Sigler: Estimation of abundance of Anoplopoma fimbria off Alaska 



595 



tice, A = 1 implies an effective sample size for the 

 age and length samples of 100 if the CV of the sur- 

 vey index is 10%. This CV is reasonable according to 

 a statistical analysis of the survey index, which found 

 that a 10% interannual change in the survey index 

 typically was statistically significant at the P= 0.05 

 level (Sigler and Fujioka, 1988). The reasonableness 

 of the effective sample size is harder to determine 

 because the age and length samples are composites 

 of multinomial samples from survey stations with 

 underlying geographic variation in age and length 

 compositions, so that the effective sample size is less 

 than the true sample size. In the above expression 

 log (p ,^j) usually found in the log-likelihood for the 

 multinomial distribution was replaced with log ( p ^J 

 p .J- This replacement removes a "nonsignificant" 

 portion of the likelihood and makes it easier to ex- 

 amine the model fit and probably somewhat improves 

 numerical performance (Kimura, 1990). 



Assuming M and A are known, this model contains 

 Y + A + 2 parameters: recruitment, TV^j ..., A^ ., the 

 initial age composition, N q^^, ..., N ^^, and the selec- 

 tivity parameters, A^q, /3, and y are unknown. The 

 quantities and /ij, ..., /}y are functions of the ob- 

 served survey indices and observed catches and of 

 the parameter estimates. Setting clL/dq equal to zero 

 and solving for q gives 



q = exp 



1'°^^ 



N. 



Y 



The /j J , . . . , jUy are computed from /; 



~ Ni ' 



by treating reported catches as exact. Although 

 clearly there will always be some error in the reported 

 catch, I concluded that this approximation generally 

 was reasonable given the comprehensive system for 

 tracking the Alaskan sablefish catch, which includes 

 processor reports, fish receipts, individual fishing 

 quota landing reports, and observer coverage. I esti- 

 mated log-parameters rather than parameters on the 

 original scale to improve reliability in the estima- 

 tion process (Kimura, 1989, 1990). 



An allowable biological catch (A5Cgg) was calcu- 

 lated from a 1-year future projection. The constant 

 fishing mortality, F^^r, , was applied, which reduces 

 the exploitable population to 40% of the unexploited 

 state (Clark, 1991). A 1 -year projection of recruitment 

 was forecast as the average of recent estimated re- 

 cruitments. The two most recent recruitments were 



not used in the average; their estimates, based on 

 only one or two years of data, are unreliable. 



Validation of estimation method 



I used Monte Carlo simulation for model validation 

 (Kimura, 1989; Press et al., 1989) to verify that the 

 age-structured analysis provides reasonable results. 

 The steps (Press et al., 1989) are as follows: 1) Fit 

 the model using real data; 2) take the estimated pa- 

 rameter values as the true values for the simulation; 

 3 ) calculate the expected data based on these param- 

 eter values, then simulate a new data set by adding 

 a particular error to the expected data based on an 

 observed mean square error ( MSE ) or any hypotheti- 

 cal value; and 4) estimate the model parameters for 

 the simulated data. If the resulting parameter esti- 

 mates and the true parameter values are similar, 

 then the estimation method is to some extent vali- 

 dated. This approach assumes that the model fits 

 the simulated data perfectly except for random er- 

 ror. If the true errors were of larger magnitude or 

 arose from a different source than that assumed in 

 fitting the model, for example in violating the as- 

 sumptions of equal fishery and survey selectivity and 

 constant growth rates, then estimation performance 

 with simulated data overstates the true model per- 

 formance. 



Log-normal error (CV=0.10 on the original scale) 

 was added to the expected abundance index, and 

 multinomial error (n =200 1 was added to the expected 

 age and length data. These values were based on 

 variability of the model residuals (difference between 

 the observed and expected data). The log-transformed 

 abundance indices are assumed to be independent, 

 normally distributed random variables, and a "mi- 

 nor dilemma" (Kimura, 1989) arises on whether to 

 simulate the abundance indices so that they are un- 

 biased on the original or the logarithmic scale. I chose 

 to simulate the abundance indices so that the sur- 

 vey index was unbiased on the original scale. 



Age data are often limited in availability, whereas 

 length data are nearly always available. Age data 

 are more desirable than length data for estimating 

 age structure because the correspondence between 

 age and length is not one-to-one. Age data are com- 

 monly inaccurate: some individuals, especially older 

 fish, are misassigned to ages around the true age. 

 Thus, inaccurate age data and length data are simi- 

 lar in that true age does not uniquely correspond to 

 assigned age or length. Parameters were estimated 

 from simulated data for both accurate and inaccu- 

 rate age data, the latter also being equivalent to 

 length data. For accurate age data, 100% offish of 

 simulated samples were correctly assigned and cat- 



