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Fishery Bulletin 90(2), 1992 



portion is the product of fishing effort and q for the 

 fishing method. Since yield is the product of q, effort, 

 and biomass, then yield-per-effort equals the product 

 of q and biomass and q is yield-per-effort divided by 

 biomass. It then follows that the cv's of q and yield- 

 per-effort are equal. The cv of yield-per-effort of the 

 Pacific halibut longline fishery is estimated to be 0.02 

 (Quinn et al. 1982), and that of Newfoundland flounder 

 trawlers on the Grand Bank (Smith 1980) is estimated 

 to be about the same. The levels used in these simula- 

 tions (0.4 and 0.2) are about an order of magnitude 

 higher than those. 



Effects of the variability in q on estimation errors 

 were investigated in three tests. All simulation con- 

 stants were as in the protracted recruitment test ex- 

 cept those related to abundance sampling. Simulation 

 control constants were cv[q] and the two constants re- 

 quired to compute the sample size. Although they were 

 probably unrealistically large, a cv[q] of 0.4 was used 

 in the first test and 0.2 was used in the second. 



First, the impact of extreme variability (cv 0.4) and 

 extremely light sampling was tested. The sample size 

 (r 3) was such that a 95% confidence interval on relative 

 abundance was within ± 50% of the expectation. The 

 extremely high cv[q] and low relative-abundance sam- 

 ple size were not reflected in error variances as much 

 as expected (Table 2), but error variances were higher 

 than those of the protracted recruitment test. Most 

 abundance estimates were biased by less than 10%. 

 Estimates were not correlated. 



Next, the cv[q] was reduced to 0.1 and the sample 

 size was increased so that a 95% confidence interval 

 on relative abundance was within ± 5% of the expec- 

 tation (r 16). The result was very similar to those of 

 the first test except error variances were much lower. 

 Biases of abundance estimates were ± 10% or less and 

 estimates were not correlated. 



There was no evidence that high variation in the 

 qs biased abundance estimates even if sample sizes 

 were insufficient, but error variances were affected. 

 Error variance was considerably reduced with reason- 

 able sample sizes. 



Bias 



The results of these experiments (Tables 1 and 2) show 

 that abundances and gear efficiencies (q's) of the 

 smallest and largest size-classes were often biased. Bias 

 did not occur with uniform, constant recruitment and 

 no sampling variation, but as process and sampling 

 variation increased, bias in estimates of the smallest 

 and largest sizes became pronounced. 



Each expected value is a proportion of calculated 

 abundance. The abundance calculation sums future 

 catches (data), last-period abundance (estimates), and 



an amount for unobserved changes (estimate). Future 

 catches and terminal abundance are thus the major 

 components of each projection. Both catch and final 

 abundance must be integrated over size. The integra- 

 tion of catch over size at each catch date following the 

 date of the expected value is required. The integration 

 of abundance over size on the date of the final relative 

 abundance sample is also necessary. All integrals are 

 approximated, so these calculations are the source of 

 the bias. The amount of error incurred at each integra- 

 tion depends on how well the trapezoidal rule approx- 

 imates the size distribution. Since the size frequency 

 within a size-class is never smooth, the approximation 

 will be in error with the amount depending on the 

 degree of smoothness within the size-class. If growth 

 is variable or the number of fish is small, clumps in 

 size frequencies can result from chance alone, but the 

 major factor is the growth and recruitment pattern 

 combination. 



Narrowing the size-classes eliminates this problem. 

 If they are narrowed enough to eliminate clumping 

 caused by the particular recruitment frequency con- 

 traction, the size frequency within size-classes will be 

 smooth and the trapezoidal approximation will be ac- 

 curate. The seasonal contracted pattern of recruitment 

 test 3 was again used to demonstrate this. An asymp- 

 totic size of 120 cm was simulated with recruitment 

 occurring at 20 cm. First, it was assumed that the data 

 were collected in 20 cm intervals so that the asjonptotic 

 size was 6 and the recruitment size was class 1. In the 

 second case, it was assumed that data were collected 

 in 2 cm groups so that the asymptotic size was 60 and 

 the recruitment size was class 10. The unobserved 

 change rate was set at 0.2 in both tests, and all other 

 simulation control variables were as in the contracted 

 recruitment test. 



Ninety-two trials were required to obtain a 95% con- 

 fidence interval half-length of 0.05 on the bias of total 

 abundance in bias test 1 with 20 cm interval data. 

 Estimates of the smallest and largest size-class abun- 

 dances were biased and the error variances were very 

 large (Table 3), particularly for the largest size-class. 

 The estimate of the survival from unobserved change 

 (z) was, however, reasonably accurate and precise. 



Only 16 trials were required to obtain a 95% con- 

 fidence interval half-length of 0.03 on the bias of total 

 abundance in bias test 2 with two-unit size-interval data 

 because the error variances were very low. Estimates 

 of the first three size groups were probably biased by 

 10% or more, but the rest were not. Only six of the 

 47 estimates were probably biased at all (0.95 level). 

 The estimate of the exponent of z was also not biased. 

 Although the matrix was too large to be included (194 

 rows and columns), there was no evidence that esti- 

 mates were correlated. 



