590 



Fishery Bulletin 98(3) 



Proportion of positive tows 



The proportion of positive tows for a particular fish species 

 was estimated after classifying each tow as either (no 

 fish caught) or 1 (at least one fish caught). For the shrimp 

 bycatch data, the model assumes that the data are inde- 

 pendent results from n successive trials of a Bernoulli- 

 type random variable with a probability p of catching a 

 given fish species. In this case, it is assumed that the fre- 

 quency distribution of observed zero and positive tows in 

 each cell follows a binomial distribution. The error term is 

 assumed to be constant and independent among the cells. 

 The binomial distribution is then defined in terms of the 

 proportion iy) of positive tows (r) to total tows In) per cell, 

 and the probability density function f(y) and associated 

 variance Var (y) function are given by 



Mean bycatch rate 



In this section only positive tows were considered. The 

 delta lognormal model assumed that for a given species 

 the number offish caught as bycatch relates to fixed vari- 

 ables: data source (commercial or research), year, season, 

 area, and depth zone. The mean bycatch CPUE given a 

 nonzero catch was also estimated following a generalized 

 linear model approach. In this case, the random compo- 

 nent for the estimated CPUE was assumed to follow a 

 lognormal error distribution within cells. The probability 

 density function is given by the normal function 



f^yy- 



yl2n:o- 



i:)' 



fory = r//!, fir) = \ hi'(l-/i)""' where r = 1, 1, 2 n 



Variy) =;/(! -fi)/n 



where // = the mean of v. The response variables v, are 

 independent for / = 1,2 n tow trials. 



The systematic component defines the set of explana- 

 tory variables .Vj,.v.„...,.v which produce a linear predictor 

 f] given by 





For the shrimp bycatch data, the linear predictor is a 

 linear function of the fixed explanatory variables dataset, 

 year, season, area, and depth zone, such that 



J] = Pq + P^  dataset + /3.,  year + p^  season + 

 p^  area + P-  depth zone, 



where the p^ are parameters to be estimated. 



The link function that relates the linear predictor (j to the 

 expected value // of observations y in each cell of the model 

 must be a monotonic differentiable function g such that 



^'Ai)= ')■ 



In this case, the logit or logistic function expresses the 

 relationship between the assumed binomial error distri- 

 bution of /^ and the given linear function of explanatory 

 variables rj, as 



')=log 



^| 



(1-^/) 



The GENMOD algorithm uses maximum-likelihood esti- 

 mates for assumed binomial distributions, which are unbi- 

 ased to a first order of approximation (McCullagh and 

 Nelder, 1989) 



where // = £[v];and 



d~ = Variy I with a logarithmic link function. 



This specification is mathematically equivalent to defin- 

 ing the random component as lognormal with the identity 

 as the link function. 



The systematic component is defined as 



LogiCPUE} = Pq + P^- dataset + P^  year + 

 /3.J  season + j3^ • area +■ P^  depth. 



where CPUE = the catch rate in numbers offish per net 

 hour for nonzero catches; 

 pQ = the overall mean; 

 dataset = a fixed effect differentiating data sources 

 from commercial shrimp fishing from 

 those in research trawls, the terms year, 

 season, area, and depth are also fixed 

 effects; and 

 the P = parameters to be estimated. 



The link function between the random and systematic 

 components is the identity function: 



')=/'■ 



Estimation of bycatch 



The overall model is then referred to as the delta lognor- 

 mal model. This model generates the estimated propor- 

 tion of positives tows ( P,,ki,„ ) and the mean bycatch rate 

 ( CPUE^^^.,,^ 1 for a given species. Estimates of bycatch are 

 calculated as the product of the proportion of positives tows 

 ( p,^j,,„ ) multiplied by the mean bycatch rate ( CP[/£„,,,,„ ) 

 multiplied by the shrimping effort t/",^,,,,) multiplied by the 

 two nets (assumed) per boat. Shrimping effort data are the 

 same as those used in the current general linear model. 

 Annual estimates of bycatch are simply the sum of bycatch 

 per cell over the season, area and depth zone strata, for 

 the commercial sector (;=1). 



Bvcatch 



1- 



,.<.^CPUE,„,.>if,,„. 



