Hampton. Growth parameters of Thunnus maccoyii from tagging data 



579 



recapture, and were at liberty for at least 250 days. 

 These criteria were satisfied by 1800 returns. The last 

 criterion was applied to eliminate the possible effect 

 that the tagging operation might have on length 

 growth and to minimize the biasing effect that seasonal 

 fluctuations in growth, if present, might have on 

 parameter estimation. 



Most fish were 50-80 cm long when tagged, with the 

 smallest 38 cm and the largest 104 cm. The range of 

 recapture lengths was 51-185cm, with most being in 

 the range 60-100 cm. The times at liberty for the 

 primary data set range from 250 days (the minimum 

 allowed) to approximately 11 years. 



Parameter estimation 



Model I: Standard von Bertalanffy model The 



form of the von Bertalanffy growth model appropriate 

 for fitting to tag-return data (indexed by i) is, as 

 described by Fabens (1965), 



minimizing 



LL = -In L = - ln(2Tro e 2 ) + 

 2 



I [dli-E(dli)] 2 



i = l 



2o e 2 



cSli = (L 0O -l j )(l 



Ki 



) + e f 



(1) 



This was accomplished for all the models described in 

 this paper, using the minimization subroutine MINIM 

 (programmed by D.E. Shaw, CSIRO Div. Math. Stat., 

 P.O. Box 218, Lindfield 2070, Aust.), which uses the 

 method of Nelder and Mead (1965). Equivalent sub- 

 routines are available in several commercially-available 

 software packages. 



Model 2: Kirkwood and Somers model Kirkwood 

 and Somers (1984) described a model which allowed for 

 individual variation in growth through an individually 

 variable L^ . Specifically, L^ was assumed to be nor- 

 mally distributed with mean /u L and variance ol 2 . 

 For given 1; and t ; , <5lj is a normally-distributed ran- 

 dom variable whose expected value is given by 



where dlj is the length increment, 1; is the release 

 length, t, is the time at liberty, and e; is a model error 

 term (or residual), all for the ith observation. The error 

 term ej is assumed to be a normally-distributed ran- 

 dom variable with an expected value of zero and 

 variance o e 2 . Thus, for given lj and tj, dlj is a norm- 

 ally-distributed random variable with an expected value 

 of(L„- 



E(dlj) = (ml -l,)(l-e- Kt i) 



(2) 



and variance by 



var(dli) = o L 2 (l-e 



-Kti\2 



(l-e- Kt and a variance of o e 2 . Estimates The negative log-likelihood function now becomes 



of L^, K, and o e 2 can be obtained by nonlinear or- 

 dinary least squares (as in Kirkwood and Somers 1984) 

 or by maximum likelihood (Kimura 1980). In the case 

 of model 1, either technique can be applied, since the 

 variance of dlj is assumed to be constant with increas- 

 ing tj. However, in the models that incorporate in- 

 dividual variability (see below), the variance of dlj 

 increases with increasing tj. This would require the 

 use of weighted least-squares if this approach was fol- 

 lowed. Since the weights would depend on the esti- 

 mated K, an iterative procedure would be necessary 

 to obtain the appropriate estimates. Therefore, the 

 maximum-likelihood method, which is far more 

 straightforward, is used to obtain parameter estimates 

 for this and the models that follow. 



For n observations of dlj and tj (i = 1 to n), the 

 likelihood function is 



L = | ] (2n o e 2 ) 2 exp 



i = l 



[dli- E(dlj)] 2 



2o 2 



and estimates of L^, K, and o e 2 are found by 



LL = £ In [2n var (dlj)] + [dlj - E(dlj)] 2 



i-i 



2 var (dlj) 



from which maximum likelihood estimates of ^ L , 

 o Loo 2 , and K may be obtained. 



Model 3: Kirkwood and Somers model with model 

 error Let us now assume that the variance of dl, is 

 comprised of components due to both the individual 

 variation in L^ and to a normally-distributed model 

 error, e ; , having a mean of zero and variance o e 2 . In 

 this case, E(dl ; ) is unchanged from Equation (2) and 

 the variance of dl; is given by 



var (dlO = o L 2 (l-e- Kt .) 2 + o t 2 



(4) 



Maximum-likelihood estimates of ju l , o L 2 , K, and 

 o e 2 can again be obtained by substituting the right 

 sides of Equations (2) and (4) into Equation (3). 



