Hampton: Growth parameters of Thunnus maccoyii from tagging data 



585 



Table 3 



Values of the 97.5 percentile of the L^ distribu- 

 tions derived for models 2-7. These values may 

 be compared with an observed maximum size in 

 the catch of 225 cm. 



Model no. 



97.5 percentile of 

 the L„ distribution (cm) 



203.4 

 216.0 

 216.5 

 194.8 

 214.4 

 215.2 



(models 5, 6, and 7). An approximate consistency test 

 for derived L m distributions is to compare an upper 

 percentile (e.g., 97.5) of the distribution with the 

 observed maximum size in a large sample of the catch 

 (Kirkwood and Somers 1984). The largest southern 

 bluefin ever measured by Japanese researchers during 

 many years of length-frequency sampling is 225 cm 

 (Yukinawa 1970, Shingu 1978). This may be compared 

 with the derived 97.5 percentile of the L^ distributions 

 for models 2-7 (Table 3). While by no means a definitive 

 test, the comparisons suggest that the L^ distributions 

 derived for models 3, 4, 6, and 7 (models that include 

 model error) are more consistent with the observed 

 maximum length than the distributions derived for 

 models 2 and 5 (models that do not include model error). 

 The estimate of o K 2 for model 5 fell substantially 

 with the addition of model error (models 6 and 7). The 

 value of o K 2 for model 7 represents a coefficient of 

 variation of approximately 7%. It should be pointed out 

 here that the log-likelihood surface was very flat with 

 respect to both o L 2 and o K 2 , i.e., relatively large 

 changes in either parameter resulted in only very small 

 changes in LL. This matter is explored further using 

 simulation techniques. 



Model selection 



The minimum negative log-likelihood function values 

 given in Table 1 are indicators of the goodness of fit 

 of the models to the data. Model 2 (3 parameters) pro- 

 vided a substantially poorer fit to the data than model 

 1 (3 parameters) and therefore need not be tested. Also, 

 the inclusion of release-length-measurement error 

 resulted in slightly worse fits to the data, eliminating 

 models 4 and 7 from further consideration. The only 

 likelihood ratio tests that are required are: (1) the null 

 hypothesis (H ) that model 1 is the correct model 

 against the alternative that model 3 (4 parameters) is 



correct; (2) the H that model 1 is the correct model 

 against the alternative that model 6 (5 parameters) is 

 correct; and (3) the H that model 3 is the correct 

 model against the alternative that model 6 is correct. 

 H was rejected for both tests (1) and (2) (P< 0.001), 

 indicating that the more complex models 3 and 6 pro- 

 vide significantly better fits to the data than model 1. 

 For test (3), H was accepted (P>0.05), suggesting 

 that the additional complexity of the extra parameter 

 (ok 2 ) in model 6 did not result in a significantly im- 

 proved fit over model 3. On this basis, model 3 was 

 adopted as the most appropriate model for these data. 



Simulations 



Assessment of model performance The results of 

 analyses of 100 simulated data sets, produced assum- 

 ing independence of L^j and Kj, are given in Table 4. 

 These suggest that all of the models, with the possible 

 exception of model 2, provide unbiased estimates of 

 fi L and fi K . This is somewhat surprising in view of 

 the wide range of estimates of these parameters ob- 

 tained from analyzing real data with the same models 

 (Table 1). In particular, one might have expected, on 

 the basis of analyses of the real data, that models 2 and 

 5 would have given biased estimates of /u L and n% 

 for the simulated data also. The estimates of these 

 parameters obtained from the real data, using the 

 above models, even lie outside the approximate 95% 

 confidence bounds calculated from the simulated data. 

 A possible explanation for this is that the simulated 

 data do not contain all the growth-related features of 

 the real data. Such unaccounted-for structure, if it 

 affected the performance of the models differently, 

 could produce such inconsistencies. This, in fact, was 

 observed in the case of the apparently biased estimates 

 given by model 1 for the real data. 



While the simulations indicate that estimates of the 

 mean parameters are relatively unbiased and precise, 

 this is not the case for estimates of their variances. In 

 particular, it is clear that reliable estimates of o K 2 can- 

 not be obtained from this data set, since estimates from 

 the simulated data ranged from practically zero to 25% 

 (expressed as the coefficient of variation). This could 

 be due in part to the loss of information on K-variability 

 incurred because of the necessary exclusion from the 

 analyses of fish at liberty for less than 250 days. 



The mean values of o L 2 are reasonably consistent 

 with the estimates obtained from the real data. The 

 estimates from models 2, 3, and 4 are positively biased, 

 while those from model 5 are negatively biased. The 

 estimates from model 7 are unbiased, but have a coef- 

 ficient of variation of 45%. The estimates of o e 2 are 

 somewhat less than those obtained from the real data, 

 except in the case of model 1. 



