Perkins and Edwards A mixture model for estimating discarded bycatch 



333 



Table 1 



Fishing effort in numbers of sets for the U.S. tuna purse- 

 seine fleet fishing in the eastern tropical Pacific Ocean, 

 1989-92. Geographic areas are defined according to Fed- 

 eral Register ( 1989 ) ( Fig. 1 ). N is the total number of sets 

 in a given area, n is the number of sets for which discard 

 weight was recorded, and n* is the number of sets for which 

 strictly positive discard was reported. 



Set type 



Area 



■V 



n* 



Dolphin 



School 



Log 



' Totals for log sets do not include sets in geographic area 2 be- 

 cause these sets were not included in our analysis. See text for 

 explanation. 



tion) for this analysis, because school sets and only 

 10 log sets (4 with estimated discard) occurred in 

 this area (Table 1). We also omitted 7 sets in which 

 the entire catch (target catch plus discard) was lost 

 owing to equipment failure. 



Modelling discard per set 



We chose a modified negative binomial (NB) distri- 

 bution known as the negative binomial with added 

 zeros (NBAZ) (Johnson and Kotz, 1969) to model dis- 

 card per set. This distribution can accommodate the 

 wide range in the proportion of zero observations, as 

 well as the relatively heavy tails in the observed dis- 

 tributions of discard for all three set types (Fig. 2). 

 (See Discussion section for two other models consid- 

 ered but rejected. ) 



The NBAZ is a mixture of a NB distribution and a 

 discrete probability mass at zero. Under this model, 

 discard per set is either exactly zero with probabil- 

 ity p or has a NB distribution with probability 1-p. 

 The NB portion of this distribution can be viewed as 

 representing strictly positive amounts of discard 

 rounded to integer values. Thus, zero values that are 

 part of the NB can be interpreted as observations of 

 small amounts of discard rounded down to zero. Zero 

 values from the probability mass can be interpreted 



as exact zeros. The probability function for this modi- 

 fied NB distribution is 



Pr{y = y) = 



i a 



) n y+ i/a)/_ L _y /a (_aL\ 



r" _v!I"(l/a) \\ + (Wj \l+ap ) 





 = 1,2,. 



(1) 



where Y is an individual observation (tons of discard 

 per set), p is the probability of an observation com- 

 ing from the "true zero" state, 1-p is the probability 

 of an observation coming from the NB state, and p 

 and a are the mean and variance parameters, re- 

 spectively, of the conditional NB. 1 



The parameter a determines the shape of the dis- 

 tribution. As a approaches zero, the conditional NB 

 distribution in the mixture approaches a Poisson dis- 

 tribution. As a increases, the conditional NB becomes 

 more skewed, with a heavier tail and higher probabil- 

 ity of a zero observation. The parameter p is a mixing 

 parameter which controls the relative importance of 

 the NB and the probability mass at zero. When p is 

 one, the distribution is a probability mass at zero. When 

 p is zero, the probability distribution becomes strictly 

 NB and expected discard per set is p (the NB mean). 



The expected value and variance for individual 

 observations from this probability distribution are 



E[Y] = (l-p)jU (2) 



var[Y] = (l-p)(p + (a + p)p 2 ). (3) 



We fit the NBAZ model using maximum likelihood 

 and allowing the three model parameters to depend 

 upon set type and geographic area. We also consid- 

 ered using tons of tuna loaded (i.e. commercial catch), 

 time of day, and month as covariates, but rejected 

 them as either unfeasible (due to sampling unbal- 

 ance) or statistically unimportant. We did not at- 

 tempt to account for any long-term (i.e. year to year) 

 trend in discard rates because the data included too 

 few years for such an analysis. 



A priori, we used the same three geographic areas 

 (Fig. 1) as those currently used to compare U.S. and 

 non-U. S. dolphin mortality rates (Federal Register, 

 1989). These roughly define the major fishing areas in 



1 The NBAZ can be equivalently reparametized in terms of a "zero- 

 truncated" NB and a probability mass at zero. The parameter 

 corresponding to p would, in that case, denote the probability 

 of a zero observation regardless of source. Thus, that form does 

 not distinguish between "true" and "rounded" zeros. 



