548 



Fishery Bulletin 100(3) 



The mean length-at-age 



lia,)-- 



L_ • 1 - exp 



-Ki 



a, -tn 



sin(2;r(a ,-w)/12) 



12 2;r sin(2;r(/o-«)/12) 



• (2) 



was modeled by a seasonally periodic von Bertalanffy 

 growth formula, generalized by the inclusion of an expo- 

 nent, r. Age (a,) was in integral units of months, with May, 

 the assumed date of birth at mean time of spawning of 

 South Australian King George whiting, being month 1. 

 Division by 12 in the mean length formula preserves the 

 usual units of ii" where age is in years. 



The seasonality function is sinusoidal, although more 

 complex seasonality functions can be postulated that al- 

 low asymmetry in growth through the year. Values of the 

 seasonality amplitude parameter ;/ > 1 imply decreasing 

 length in the yearly time of minimum growth (Pauly and 

 GaschiitzM. We therefore constrained u < 1, assuming that 

 no shrinking in length occurs. With sine, the phase pa- 

 rameter (ft)) gives the month of maximum growth, where 

 months of age 1, 13, 25, etc. denote the birth month. May. 



The likelihood standard deviation (a) was modeled as an 

 allometric function of mean length: 



Parameters were estimated by minimizing the negative 

 sum of log-likelihoods with the AD Model Builder estima- 

 tion software (Otter Research Ltd., Sidney, B.C., Canada): 



O 



-£ln(L, 



(5) 



cr(a, ): 



Ha, 



(3) 



Initial parameter values for all models tested were <q = 0, 

 u = 1,(0=9, S(, = 0.1, s, = 1, r = 1, with L„ = 480, K = 0.36 

 for females, and L^ = 420, K = 0.6 for males. 



Confidence bounds on parameters for each data set were 

 derived by assuming the standard asymptotic normal ap- 

 proximation. These were numerically calculated by AD 

 Model Builder routines as the diagonal elements of the 

 covariance matrix, the negative inverse of the Hessian 

 matrix at the likelihood maximum. 



Growth model choice 



The full, seasonal, generalized von Bertalanffy model des- 

 cribed above has eight parameters. A best-fitting model 

 with the minimum number of parameters was sought. 

 In cases where a model is overparameterized, high cor- 

 relations between parameters appear and the confidence 

 bounds widen. To test for a diagnosis of overparameter- 

 ization and then to correct it by fixing otherwise freely 

 estimated parameters, we applied the following algorithm 

 based on standard methods of model selection: 



This power function for standard deviation in terms of 

 mean length, applied by Francis (1988) to fitting tag- 

 recovery length increments, has the desired property that 

 once growth stops, the standard deviation in lengths-at- 

 age also ceases to change. Similarly, in winter months of 

 slowed (or zero) mean length increase, change in standard 

 deviation slows correspondingly. 



The left-truncated normal likelihood, which applies to 

 samples from commercial and recreational fishermen, 



exp 



aia) 



-t-oo 



f ^ 



J oia, 



(4) 



■exp - 



dl , if/, >LML, 



[LML, 



0, if/, < LML, 



postulates a probability cut-off to zero for landed samples 

 less than LML and a normal probability, integrating to 1, 

 for the range of legal lengths. LML is subscripted by the 

 fish sample data point (/) to indicate that LML is either 

 280 or 300 mm depending on the date of capture of the 

 fish. For research samples, for which all lengths could be 

 observed, the full untruncated normal likelihood (Eq. 1) 

 was used. 



1 Search for parameters that frequently show high cor- 

 relations with other parameters, have high variances, 

 or in estimation hit preset upper and lower bounds, 

 beyond which biologically unrealistic values are being 

 inferred. 



2 Set those parameters to fixed likely values. This 

 assumes a biologically likely value can be postulated. 

 If not, use a mean value among the range of estimates 

 obtained when the parameter is allowed to vary freely. 

 A parameter may also be selected if it varies little 

 among the range of data sets for which estimates are 

 obtained. 



3 Test for the change in negative log-likelihood to deter- 

 mine whether the reduced model is significantly less 

 well fitting by using chi-square likelihood ratios. 



4 Check that confidence intervals and correlations are 

 reduced. 



In addition, we seek model that converges reliably for 

 all data sets. Satisfactory convergence should not depend 

 on the initial parameter values chosen, but in some cases 

 a new set of initial values can be tried for any given non- 

 converging data set. Convergence can also depend on the 

 minimization algorithm or subroutine employed. However, 

 for purposes of model selection, failure to converge for 

 some data sets analyzed is an indication that the likeli- 

 hood surface for that model is less smooth and the true 

 global maximum may not always be obtained. Reliable 

 convergence is particularly desirable when 1) bootstrap- 

 ping, where nonconvergence leaves resampled data sets 



