64 



Fishery Bulletin 101(1) 



on measurement error would come from fish with short 

 times at liberty because in these cases the amount of pro- 

 cess error would be small. However, as explained above, 

 fish with times at liberty of less than 270 days were ex- 

 cluded to eliminate seasonal effects. Process and measure- 

 ment errors are partially confounded in the model. The 

 estimation procedure could not distinguish an additional 

 constant component to the linear, temporally increasing 

 process error from the measurement error for recaptured 

 fish measured by scientists. For the 1980s data, the esti- 

 mation procedure was able to estimate a nonzero value for 

 the scientist measurement error. The results for the 1980 

 returns suggest that the measurement error for the scien- 

 tists was about 509^ of that for fishermen. 



For both the 1960s and 1980s data, the estimation 

 procedure is able to distinguish between fishermen and 

 scientist measurement error. The assumption that the 

 measurement error of fishermen and scientists are the 

 same is rejected by a statistical test (i.e. the second last 

 line of both Table 1, A and B ). 



Examination of the residuals indicated no systematic 

 lack of fit in either the 1960s or 1980s data (Fig. 1, A-D). 



Shape of the likelihood function 



It is worth noting that the likelihood function is bimodal 

 with respect to the parameter determining the length 

 at which growth changes between the two phases (L*). 

 Figure 2 shows the negative log-likelihood (-LL) values 

 as a function of L* for the 1960s and 1980s data for the 

 best-fit model. For the 1960s data, the -LL function has 

 an absolute minimum at L" = 74 cm and a second, local 

 minimum at L* = 91 cm. For the 1980 data, the absolute 

 minimum occurs at L* = 85 cm and the second, local mini- 

 mum at L' = 120 cm. 



Monte Carlo and bootstrap simulations were conducted 

 to evaluate the bimodal nature of the likelihood function. 

 The Monte Carlo simulations were done by assuming the 

 two-phase VHf! growth model and were conditional on the 

 SBT release lengths. Both types of simulations confirmed 

 that the minima with the lowest absolute value for the 



negative log-likelihood function would switch between 

 individual realizations within the simulations. Only the 

 results of the bootstrap simulations are presented in this 

 article. These were based on 1000 individual simulations 

 for which the tag increment data were randomly sampled 

 with replacement. For each individual simulation, the val- 

 ue of L* that yielded the absolute minimum value for the 

 negative log-likelihood function was determined. For the 

 1960s data, the best-fit estimate of L* was near 74 cm in 

 930 of the simulations and 91 cm in 70. For the 1980 data, 

 the absolute minimum in the negative log-likelihood func- 

 tion occurred 767 times when L* was near 85 cm and 233 

 times when L* was near 120 cm (Fig. 3). Thus, although 

 the lower value for L* was the most likely for both the 

 1960s and 1980s data, the 95% confidence intervals based 

 on the bootstrap results would encompass both values. 

 The estimated value of the other parameters determining 

 the expected growth curve are correlated with that of L*. 

 Thus, the alternative minima in the log-likelihood func- 

 tion are associated with substantially different estimates 

 for the k and L parameters (Table 2). This in turn has 

 implications for possible biological interpretations of the 

 parameter estimates (see below). 



Joint analyses of the 1960s and 1980s data 



Results of jointly modeling the 1960s and 1980s tag 

 return data to test for common parameters are presented 

 in Table 3. The model error (cr^^) was the only parameter 

 found not to be significantly different in the combined 

 analyses. Having a single parameter value for the model 

 error component had virtually no effect on the parameters 

 determining the expected growth rates, compared to those 

 estimated in the separate analyses. 



Comparison of 1960s and 1980s growth rates 



The fact that all of the parameters that describe the 

 cxpecti'd growth rates significantly differ for the 1960s 

 and 1980s data indicates that SBT growth rate changed 

 between these two periods. For the best-fit solutions, the 



