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Fishery Bulletin 11 6(2) 
bounds of the 95% CIs of L, and K were respectively 
the 25 th and 975 th quantiles of the 1000-bootstrap-de¬ 
rived values of L„ and K. 
fitting lines represented increasing probability of ma¬ 
turity with fish length). Then, we estimated the age- 
specific probability of maturing (m): 
Age and length at 50% maturity Given that the ma¬ 
turity state is a binary variable and that probability 
of maturity tends to increase with length and age, we 
used logistic regression to estimate the age and length 
at which the probability of being mature is 50% (A 50 
and L 50 ). The equation that we used was 
logit(p) = In | j = P 0 + (3) 
where x = length or age; and 
Po and Pi are model coefficients. 
/Iso or L 50 are the solutions of Equation 3 for 
p- 0.5, that is, — Pc/Pi- 
We derived the 95% CIs of A 50 or L 50 by using a 
bootstrap method (Manly, 1997). We generated 1000 
bootstrapped data sets of equal sample size to those of 
the original data set and then fitted a logistic regres¬ 
sion to the bootstrapped samples to derive estimates of 
A 5 0 or L 50 . The lower and upper bounds of the 95% Cl 
of A 50 or L 50 were respectively the 25 th and 975 th quan¬ 
tiles of the 1000-bootstrap-derived values of A 50 or L 50 . 
Probabilistic maturation reaction norms The PMRN ap¬ 
proach involves estimating age- and length-specific 
probability of maturation with the use of a logistic re¬ 
gression (Heino et ah, 2002). Because newly mature 
T. japonicus cannot be distinguished easily from those 
previously mature, we followed the demographic ap¬ 
proach developed by Barot et al. (2004a, 2004b). This 
method is based on calculating change in length-specif¬ 
ic maturity over a time interval, here 1 year, aligned 
by the average length increment over the time interval 
(AL). Specifically, we used data for a given age, sex, 
and area to fit a logistic regression with maturity sta¬ 
tus as a response and length as a predictor, separately 
for each age group: 
logit(O t ) = In 
f-M 
U-oJ 
- Po+Pi^f 
(4) 
Equation 4 is equivalent to fitting Equation 3 to 
age-specific data. 
O t = the maturity ogive for age t; 
L t = length of fish at age t; and 
Po and Pi = the regression intercept and slope. 
We fitted such a logit function of O, for the ages in 
which both immature and mature individuals were 
present (he., £=1 and t= 2 of both sexes and areas). Be¬ 
cause of the relatively low number of age-specific sam¬ 
ples (e.g., no. = 19-58 per age-sex-location group), the 
logistic regression did not provide a good fit for some 
groups (i.e., age 1 of both sexes at Tsukuan, P=0.05- 
0.08). Nonetheless, we continued further analysis be¬ 
cause these fits were reasonable on the basis of visual 
inspection (i.e., we accepted the model fits when the 
m(L t ) = 
O t (L t )-O t ^(L t -AL) 
1- O t _ 1 (L t - AL) 
(5) 
Here the probability of maturing (m) is estimated as 
the fraction of immature fish at age t- 1 that grew in 
length AL and reached maturity at age t. An underly¬ 
ing assumption of Equation 5 is that immature and 
mature fish of the same ages have the same growth 
and survival rates (Barot et al., 2004a, 2004b). Our 
data indicate that the lengths at ages 0, 1, and 2, re¬ 
spectively, did not vary between immature and mature 
fish of either sex in Kengfang but that the lengths var¬ 
ied between immature and mature male fish at ages 
0, 1, and 2 in Tsukuan (Suppl. Table 3 (online only); this 
finding also is based on f-test results: P=0.01, 0.02, and 
0.003 for ages 0 [re-20], 1 [re=49], and 2 [re=32], respec¬ 
tively). The observed differences in length at age be¬ 
tween immature and mature fish indicate that our data 
violate this assumption. However, reports of previous 
studies have suggested that the PMRN approach was 
relatively robust even when this assumption is mod¬ 
estly violated (Barot et al., 2004a, 2004b). 
The trajectory of probability of maturing generally 
indicates a sigmoid curve with increasing lengths. 
Therefore, we fitted a logistic regression with m as 
response and length as a predictor and estimated the 
length at m- 0.5 (i.e., Lp 50yt ): 
logit(m(L t )) = p 0 + [3 1 xL t , (6) 
with Lp 50jt obtained as the solution of Equation 8 for 
m(L t )-0.5. 
We estimated the 95% CIs of Lp 50it using the boot¬ 
strap method. We generated 1000 bootstrapped data sets 
of age t and age t- 1 (each with equal sample size of the 
original data set) and used these data sets to derive 1000 
estimates of Lp 50>t . The lower and upper bounds of the 
95% Cl of Lp 50yt were the 25 th and 975 th quantiles of the 
1000-bootstrap-simulated values of Lp 50jt . 
Randomization tests Because parametric tests for com¬ 
paring the von Bertalanffy growth coefficients (i.e., L TO 
and K) and maturation indices (i.e., A 50 , L 50 , and PM- 
RNs) between the 2 sampling areas are cumbersome 
or unavailable, we used randomization tests (Manly, 
1997; see also Barot et al., 2004b). Specifically, we per¬ 
muted the data column of “area” and evaluated the be- 
tween-area differences in the sex-specific estimates for 
each of these growth and maturation indices with 999 
replicates. The sorted sequence of the 999 between- 
area differences of estimates approximated the range 
of all possible values of between-area differences for 
estimates under the null hypothesis (i.e., with no dif¬ 
ferences between the areas). We then evaluated the 
probability of the observed between-area difference 
for a sex-specific estimate (i.e., P-values) as 1-P(X<D), 
where D is the observed between-area difference. We 
