Parrack: Estimating stock abundance from size data 



31 1 



was not large; the significance levels for the hypotheses 

 of bias < 10% were >0.99995 for all estimates. Preci- 

 sion was not a problem, although error variances were 

 not zero. 



Since sampling variation was zero and the estima- 

 tion model encompassed all of the population char- 

 acteristics simulated, the estimation bias and impreci- 

 sion were unexpected. The only possible source of that 

 error are the integral approximations required in 

 estimation. 



The estimate of the unobserved change rate (z) was 

 biased high by about 60% and its error variance was 

 large. It entered estimation in an exponent, so the term 

 in the model was the exponent of z (the reciprocal of 

 "survival" from unobserved change), not z. The error 

 term was again computed on the exponent of the 

 estimate of z instead of z. The estimated bias was ten 

 times lower and the error variance was several orders 

 of magnitude less. This result proved consistent in all 

 tests of population processes. 



Partial correlation coefficients between parameter 

 estimates did not exhibit meaningful trends. Although 

 some adjoining abundance estimates were correlated 

 (probably because the abundances were), evidence of 

 other correlations were absent. Estimates of z were not 

 correlated with the estimates of the q's or abundances; 

 estimates of the q's were not correlated with abun- 

 dance estimates. This result proved consistent. The cor- 

 relation matrices for this and following tests are not 

 shovra for the sake of brevity but are presented in Par- 

 rack (1990). 



The two seasonal recruitment tests (protracted and 

 contracted patterns) show increased bias and impreci- 

 sion. As the recruitment frequency contracted, bias and 

 error variance-of-abundance estimates of the smallest 

 and largest size-classes increased. This problem was 

 worst for the largest size-class. Estimates of the q's 

 also degraded. 



Unobserved change rate The estimator assumes 

 that the rate of change due to phenomena that cannot 

 be observed (natural death, migration, unrecorded 

 catch) is constant over periods. Since the assumption 

 is undoubtedly false, estimation errors resulting from 

 assigning a U(0.1,0.4) random variable to z for each 

 period were investigated. Other simulation character- 

 istics were as in the seasonal protracted recruitment 

 test. The 95% confidence intervals on the difference 

 of abundance estimation bias between this test and the 

 protracted recruitment test included zero for size- 

 classes 3 and 4, most others, and total abundance. 

 Error variances were likely equal for size-classes 3, 4, 

 and total abundance (SL 0.005, SL<0.000, SL<0.000). 

 Correlations between estimates were low. A fourfold 

 random variability in z did not affect estimation at all. 



Growth Three tests consider highly variable growth. 

 The cv's of asymptotic size and k were 0.4. Test 1 

 simulated the same growth parameters as the pro- 

 tracted recruitment test (k 0.17), test 2 considered 

 growth twice as rapid (k 0.34), and test 3 growth twice 

 as slow (k 0.085). All other simulation control variables 

 are the same as the protracted recruitment test, so the 

 results are comparable. 



The results of all three tests were very similar. All 

 reflected the high variation of asymptotic size: the 

 parameter vector included size-classes larger than the 

 asymptote. Abundance and q estimates of these classes 

 (12 and larger) were worthless; huge bias and impreci- 

 sion occurred. Abundances of smaller size-classes in all 

 three tests were more precise than in the protracted 

 recruitment test where growth was not variable. Biases 

 and error variances of abundance and q estimates for 

 size-class 1 1 and smaller were very similar in the three 

 tests; performance seemed unaffected by growth rates. 

 The exponent of z was again estimated much better 

 than z in all three tests; estimates were precise al- 

 though significant bias was present in the case of rapid, 

 variable growth. Evidence of correlated estimates was 

 absent. The introduction of an extremely high level of 

 variation on individual growth parameters did not 

 negatively affect estimates. 



Data estimation and sampling 



Errors attributable to sampling and the compilation of 

 various input statistics were studied in seven tests. 

 Catches are rarely censused as assumed by the esti- 

 mator; estimates are usually the available statistics. 

 The estimator models the dates of each catch, yet catch 

 statistics are usually summed over an interval of dates. 

 Growth rates are assumed to be known, but that is 

 never possible; growth parameters must be estimated. 

 Last, the variability in sampling-gear efficiency coef- 

 ficients, and thus in the abundance indices, is also a 

 source of uncertainty. 



Most of the simulation control variables in these 

 seven tests were the same as in the protracted recruit- 

 ment test. Asymptotic size was 11.95, growth k was 

 0.17, the unobserved loss rate (z) was fixed at 0.1, and 

 the seasonal, protracted recruitment pattern was 

 employed; thus recruitment levels varied 20-fold be- 

 tween periods. Catching was simulated differently than 

 in the protracted recruitment test. Catching was con- 

 tinuous (see Appendix 1, step 4) instead of a single sub- 

 traction at midperiod, and the fishing mortality rate 

 (F) was a U(0. 1,0.4) random variable. 



Catch dates A single scenario was used to investi- 

 gate the importance of recording each catch date and 

 modeling each catch separately. The summed catch 

 over each period was assumed to be known, but not 



