Parrack: Estimating stock abundance from size data 



309 



number of fish sampled for growth (g) was specified 

 in the simulation indirectly as a probability level and 

 limit of a confidence bound. For a 1 - a level confidence 

 interval on A of bound length 2<t>A, 



0A = Z(l-a/2)o[A] 



so 



= Z(l-a/2) \/A2(cv[A]2 + cv[e]2)/g, 



g = Z2 I 1 - - 1 (cv[A]2 + cv[e]2) - (D2. 



On each sampling date r, relative abundance samples, 

 l<k<r, were simulated as 



Yt,s = Z Yt.s,k - r = X qs.k Nt,s - r, 



k=l k=l 



qs.k ~ N (q^, cv[q]2 q^^), 



process variables (z, growth parameters and recruit- 

 ment magnitude, duration, and timing), but time trends 

 were not. A different unobserved change rate was 

 drawn for each time-period (zt~NO^, o2[z])), but the 

 expectation and variance were constant over time and 

 size. Different growth parameters were drawn for each 

 fish, but the expectations and variances were the same 

 for all fish. Recruitment magnitudes for each time- 

 period were drawn from a uniform distribution so time 

 trends were not simulated. A different recruitment 

 peak (i.e., the expectation of recruitment date) was 

 drawn for each time-period from a common expecta- 

 tion and variance. A duration of recruitment (i.e., 

 variance of recruitment date) was drawn for each time- 

 period, but with the same expectation and variance. 

 Random variation was simulated in sampling variables 

 (catch estimates, growth parameter estimates, and the 

 Qs), but biased estimates were not simulated. Al- 

 though a different vector of sampling efficiencies (the 

 Qs) were drawn for each sampling date, the expecta- 

 tion and variance for each size was temporally constant. 



where cv[c[] and the q^ were simulation constants. The 

 qs were 0.025, 0.05, 0.175, 0.225, 0.2425, and 0.25 

 (smallest to largest size-class) in most simulations. 

 Several other variations were tried, and it was found 

 that these constants did not affect results at all. Cv's 

 of q were 0.4 or less. The observation and its variance 

 were calculated as the maximum likelihood estimates, 



Yt,s = 1 Y,,,,k/r 



k 



s2[Y,„3] = I (Yt,s,k-Yt,s)^/(r2-r). 



k 



The sample size (r) was fixed indirectly by two control 

 variables, the probability level and confidence-bound 

 width for Yt,s where a2[Yt,s] is as (5). If a 1 -a level 

 confidence interval on Yj g was to be of bound length 

 2»<J)«Nt,s*qs then: 



<»Nt,sqs = 1 - Ho[Yt,s], 



r = Z2(l-a/2)cv[qj2 - (1)2. 



Sampling error entered the simulation as variation in 

 qs, not as variation in the Ytsi the variance of the 

 abundance index was not an input. 



These simulations encompassed many possibilities, 

 but not all. Random variation was simulated in all 



Results 



The Monte Carlo tests fall into two categories: those 

 that investigate the influence of population process 

 variability on estimation errors, and those that test the 

 effects of sampling and data estimation. Process vari- 

 ability includes recruitment phenomena, growth rates, 

 and unobserved change due to emigration, immigra- 

 tion, natural death, and unrecorded catch. Sampling 

 variation and data estimation includes four topics: 

 catch estimation error, unrecorded dates of catch, 

 growth parameter estimation error, and variability in 

 sampling-gear efficiency coefficients, and thus in the 

 abundance indices. Each of these items were studied 

 separately in 14 tests. 



Population process variability 



For these tests, catches, dates of catch, and popula- 

 tion growth parameters were considered to be known, 

 and sampling gear efficiencies (the q's) invariant so that 

 all sampling variation was absent. Catches were taken 

 at midperiod. The probability of death due to catching 

 in each period was an 11(0.05,0.2) random variable, and 

 asymptotic size was 11.95 units (i.e., 119.5cm, with 

 10 cm intervals). 



Recruitment patterns Uniform recruitment test re- 

 sults (Table 1) show little bias and high precision in 

 estimates of abundance and q's. Significance levels for 

 the hypothesis of bias = 1.0 (unbiased) versus biasi^l.O 

 (biased) were <0. 00005 in almost every case, but bias 



