Hearn and Polachek: Long-term growth rate changes in Thunnus maccoyii 



63 



also be noted that the two-phase VBG model with common 

 L^ (row 2, Table lA). was very similar to the common k 

 parameterization, reflecting the high correlation between 

 L„ and k in the VBG models. For the 1960s, the continu- 

 ous two-phase VBG model was rejected, P< 0.005 (row 5, 

 Table lA). 



For the 1980s data, the best-fit model based on the AIC 

 values was for the two-phase VBG model with a common 

 value for the k parameter in both phases (row .3, Table IB). 

 The estimate of the size at which the change between the 

 two phases occurs was 85 cm (compared to the estimate 

 of 74 cm for the 1960s data). As with the results of the 

 1960s data, the common-/^ model, common- L model, and 

 the full two-phase model yielded very similar values for 

 both the likelihood and parameter estimates in the second 

 phase, but not for those in the first phase. This similarity 

 reflects the high correlation between k and L in the VBG 

 model, so that over the limited size range below L* nearly 

 identical growth rates can be achieved in the common-^ 

 model by decreasing the value of L^j. For the 1980s (as 

 with the 1960s), the continuous two-phase VBG model (7 

 parameters), was rejected, P< 0.005 (row 5, Table IB). 



For both the 1960s and 1980s data, the two-phase model 

 provided a substantially and significantly better fit to 



the tag return data than a simple VBG model. This can 

 be seen in Table 1 (A and B) by comparing the negative 

 log-likelihood and AIC values for the simple VBG model 

 (row 4) with any that include a two-phase component, 

 particularly the continuous rate two-phase VBG model. 

 We also fitted a smooth Richards' (1959) growth model (a 

 generalization of the VBG model) to the data, which fitted 

 better than the simple VBG model, but worse than the 

 two-phase VBG models. 



Note, however, that the log-ratio test and AIC criterion 

 may not be fully applicable for testing the differences 

 between the simple and two-phase VBG models because 

 the simple VBG model can arise in more than one way 

 as a submodel of the two-phase model (e.g. with common 

 L . and k parameters or from L* equaling zero or infinity) 

 (Davies, 1977, 1987). Nevertheless, the large magnitude 

 of the differences in the log-likelihood values indicates a 

 significance difference. 



For the 1960s data, it should be noted that the scientist 

 measurement error (crj was estimated to be essentially 

 zero when it was included as an explicit term in several of 

 the models. In these cases, we refitted the models exclud- 

 ing this parameter. Common sense dictates that measure- 

 ment errors would not be zero. The most informative data 



