Kimura et al. Tag-recapture data analysis of Pacific cod growth 



273 



and all observational errors e,, and e 2 , are assumed to 

 be zero. 



Simulation 3 Both observational errors and varia- 

 tion in L„„ are present in this model, with e^, £ 2l , £ 3, 

 ~ N(0,o 2 ). 



For these simulations, we assume that the true popu- 

 lation parameters are (L = 61.23, K,,=0.296), param- 

 eters previously estimated for female Pacific whiting 

 (Kimura 1980). Measured in years, t,, and d, were uni- 

 formly and independently distributed over the inter- 

 val [1, 5], assuming 365d/yr. New values for t b and d, 

 were generated for every replication of the simulation. 



Analytic standard errors of parameter estimates 

 were estimated using nonlinear least-squares meth- 

 ods, and the variance estimate described above for 

 James' method. In addition, the sample standard er- 

 rors (i.e., the sample standard deviation of the repli- 

 cate parameter estimates) are also provided. The rea- 

 son for this is that the simulations included variation 

 in t h and d, that would not be included in the analytic 

 estimates of standard error, and the probability that 

 the analytic standard errors are themselves biased due 

 to failure of assumptions. For the actual datasets, ana- 

 lytic standard errors were used. For simultaneously 

 considering bias and variance, we include estimates of 

 mean square error (MSE) calculated in the usual 

 way: I (estimate - true fin. 



Following a presentation of simulation results, we 

 analyze tag- recapture data collected for sablefish Anop- 

 lopoma fimbria from the Gulf of Alaska and off the 

 U.S. west coast, and Pacific cod Gadus macrocephalus 

 from the eastern Bering Sea (Fig. 1). Because sable- 

 fish and Pacific cod are difficult species to directly age, 



^^^V*^*** 



Figure 1 



Tag and recovery areas for Pacific cod in eastern Bering Sea (BS). sablefish in Gulf of 

 Alaska (GOA). and sablefish off U.S. west coast (WC). 



their growth-curve parameters estimated from tag-re- 

 capture data are of special interest. The von Bertalanffy 

 parameters estimated from tag-recapture data were 

 compared with parameters estimated from length-at- 

 age data based on direct ages (i.e., ages from directly 

 counting annuli). For sablefish, we use length-at-age 

 datasets based on break and burn ages (Chilton & 

 Beamish 1982) and parameters estimated using 

 nonlinear least-squares (Kimura 1980). For Pacific cod, 

 we compare tag-recapture results with published von 

 Bertalanffy parameter estimates (Thompson & Bakkala 

 1990). 



Results 



Simulation results 



For the simulation results given here, the tag-recap- 

 ture sample size was n=300, simulations were repli- 

 cated 200 times, and the normal errors (£ h , e 2l , e 3l ) 

 were all independently distributed with o 2 =25. There- 

 fore, the sample size was realistically small and the 

 error variances were substantial. For each estimation 

 method and parameter, simulation results (Table 1) 

 were summarized by four entries: sample mean of 

 the estimated parameter, mean analytic standard er- 

 ror, sample standard error (i.e., the standard devia- 

 tion of estimated parameters), and mean square error. 

 The unweighted Fabens' estimate of L is biased low 

 in Simulation 1, biased high in Simulation 2, with 

 biases apparently canceling each other in Simulation 

 3. For the weighted Fabens' estimates, the L estimate 

 is unbiased for Simulation 1, but 

 biased high for Simulations 2 and 

 3. James' estimate of L is unbi- 

 ased in Simulation 2, and only 

 modestly biased in Simulations 

 1 and 3. Biases in L and K, esti- 

 mates are in the opposite direc- 

 tion, as might be expected from 

 the negative correlation between 

 these parameter estimates. 



Standard errors are uniformly 

 smallest for the unweighted 

 Fabens' method, in the middle for 

 the weighted Fabens' method, 

 and largest for James' method. 



Performance as measured by 

 mean square error was entirely 

 dependent on the assumptions of 

 the simulation. For Simulation 1 

 (i.e., observational errors) the 

 weighted Fabens' method MSE 

 was smallest; for Simulation 2 



