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Fishery Bulletin 91(4), 1993 



estimate fractional age based on an assumed birth date; 

 therefore, all ages are integers. Maximum-likelihood 

 estimates (MLE) were obtained according to the proce- 

 dure described by Kimura (1980). The objective func- 

 tion ( 0) minimized to obtain the parameter estimates 

 was the negative log-likelihood of the VB model: 



„, ro ,, ik-[i.(i-^-v)]} ! 



0= Nlo&.(2na-) + is i (1) 



2 2a 2 



where: /, = fork length of individual i, 



t, = presumed age of individual i, 

 TV = number of individual measurements, 

 L,= estimated asymptotic fork length, 

 K = estimated growth coefficient, 

 t u = estimated hypothetical age at zero length, 

 a 2 = estimated variance associated with the 

 length-at-age. 



The objective function was minimized by using the 

 Quasi-Newton algorithm in the NONLIN statistical 

 module of the SYSTAT microcomputer program 

 (Wilkinson, 1989). Asymptotic standard errors and pa- 

 rameter correlations were obtained from the Hessian 

 matrix once the iteration process was complete (see 

 Wilkinson, 1989). As suggested by Kimura (1980), the 

 initial values supplied for the parameters were ob- 

 tained from the Walford plot (Ricker, 1975). 



Comparison of growth curves with tag-return data 



We used the growth curves fitted to the two data sets 

 and the observed length increments from the available 

 tag-return data to test our assumption that the length- 

 frequency modes represent year classes and vertebral- 

 rings represent annual features. It must be acknowl- 

 edged that there is no statistically correct means of 

 comparing the models and the length increment data, 

 because the growth curves derived from length fre- 

 quencies and vertebral-ring-counts are based on age- 

 length data. The only statistically correct predictions 

 possible from these growth models are predictions of 

 length (the dependent variable) from age (the inde- 

 pendent variable). However, the length increment data 

 can be related to the growth curves by first generating 

 predictions from the fitted growth models which are 

 then compared to the length increment data. This was 

 accomplished by predicting lengths-at-recapture by us- 

 ing the standard Fabens (1965) length increment 

 model: 



l r =1, + (L -l i ){l-e KM ) 



(2) 



where: /, = length at release of individual i, 

 A„= time at liberty of individual i, 

 l r , = estimated length-at-recapture of individual 



and L. and K are the VB parameters estimated from 

 the length-frequency or vertebral-ring-count data. As 

 noted above, this procedure is not correct in the strict 

 statistical sense because the growth models are not 

 used to predict length from age (see Francis, 1988). 

 However, for comparative purposes, the second method 

 should reveal any gross departures from the "annual 

 features" assumption applied in fitting the growth mod- 

 els. Some justification for this is presented later. 



Results 



Growth analysis based on length-frequencies 



The most parsimonious model structure for the alba- 

 core length-frequency data set included seasonal growth 

 and age-dependent standard deviation in length-at-age. 

 The incorporation of first length bias did not signifi- 

 cantly improve the fit. Parameter estimates and esti- 

 mates of the means and standard deviations of length- 

 at-age are given in Table 1. Note that we use the term 

 "relative age class" to denote that estimates of abso- 

 lute age are based on the assumptions that the length 

 modes represent annual cohorts and t =0. 



The seasonal growth phase estimate of 0.216 indi- 

 cates that fastest growth occurs in February, and slow- 

 est growth occurs six months later in August at the 

 end of the austral winter. The seasonal growth ampli- 

 tude estimate (0.949) is near the upper limit of 1.0, 

 indicating that growth is almost non-existent at that 

 time of the year. Standard deviation in length-at-age 

 increased progressively with age for the nine signifi- 

 cant age classes detected. 



The predicted aggregate length-frequency distribu- 

 tions fitted the observed distributions very well over 

 the entire range of sizes (Fig. 2), and the predicted 

 modes closely matched the actual modes in most 

 months. The predicted modal distribution pattern in- 

 dicates that there were usually four prominent age 

 classes in troll catch samples. 



Growth analysis based on vertebral-ring 

 counts 



The fork lengths of the albacore sampled ranged from 

 44 to 110cm (Table 2), which corresponds to the size 

 range of albacore caught in the surface and longline 



