McGarvey and Fowler: Seasonal growth of Sillaginodes punctata 



549 



unestimated, and 2) for integration of growth into overall 

 fishery stock assessment estimators, where finding more 

 appropriate initial values for the growth parameters may 

 be nontrivial, and the cause of nonconvergence of the 

 overall estimation may not be easily traced to the growth 

 submodel. 



In addition to reduced versions of the model presented 

 above, we fitted a related model proposed by Akamine 

 (1993), which incorporates into the Richards (1959) 

 growth curve a sinusoidal seasonality function like that 

 of Equation 2. 



Weight-at-length 



Mean (corrected) weight versus total length was modeled 

 by an allometric relationship: 



m,) = ail' 



(6) 



A normal likelihood was again used. The standard devia- 

 tion of the likelihood (i.e. of the fitted spread of observed 

 weights about the mean given in Eq. 6) was assumed to 

 vary linearly with length: 



^Jl,'l=(^uO+^,J>' 



(7) 



Parameter confidence bounds were estimated by a boot- 

 strap of 1000 runs. 



Results 



Growth model choice 



The generalized von Bertalanffy model (Eq. 2, abbreviated 

 as gVB) gave the best fit with five of six data sets (Table 2). 

 However for both gVB and the other 8-parameter model, 

 Akamine-Richards (AR), correlations among parameters 

 were unacceptably frequent and high (Table 2). Other 

 evidence of overparameterization included frequent 

 occurrence of wide confidence bounds (SE >209c. Table 2). 

 High correlations and parameter uncertainties were most 

 widespread for Sg and s,. The exponent, r. and seasonality 

 amplitude, u. also occurred frequently with high uncer- 

 tainty, and tf^ and r often were found in high correlations; 

 Iq hit both upper (7.99) and lower (-30) confidence bounds 

 for some models with west coast data sets. As with stan- 

 dard von Bertalanffy models, ?„ quantifies the age (here, in 

 months) at which length extrapolates to 0. The occurrence 

 of ;/ estimates hitting their upper bound of 1 is not due 

 to model overparameterization but reflects the biological 

 modeling decision to exclude shrinking. 



The Akamine-Richards ( AR) model varied widely among 

 data sets in its relative closeness of fit, failed to converge 

 for West Coast females, and did not yield a positive defi- 

 nite hessian for Gulf St. Vincent females. It was less well 

 fitting than the gVB for all but one data set. The gVB 

 model was therefore chosen as the better 8-parameter 

 model, and subsequent reduced models were based on it 

 in preference to AR. 



From indicators summarized above, four parameters for 

 defining overall mean length K, L^, t„ and r, appear to be 

 too many. Two obvious candidates for fixing to constant 

 values were t^ and r. These occurred frequently in indica- 

 tors of model overparameterization. Both have intuitive 

 biological default values at which they can be fixed, and 

 1 respectively. Thus two 7-parameter models were run, 

 with /„ and r fixed, the latter (with r=l) reducing to a 

 seasonal von Bertalanffy curve (Somers, 1988; Hoenig and 

 Hanumara, 1990). 



The fit with 1^ = fixed was not significantly different 

 from the full 8-parameter gVB for three of six data sets. 

 This fit was tested by chi-square likelihood ratios at 95% 

 confidence, indicated (Table 2) by successive likelihood-ra- 

 tios of 1 .92 or less, for comparing fits of hierarchical models 

 differing by one in number of freely estimated parameters 

 (Rice, 1995). The regular seasonal von Bertalanffy model 

 (r=l, abbreviated as "reg VB") fitted less closely than gVB 

 with t^1 = for all six data sets; significantly worse for all 

 Spencer Gulf and West Coast data sets. In addition, not 

 shown in Table 2, the estimates of tg were frequently far 

 from the realistic biological range near 0, often estimated 

 at 10 to 20 months above or below. The frequency and 

 magnitude of high correlations were substantially reduced 

 with both 7-parameter models. Thus we choose to fix t^ = 

 and let r freely vary. 



The occurrence of very high correlation between Sf, and 

 Sj for all data sets and models (Table 2) suggested fixing 

 one of them, in particular, the exponent, Sj. We set Sj = 0.3 

 which fell near the average among the range of estimated 

 values. The outcome was that t^ (now free) hit its bound 

 in three of six cases, thus exacerbating that pathology. 

 When both ^q = and s^ = 0.3 were fixed, the fits were uni- 

 formly poor (Table 2). Thus, the high correlation between 

 the two allometric standard deviation parameters met 

 with no obvious solution. However, these posed no wider 

 problem because these parameters did not interact with 

 those describing the mean length at age and the correla- 

 tion between pairs of allometric parameters (such as a 

 and p in the weight-length model, Eq. 6) is common and 

 often unavoidable. Because a significantly better fit for the 

 length-at-age standard deviation is sought, we let both Sq 

 and Sj vary freely. 



The gVB with ^q = fixed provided an optimal trade-off 

 in reduced overparameterization and good fit: it achieved 

 the objective of substantially reduced interparameter 

 correlations without significantly worse fits in three of 

 six cases. The remaining three data sets (Spencer Gulf 

 females and males, West Coast females) had yielded un- 

 realistic estimates of tg in full model gVB (of-17.9, -21.7, 

 and the upper bound of 7.99 respectively). This model was 

 therefore chosen and results from it presented below. 



Growth: length-at-age 



Estimated length at age showed seasonal periodic trends 

 for the three regions and two sexes (Figs. 2-4). Estimates 

 of;/ (seasonality amplitude) were constrained at the maxi- 

 mum allowed value of u = 1 for three of six data sets ( Table 3 ). 

 The peak month of maximum growth occurred in mid- 



