62 



Fishery Bulletin 101(1) 



SI, = E\a] + e, + e„ 



(7) 



where f, is due to measurement error in the observed 

 growth increment (i.e. the combined effect of any errors 

 in measuring the lengths at the time of release and recap- 

 ture) and e, is due to process or model error The latter may 

 be a function of /j, St, SI, and the model parameters. 



For the measurement error component, we allowed for 

 different variances, depending upon whether the recap- 

 tured fish was measured by an independent and scientifi- 

 cally trained individual or by a fisherman. Scientifically 

 trained individuals (i.e. scientists) included fishery observ- 

 ers, port samplers, and CSIRO staff We assumed that the 

 measurement error was normally distributed, with mean 

 zero and variance o^-, where .v is one of /"or s for recaptured 

 fish measured by fishermen or scientists. 



The choice of the functional form for the process error 

 in growth models is a complex issue. One approach has 

 been to consider that process error stems from variability 

 among individuals in the expected value of the growth 

 parameters (e.g. Sainsbury, 1980; Hampton, 1991; Wang 

 et al., 1995). This approach in the case of the two-phase 

 VBG model would result in many potential structures for 

 the process error component because there could be indi- 

 vidual variability in the expected value of any single or 

 possible combination of parameters (of which there are 25 

 combinations). There is little theoretical basis for deciding 

 which of these 25 combinations to use. As an alternative, 

 we selected a more empirical approach. A function that 

 increases with longer times at liberty seemed appropriate, 

 and was also consistent with preliminary analyses. We ex- 

 plored both linear and quadratic functional relationships 

 between the times at liberty and the process error com- 

 ponent. The quadratic term was found to be insignificant, 

 and therefore we chose to report only results for a simple 

 linear functional relationship, namely 0;;^St. Hence the cor- 

 responding variance of the expected gi'owth increment of 

 fish ( is V( ^^ ) = a^- + a,~ St^. It should be noted that without 

 independent data on measurement error any constant 

 component in process error would be totally confounded 

 with the measurement error term in the model. Therefore, 

 a^ should be considered as a combined measurement and 

 process error term. Both a^ and (T„, were estimated empiri- 

 cally by maximum likelihood tag increment data. 



Assuming a Gaussian error distribution, the likelihood 

 function is 



=ni--,.^.p(.it«!j 



(8) 



The estimates of the parameters are found by minimizing 



■ln(L) = ^^ 



,„,p.„.,„.it«a! 



(9) 



D.E. Shaw, CSIRO Div Maths, and Stats.), which uses the 

 Nelder and Mead (1965) method. 



Model selection The estimation of the full two-phase 

 VBG models across both tagging periods contains 16 

 parameters (five model parameters plus three variance 

 parameters for each time period). We examined a variety 

 of alternative hypotheses to test whether the number 

 of parameters could be reduced by eliminating some or 

 equating them. For the model parameters, we considered 

 whether the L„ or k terms were equal either between time 

 periods or between the first and second phases within a 

 time period. We also considered the simple VBG model, for 

 which L* doesn't exist. 



For the L '■ parameter, we considered whether the esti- 

 mates were different between the two time periods. We 

 also examined models in which the value of L* was de- 

 termined by assuming that the expected growth rate for a 

 fish of length L* was equal for both growth phases (i.e. by 

 assuming that the changes in growth rates as a function of 

 length is a continuous function). Under this assumption 





(10) 



The minimum value was obtained for all models by using 

 the minimizing subroutine MINIMD (programmed by 



This model is referred to as the continuous rate two- 

 phase model in the "Results" section. However, this model 

 is not smooth because it has a discontinuity in the deriva- 

 tive of the growth rate atL*. For the variance parameters, 

 we considered whether any of them could be eliminated 

 and also whether a^ = Of. We used the log-ratio test and 

 AIC criterion (Akaike, 1974) to identify the most parsimo- 

 nious model. 



Results 



Best fits to the 1960s and 1980s data 



Table 1 contains the maximum likelihood solutions for 

 various assumptions when fitting growth models to either 

 the 1960s or 1980s tag-return data separately. Using the 

 AIC criteria, we found the best-fit model for the 1960s tag- 

 return data was one with linear growth in the first phase 

 and with the change between the two phases at approxi- 

 mately 74 cm (row 1, Table lA). The fit to this model com- 

 pared with all other parameter combinations yielded both 

 the lowest AIC and negative log-likelihood values. The fit, 

 however, was only marginally better then the fit (row 3, 

 Table lA) to the two-phase VBG curve with common k 

 parameters (e.g. where the difference in the negative log- 

 likelihood values is 1.21). Except for the first phase growth 

 parameters, the estimates for the other parameters are 

 nearly identical between these two models. This similar- 

 ity reflects the fact that growth is nearly linear over the 

 initial part of a VBG curve. Thus, by having a relatively 

 high L. I (271 cm), essentially similar gj-owth rates can be 

 achieved up through the 74 cm size range when ^j = ^-2, as 

 compared with linear growth in the first phase. It should 



