, these simpler forms give results that are 

 identical to those from using Equation 4. 



With no areal dependence, the MLE for the prod- 

 uct ( l—p)p is simply the sample mean: 



E[Y] = y = (\/n)^y k , 



(6) 



where the set type subscript i is suppressed for clar- 

 ity. Similarly, with complete areal dependence, the 

 MLE for each area reduces to the sample mean in 

 that area, and the "pooled" estimate is computed by 

 using Equation 5. In both of these cases, the vari- 

 ance for the MLE of ( 1 -/)»// can be estimated by us- 

 ing the sample variance of the data. 



When only the mixing probability p depends on 

 area, the MLE for mean discard per set in area,/' is 

 slightly more complicated, and reduces to 



