Parrack: Estimating stock abundance from size data 



307 



A significance level for bias larger than 10% was 

 found by computing the probability of the standard 

 normal random variable as follows: 



significance level (««:;[g«;^) = 

 z 



J fpdp, z = (G-o.9)-[s(e)/\^]. 



— oo 



significance level {«0:;:[|H 5;}) = 



oo Z 



J" fp dp = 1.0 - J U dp. = 



YEARS 



Figure 1 



Frequency of recruitment dates from one trial of uniform 

 recruitment simulations. 



(G-1.1) - [S(e)/^]. 



The results between tests were statistically compared 

 by placing confidence intervals on the difference 

 between the biases (Law and Kelton 1982:319) and 

 using the variance ratio test {F test) to compare error 

 variances. 



In each Monte Carlo test, the intent was to complete 

 trials until the estimate of bias was within a given 

 bound with a prescribed probabihty (Law and Kelton 

 1982). Several parameters were estimated, so several 

 biases were involved. It was too costly to confirm that 

 all bias estimates were trustworthy and many param- 

 eters were not of primary interest, so the error of last- 

 period total stock size (E[N(T.)]) was used as the 

 reference statistic. Trials were completed until 



1.962 . s2(£[N(T.)]) - n < <D2, 



where <t> was usually small. The 95% confidence bound 

 half-lengths for all parameters were computed to in- 

 dicate how well bias was estimated for each parameter. 



The method of Schrage (1979) was used to generate 

 uniform random variables because it is portable and 

 knowTi to perform well (Law and Kelton 1982:227- 

 228). Normal random variables were generated by the 

 polar method (Law and Kelton 1982:259). The method 

 of Scheuer and Stoller (1962) was used to generate cor- 

 related bivariate normal random numbers. 



In most trials, the lives of 20,000 fish were individual- 

 ly simulated over 20 time-periods. A history of abun- 

 dance and catch was created, then abundance sampling, 

 catch estimation, and growth parameter estimation 

 was simulated. Each fish possessed a unique growth 



pattern and recruitment date and independently en- 

 countered unobserved events and fishing death. The 

 result of these encounters, growth rates, and recruit- 

 ment dates were tabulated into size-class and date- 

 specific matrices of numerical abundance and catch. 

 The sequence of events of the population simulation is 

 diagrammed in Appendix 1. A detailed description of 

 the simulation and justification of control variable 

 levels is given by Parrack (1990). 



Von Bertalanffy growth was simulated by fixing m 

 of equation (1) null. For each fish, A and k of (1) were 

 drawn as normal random variables. The expectations 

 were set near those estimated for many stocks, in- 

 cluding Pacific cod (N.J.C. Parrack 1986), and their 

 coefficients of variation (cv) were set as high or higher 

 than common in other studies (<0.4). 



Two kinds of recruitment were considered, uniform 

 and seasonal. The uniform pattern (Fig. 1) simulated 

 continuous recruitment of constant magnitude. The 

 date each fish recruited to the minimum size category 

 was drawn as a U(l,20) random variable. Seasonal 

 recruitment dates were drawn from normal distribu- 

 tions so that recruitment magnitudes varied U(l,20) 

 between periods and so that a typical "pulse" of yoimg 

 fish recruited once each period, with some recruitment 

 occurring continuously. The recruitment peak was 

 simulated to occur randomly during April, May, and 

 June by drawing the expected recruitment date for 

 each period U(0.25,0.50). Protracted and contracted 

 seasonal recruitment patterns were considered. Sea- 

 sonal protracted recruitment was simulated by draw- 

 ing the standard deviation of recruitment dates 

 U(0.20,0.33) so that 80% of recruitment occurred ran- 

 domly within ±3-5 months of the peak (Fig. 2). Sea- 

 sonal contracted recruitment was simulated by draw- 

 ing the standard deviation U(0.13,0.26) so that 80% 



