Hampton, Growth parameters of Thunnus maccoyn from tagging data 



581 



to be normally distributed, with 



E(l 2i ) = 

 K.-E(l,)] 



and 



1 + 



°K 2 tj 





+ E(li) (7) 



where o el 2 and o e2 2 are independent model error vari- 

 ances. Maximum-likelihood estimates of p L , o L 2 , K, 

 to, o e i 2 , and o e2 2 could, in theory, be found by "mini- 

 mizing 



nl 



LL = X 



i=l 



In [2n var (dlj)] [dl ; - E (dl;)] 2 



2 var (dlj) 



var (l 2i ) = 

 C : o L J + C 2 [ MLoo -E(l;)] 2 + o e 2 + 0m £ E(e- 2 Kti), (8) 



where 



Mk 



E(e- 2Kt .) = 



1 + 



2o K 2 tj 



Given the estimate of o m 2 derived for model 4, max- 

 imum-ikelihood estimates of ml , o L 2 , mk. o K 2 , and o e 2 

 can be obtained as before. 



Estimation of t An estimate of t is required for 

 many applications of the von Bertalanffy model, but 

 this parameter cannot be estimated from tag-return 

 data alone. To estimate t , one or more observations 

 of age-at-length are required. Kirkwood (1983) de- 

 scribed a maximum-likelihood method for determining 

 t , along with L^ and K, if supplementary age-length 

 data are available in addition to tag-return data. Such 

 data can easily be accommodated in the models de- 

 scribed above. Consider the case where nl tag returns 

 (dl; and dtj) and n2 age-length observations (lj and tj) 

 are available. We now have two random variables, dlj 

 and lj, which have normal probability density func- 

 tions conditioned on lj (release length) and dt,, and tj, 

 respectively. Their expected values, for example in the 

 case of model 3, are given by 



E(dlj) = (ml -liXl-e-* 5 *!) 



E(lj) = MLoo (i-e-K(tj-t )) 

 and their variances by 



var(dl,) = o L 2 (l- e - Kdt i) 2 + 0el 2 



var(lj) = OLjCL-e-KCj-to)) 2 + o e , 2 



+ | In [2n var (lj)] + [lj-EQj)]' 

 2 2 var (lj) ' 



j=i 



However, the estimation of six parameters (or seven 

 in the case of models 6 and 7) may prove unrealistic 

 in many cases. In order to obtain an approximate 

 estimate of t , I have employed a simpler procedure. 

 It involves fixing the growth parameters estimated 

 from tag-return data and estimating t by minimizing 

 only the second summation in Equation (9), using the 

 same approximate age-length data as Kirkwood (1983). 

 I did not attempt to estimate simultaneously all 

 parameters using the combined length-increment and 

 age-length data because of the approximate nature of 

 the latter data. 



Model selection 



An important part of this study was to select the most 

 appropriate growth model for use in southern bluefin 

 tuna stock assessments. Although L provides a means 

 of comparing the goodness of fit of the various growth 

 models with the tagging data, it is not immediately 

 clear whether the more complex models result in a 

 statistically-significant improvement in fit to the data. 

 Likelihood ratio tests (Mendenhall and Scheaffer 1973, 

 Kendall and Stuart 1979, Kimura 1980) can be used to 

 address this question. 



Likelihood ratio tests Let A be defined by 



, L(* ) 



L(+ a ) 



where L(<t> () ) and L(<)> a ) are the maximum-likelihood 

 function values under the null hypothesis (the simple 

 model is correct) and the alternative hypothesis (the 

 more complex model is correct), <j> is the set of max- 

 imum-likelihood estimates of r parameters under the 

 null hypothesis, and <f> a is the set of maximum- 

 likelihood estimates of r + s parameters under the alter- 

 native hypothesis. Under certain regularity assump- 



