62 
Fishery Bulletin 106(1 ) 
the predicted AL. Given that the first year of growth is 
always assumed to equal the average increment, there 
is an increased chance that the overall variability of the 
residuals will be underestimated. However, in practice, 
bias appears to be small as long as the data available 
from greater than a single year overlap the available 
data from durations less than a year in terms of the 
initial shell lengths. 
Alternative growth models 
To provide a comparison with the inverse-logistic model 
both the von Bertalanffy (Fabens, 1965) and Gompertz 
curves (Troynikov et al., 1998) were fitted to the tagging 
increment data from southwest Tasmania: 
AL = {L^-L t )[l-e~ K * t \ (4) 
where L ^ - the asymptotic maximum size; and 
K = the von Bertalanffy growth rate coefficient; 
and 
with 
A L = L„ 
h. 
L 
exp(-gAf) 
~L t , 
(5) 
only on the seasonal changes in variability, thus reduc- 
ing the number of parameters to seven: 
°L t 
= Maxo L 
A t +C a sm{2n(t R -p))- 
C a sin{2n(t T -p )) 
(7) 
An alternative approach to implementing this structural 
change would be to set the L| 0 and L| 5 parameters in 
the denominator of Equation 2 to values much larger 
than the maximum observed initial size. This change in 
the denominator leads to the exponential term becoming 
insignificant so that the denominator contracts to one, 
the division thus has no noticeable effect, and Equation 
2 becomes equivalent to Equation 7. 
When only annual data are available, the seasonality 
terms could be ignored and thus a six-parameter model 
could be used. After the six-parameter model was fitted 
to real data, it became clear that the L| 5 value was 
often close to the maximum size of abalone found in 
Tasmania; therefore it was possible to generate a five- 
parameter model by replacing the L| 5 parameter with a 
constant 210 mm (the size of an abalone that was never 
tagged but sometimes found in nature). In addition, the 
L| 0 value was often close to the L'<£ 5 value. By replacing 
the former with the latter it was possible to generate a 
four-parameter model: 
where g - the growth rate parameter in the Gompertz 
equation. 
Likelihoods 
°L t = 
Maxo L x At 
Ln(19) — 9 a 
1 + e 
2F0-L- 
( 8 ) 
At each geographical site, normal likelihoods with non- 
constant variances (Eqs. 2 or 3), were used to fit the 
inverse-logistic model to the n available data points. The 
negative log-likelihood was minimized to determine the 
optimum parameter estimates: 
-veLL = - Ln 
L f = l 
AL-AL 
42 no L 
2a 2 
Lf 
( 6 ) 
The nonlinear solver in Excel 2003™ (Microsoft, Seattle, 
WA) was used to fit all models. 
Alternative model arrangements 
The full seasonal model has nine parameters, but alter- 
native model structures are possible that use fewer 
parameters. The alternative model structures suggested 
relate to the description of the variability about the 
expected curve. With the seasonal growth description, 
instead of using Equation 2 to describe the expected 
residual structure with the Tasmanian data, an accept- 
able alternative was to ignore the denominator and focus 
The three alternative annual models (4, 5, and 6- 
parameter models) were fitted to the available data 
from the three regional groups of sites from around 
southern Tasmania. A comparison of the relative fit of 
each model was made by using Akaike’s information 
criterion, AIC = -2 LL + 2k, where LL is the log-likeli- 
hood and k is the number of parameters. In addition, 
the Bayesian Information Criterion BIC = -2 LL + 
kLnin) was also used, where n is the total number of 
observations (Burnham and Anderson, 2002). For each 
of these statistics, the model with the smallest value 
is to be preferred (Quinn and Deriso, 1999). The AIC 
only includes the log-likelihood and the number of 
parameters, whereas the BIC also includes the natural 
log of the sample size. Where the sample size is greater 
than 7 [Ln(7.389)=2], the BIC penalizes the addition 
of extra parameters more than the AIC. Thus, with 
the sample sizes observed in the tagging data (Table 
1) the BIC would be expected to recommend more par- 
simonious models (those with fewer parameters) than 
would the AIC. 
In addition to comparing the AIC and BIC values for 
the different models, likelihood ratio tests were con- 
ducted to compare the alternative model fits by using 
different numbers of parameters (Quinn and Deriso, 
1999; Haddon, 2001). Given the log-likelihood for each 
model fit, the likelihood ratio test is 
