Ballagh et al. : Methods for determining length-at-age for two large scombrids 
93 
ratio tests on pairs of individual methods. A Bonferroni 
adjustment was used for the multiple comparisons to 
account for inflation in the probability of a type-I error 
by adjusting the significance level: 
where a = significance level; 
a Adj = adjusted significance level; and 
n = number of multiple comparisons. 
Selectivity effects 
Mean back-calculated length-at-age from all annuli was 
compared to mean back-calculated length-at-age from 
the last annulus to infer if there were any selectivity 
effects that had biased the sample of each species, 
or if Lee’s phenomenon (Ricker, 1969) was present. 
Lee’s phenomenon, whereby lengths at early ages back- 
calculated from younger fish are greater than lengths 
at the same age estimated from older fish, has been 
shown to bias estimates of growth based on back-cal- 
culation of length-at-age by using all annuli (Vaughan 
and Burton, 1994). This bias can be a result of large 
otoliths in slow growing fish confounding the relation- 
ship between length and otolith size and resulting 
in relatively small back-calculated length-at-age for 
younger ages (Campana, 1990). However, it can also 
result from selectivity bias in the sample whereby the 
faster younger fish are disproportionately selected 
for by the fishing gears (Lucena and O’Brien, 2001; 
Ballagh et al., 2006) or where size-selective mortality 
has removed fast growing individuals from older age 
classes, thereby biasing the population demographics 
(Fossen et al., 1999). Differences in the mean back- 
calculated lengths-at-age from all annuli and the last 
annulus can therefore be used to infer any biases such 
as selectivity on sampling (Vaughan and Burton, 1994). 
Analysis of variance (ANOVA) and Student’s t-test were 
used to compare mean back-calculated length-at-age 
from all annuli to mean back-calculated length-at-age 
from the last annulus only. 
Substitution of data 
Back-calculated lengths-at-age from all annuli were 
compared to observed lengths-at-age by using ANOVA 
and t-tests. Comparisons were made to determine the 
ages for which there were no differences between back- 
calculated and observed length-at-age to test the hypoth- 
esis that supplementing back-calculated length-at-age 
with observed or adjusted length-at-age can provide 
estimates of growth parameters that are more appro- 
priate representations of the population (Ballagh et ah, 
2006). Observed and adjusted length-at-age data were 
substituted with back-calculated length-at-age data 
(BC-all and BC-last) for the first few consecutive ages 
that were significantly different from observed length- 
at-age. This procedure resulted in several combinations 
of data from different methods for determining length- 
at-age (back-calculated, observed, and adjusted), and is 
hereafter referred to as substitution. 
VBGF curves were fitted to the substituted length-at- 
age data sets by using nonlinear regression. Likelihood 
ratio tests were performed to test for differences in 
growth among the different substituted combinations 
of length-at-age data. Where differences were found, 
multiple comparisons by using likelihood ratio tests 
with Bonferroni adjustments (Eq. 4) were performed on 
pairs of substituted data combinations. 
Results 
Growth 
A high degree of variability was observed among the 
VBGF parameter estimates from the different methods 
for estimating length-at-age for both species and sexes 
(Table 2). The two back-calculation methods consistently 
yielded lower estimates of and higher estimates of 
K and t 0 than the observed and adjusted methods. Less 
error was also observed in the estimates of and t 0 for 
the back-calculated methods than for the observed and 
adjusted methods. 
Comparison of the VBGF growth estimates by using 
overall likelihood ratio tests revealed significant differ- 
ences in growth estimates among the methods for both 
species and sexes (S. commerson females: ^ 2 = 678.9, 
df=12, P<0.0001, males: * 2 =704.4, df=12, P<0.0001; 
S. semifasciatus females: ;^ 2 =849.4, df=12, P<0.0001, 
males: ^ 2 =913.1, df =12, PcO.0001). Multiple compari- 
sons revealed some consistent patterns in growth esti- 
mates among methods (Table 3). Significant differences 
were observed between the back-calculation methods 
(BC-all, BC-last) and the adjusted and observed meth- 
ods (Adj-cat, Adj-frm, Obsv), and no significant dif- 
ferences were found between the two back-calculation 
methods for both species. There were no significant 
differences between the two adjusted methods for both 
sexes of S. cornmerson and female S. semifasciatus , 
whereas the two adjusted methods differed significantly 
for male S. semifasciatus. No significant differences 
were found between the adjusted and observed methods 
for both species, with the exception of the formula ad- 
justed and observed methods for female S. commerson. 
Comparison of VBGF parameter estimates by using 
likelihood ratio tests revealed further differences 
(P<0.05, df=l) between the methods that differed in 
the overall likelihood ratio test comparisons. For S. 
commerson, all VBGF parameters differed significantly 
between individual methods with the exception of the 
formula adjusted and observed methods for females, 
for which none of the VBGF parameters differed sig- 
nificantly. For S. semifasciatus, significant differences 
were observed for all VBGF parameters in comparisons 
between the back-calculation methods and the adjusted 
and observed methods. No significant difference was 
observed between any of the VBGF parameter esti- 
mates for comparisons between observed and formula- 
