Haddon et al.: Using an inverse-logistic model to describe growth increments of Haliotis rubra 
63 
Xdf ~ -2 (LL r - LL f 
(9) 
where LL, 
LL, 
- the log-likelihood for 
the model with fewest 
parameters; and 
- the log-likelihood of the 
model with most param- 
eters. 
In the chi-squared statistic, x 2 df> df is 
the difference in number of parameters 
(in this case comparing 4 with 6 and 
5 with 6 means df takes the value of 
either 2 or 1, respectively). 
Defining the growth transition matrix 
The data from all sites includes 
instances of negative increments (Fig. 2) 
and hence it is not surprising when the 
predicted increments also feature nega- 
tive values. This implies that abalone 
can decrease in size during a time step. 
However, abalone tend not to exhibit 
negative growth; instead these negative 
increments are assumed to be the result 
of measurement error. The transition 
probabilities are simply the cumulative 
normal distribution to the upper size limit of each size 
class minus the cumulative normal distribution to the 
lower size limit of each size class in turn: 
65 75 85 95 105 115 125 135 145 155 165 175 185 
Initial shell length (mm) 
Figure 2 
Plot of growth increment (mm) after approximately one year against ini- 
tial length (mm) for blacklip abalone ( Haliotis rubra) from the southwest 
region of Tasmania. The vertical dashed lines represent the boundaries of 
the 10-mm size classes and the curved line and black squares represent 
the trend and mean growth increments, respectively, for each class of 
initial sizes. Mean values are only shown for initial size classes repre- 
senting more than one observation. Negative increments were included 
in the mean estimates. 
G -r | 
2 
U-L. 
LJ) 
j2no J L . 
dL 
L. = L, 
OJ , 
Li 
L, 
= the standard deviation of the normal distri- 
bution of growth increments for the initial 
size class j. 
= the expected average final size for initial size 
class j, which equals + A L ( •, where A L ; • 
is the average expected growth increment 
for initial size class j. 
Summing the smallest size class to -oo and the largest 
size class to +°° effectively makes both of these size 
classes plus-groups that ensure that the transition prob- 
abilities for all n size classes sum to one. 
L . + LW 
G, ' J J 
Li-L. 
hj) 
LW^< 
G, - J J 
rl- 
Lj-L. 
, 1 i, 
dL L Min < L- < L Max , ( 10 ) 
2 al 
L i - 2 
LW 
dL L, = L. 
where G- • = the transition probability of an abalone 
growing from size class j into size class 
i; 
L ( = the mid-size of size class i; 
LW - the size class width; and 
Length at age 
Because of the potentially indeterminate nature of the 
growth description, there is no analytical version of the 
growth equation that can provide a length for a given 
age. Instead, growth needs to be simulated to estimate 
length at age. That is, an initial length is assumed and 
then the predicted growth increment in a given time 
interval (seasonally short or annual) is estimated; this 
is then added to the initial length and the process is 
repeated to generate a predicted length at age. The 
simulated growth increments may include stochasticity 
(guided by the non-constant variance with initial length) 
and lead to a scatter of predicted sizes. Alternatively, 
growth can be simulated by using the growth transi- 
tion matrix from the model fit. For annual growth only 
one transition matrix is required, however, to describe 
seasonal growth there would need to be an array of 
