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Fishery Bulletin 90(2). 1992 



These results imply that little will be gained from the 

 extensive age sampling programs that are the founda- 

 tion of VPA-based methods. They are not needed if 

 size-based methods are used; only size samples and the 

 rate of growth are required. Light, periodic growth 

 sampling is sufficient to monitor possible growth rate 

 changes through time. Also, since growth rates are re- 

 quired instead of ages, mark-recapture methods can be 

 used to obtain growth measures if hardpart interpreta- 

 tions (age is not observed on hardparts; instead, char- 

 acteristic marks are interpreted as annular occur- 

 rences) are difficult or expensive to obtain. 



The method of abundance estimation developed in 

 this study, a meticulous bookkeeper of size data as is 

 the method of Beddingtc. and Cook (1981), is primitive 

 compared with other size-based methods (Foumier and 

 Doonan 1987, Schnute et al. 1989, Sullivan 1989, Sul- 

 livan et al. 1990). Its degree of success in estimating 

 abundance suggests that complete population-model 

 structures are unnecessary. Estimates were usefully 

 accurate and precise even with very high process vari- 

 ability. Very pronoimced individual growth variation 

 did not cause estimation problems. These results show 

 that precise, accurate abundance estimates are possible 

 with any recruitment pattern imaginable. It was a par- 

 ticular surprise to find that temporally variant (four- 

 fold) unobserved change rates ("natural mortality" of 

 Ricker (1948) but including migration and unrecorded 

 catch) did not affect estimation at all. That result is 

 reassuring, since the rate is probably extremely vari- 

 able in nature. 



Sampling problems did not seem to degrade estima- 

 tion either. The level of catch estimation error proved 

 unimportant and there was no indication that exact 

 catch dates need to be recorded. Highly variable sam- 

 pling efficiencies (qg) did not cause estimation prob- 

 lems, particularly when sample sizes were adequate. 

 Highly variable individual growth rates (20%) and 

 significant growth measurement error (15%) did not 

 adversely affect abundance estimation when sampling 

 was sufficient. Very large growth-parameter measure- 

 ment error (40%) and small sample size destroyed per- 

 formance; although bias was not a problem, extreme 

 error variances and correlated estimates were. 



It is of particular interest that this was the only test 

 where estimates of the unobserved change rate (z) and 

 sampling efficiencies (q) were highly correlated. The 

 lack of a pronounced correlation between sampling 

 gear efficiencies and the unobserved change rate in all 

 other tests except this one was unexpected; similar 

 studies of VPA-based methods (Paloheimo 1980, Collie 

 and Sissenwine 1983) found such correlation a major 

 characteristic. It thus seems possible that ageing 

 errors, or the violation of a connected VPA assump- 

 tion, contributed to correlation in those studies. 



Abundances of most size-classes were estimated 

 precisely with little or no bias, but biased and imprecise 

 abundance estimates occurred in three circumstances. 

 First, abundances of very small fish that were recruited 

 between the next-to-last and last relative abundance 

 sample were estimated poorly. A recruitment group 

 had to be present in the relative abundance samples 

 twice to be estimated with a useful degree of certain- 

 ty. In practice, this problem is easily fixed if obtaining 

 certain estimates of recent recruitment of small fish 

 is important enough to justify the cost of additional 

 samples during the last period. Since the estimator is 

 not based on equal time units, only dates, additional 

 sample(s) vdll monitor the size-classes of interest 

 several times instead of just once. Second, wide size- 

 classes caused bias and imprecision, particularly for 

 larger sizes. This bias was easUy eliminated by narrow- 

 ing size-classes. Last, calculations of historical abun- 

 dances were in large error. It is well known that VPA 

 calculations are poor for the most recent period of data 

 and improve as dates decrease. Though they are not 

 germane to current production levels, estimates of the 

 oldest stock sizes are the most certain ones in VPA. 

 The exact opposite is true for this size-based method. 

 Estimates of historical abundances obtained in the solu- 

 tion calculation should not be used; error variances of 

 these computations are very large. Since the estimates 

 of the final-period abundances are accurate and precise, 

 this is probably not a problem even if historical stock- 

 size estimates are needed. Although the procedure was 

 not tested, these estimates might be obtained by start- 

 ing with the initial four periods of data, estimating the 

 fourth period abundance vector, and then progressing 

 forward one period at a time. Abundance in the first 

 three periods cannot be estimated but subsequent abun- 

 dances can. The relation between the number of periods 

 in the data and estimation errors was not investigated, 

 but the authors- experience with VPA-based methods 

 indicates little, if any, would be gained with a longer 

 time-series. 



This study shows that a priori knowledge of the un- 

 observed change rate (z) is not required to accurately 

 and precisely estimate abundance with this size-based 

 method, yet it is well known (Paloheimo 1980, Collie 

 and Sissenwine 1983, Deriso 1985, Pope and Shepherd 

 1985) that such knowledge is necessary when apply- 

 ing VPA-based procedures. 



This study suggests that the unobserved change rate 

 (z) will often be estimated with bias, yet z should be 

 included in the vector of estimates anyway. Monte 

 Carlo tests of the Beddington and Cook model estab- 

 lished that simultaneous estimation of a natural mor- 

 tality schedule (analogous to the unobserved change 

 rate in this study) is necessary to avoid biased abun- 

 dance estimates (de la Mare 1988). If z is fixed instead 



