Parrack: Estimating stock abundance from size data 



313 



the dates of the catches. The accumulated catch each 

 period was assigned to the midpoint of each period for 

 estimation. The results (Table 2) were almost identical 

 with those of the protracted recruitment test (Table 

 1). The 95% confidence interval (Welch 1938) on the 

 difference between total-abundance estimation bias of 

 the protracted recruitment test and this test included 

 zero (-0.0014 to 0.0372). The error variances were 

 very similar (0.0072 and 0.0063). Estimates were not 

 correlated. The absence of exact catch dates did not 

 affect estimation. 



Catch estimation error The effects of estimating 

 catches rather than enumerating them were investi- 

 gated by drawing size-class-specific catch estimates as 

 normal random variables with expectation C(t, s) and 

 variance (cv[C] ■ C(t,s))2. This simulated unbiased 

 catch estimation and estimation error proportional to 

 catches. A large degree of catch estimation uncertainty 

 was imposed (cv[C] = 0.40). Simulation control variables 

 were the same as in the catch date test and the pro- 

 tracted recruitment test. Results were also similar. The 

 bias of total abundance estimates was about the same 

 for all three tests and the error variances were nearly 

 so. Correlated estimates were not evident. Confidence 

 intervals (95%) on the difference in bias between this 

 test and the protracted recruitment test included zero 

 for all size-classes and total abundance. The error 

 variance for size-class 3 was different (SL<0.0005) and 

 might have been different for size-class 4 (SL 0.052), 

 but probably not for total abundance (SL 0.142) and 

 all others. Imprecise catch estimates did not impact 

 bias or error variance. 



Growth parameter estimation error The effect of 

 imprecise growth parameter estimates was also con- 

 sidered. Estimates of growth parameters were simu- 

 lated as normal correlated random variables with 

 expectations equal to those of the population. As ex- 

 plained in the Monte Carlo methods section, the vari- 

 ance of a growth parameter estimate is composed of 

 two parts: process variation due to variant individual 

 growth, and growth measurement error. Simulation 

 control constants were therefore the cv of A and of k, 

 the growth measurement error cv, the two constants 

 required to compute the sample size used to estimate 

 the growth parameters, and the correlation coefficient 

 between estimates (-0.95). Simulation constants were 

 as in the catch date test except those related to growth 

 parameter estimation. 



Three tests were carried out, two without process 

 variation. First, the effect of two measurement error 

 cv's was studied in the absence of growth variability. 

 The sample size was set at one fish in these two tests 

 so affects due to measurement error would be magni- 



fied. Then, the combined effect of process variation and 

 estimation error was considered. 



In the first test with extremely imprecise growth 

 parameter estimates (cv 0.4), Monte Carlo trials were 

 carried out until it became obvious that little more in- 

 formation would be gained with further computations. 

 Error variances were huge (Table 2). Only the expo- 

 nent of z was reasonably estimated. Many estimates 

 were correlated, particularly those of z with those of 

 sampling-gear efficiency coefficients. Even without in- 

 dividually variant growth rates (an unlikely prospect), 

 large growth-parameter measurement error created 

 significant uncertainty. 



The second test simulated 15% measurement error. 

 A 95% confidence interval on the difference between 

 the bias of total abundance estimates between this and 

 the protracted recruitment test included zero, but the 

 error variances were probably different (SL< 0.0001); 

 most error variances were higher. Bias was imaffected 

 although error variance approximately doubled. The 

 estimates did not seem correlated. The introduction 

 of a 15% growth measurement error increased error 

 variances but did not affect bias. 



The third test simulated both process error (cv 0.2) 

 and 15% growth measurement error, but with a sam- 

 ple size such that 95% confidence intervals on the 

 estimate of the expectation of growth parameters were 

 with precision ±2% (g = 601 fish). The 95% confidence 

 interval on the difference in bias of total abimdance 

 estimates between this test and the protracted recmit- 

 ment test included zero ( - 0.0220 to 0.0238) although 

 error variances perhaps differed (SL=:0.05). Estimates 

 were not correlated. Apparently 15% (or less) measure- 

 ment error, even with natural growth variation, min- 

 imally affects estimation. 



Gear efficiency variability The estimator is derived 

 from the density function of relative abundance obser- 

 vations (Y), but the effect of Y variability on estima- 

 tion error was not of large interest. The variance of 

 Y is o^lYt s] = Nt s2cv[q]". The dominant term is the 

 square of abundance, so as abundance increases, o- 

 [Yts] increases. This may be dampened a bit by an in- 

 crease in q with size, but the dominant factor in the 

 variance expression for the observations is abundance. 

 Abundance levels cannot be controlled or anticipated 

 beforehand, so knowledge of the effect of Y variabil- 

 ity is of little value. Knowledge of the effect of q vari- 

 ability is useful, however, since care may be taken in 

 the selection and design of sampling gear. 



Studies that document the statistics necessary to 

 calculate the variability of relative abundance sam- 

 pling-gear efficiencies are not common. Studies of 

 commerical fishery statistics offer different but useful 

 information. Yield is a portion of biomass; the pro- 



