18 



Fishery Bulletin 102(1) 



relationship between dRIRL and dRIRS, i.e. Equation 7 

 or 8. 



Confidence intervals To evaluate uncertainties of coef- 

 ficient values and model selection, we estimated the 95 r /f 

 confidence intervals of all coefficients — i.e. a , b , s max , a-, 

 ji k , yj, and y 2 — based on profile likelihood. For example, the 



95% confidence interval of a,, — a, 



-was estimated as an 



interval that suffices in the following equation: 



2\ max log,. L(a ,b , s max , a , , ft , y 1 , y, ) 



max log,. U d , 4, s mBX , a Jt ft, y u y 2 



K=a 096 )}<^(0.05), 



(11) 



where .v^lO.OS) = value of a chi-squared distribution at an 

 upper probability of 0.05 with one degree 

 of freedom, i.e. 3.84. 



The characteristics of the interval are explained by 

 Burnham and Anderson ( 1998). 



We used Microsoft Excel (Microsoft Corp., Redmond, WA) 

 as the analysis platform, and Solver (Microsoft Corp., Red- 

 mond, WA) as the nonlinear optimization tool. 



Model selection 



We used three procedures for model selection to achieve 

 the best model. First, we constructed an a priori set of base 

 models based on biological variables; then we selected the 

 best base model. Fixation of the base model drastically 

 decreases possible candidate models to be tested. To test all 

 possible combinations of independent variables and model 

 forms is quite impractical. Second, we excluded insignificant 

 factors from the best base model. Third, we checked the sig- 

 nificance of environmental factors that were not included in 

 the base models. If one was significant, we included it in the 

 best base model. All of these procedures were performed by 

 AIC. The construction of the a priori set of candidate models 

 is partially subjective, but it is an important part of the 

 model construction (Burnham and Anderson, 1998). 



Seasonal growth in bivalves is influenced by water 

 temperature and food supply (Bayne and Newell, 1983). 

 The growth rate of Corbicula fluminea changes with age 

 (McMahon, 1983). Therefore, we constructed base models 

 combining water temperature, water fluorescence, and 

 categorical variables indicating age for the independent 

 variables of Equation 6. We tested two types of categoriza- 

 tion of age. The first segregates ages based on real age, i.e. 

 two categories: 0+ or 1+. The second segregates ages in rela- 

 tion to winter, i.e. three categories: before the first winter, 

 from the first to the second winter, and after the second 

 winter. For the real-age categorization, age was segregated 

 based on 1 September, because the spawning season was 

 in August 1997. For the winter-base age categorization, we 

 segregated ages based on 1 January. No biases should have 

 occurred because of the segregation date of the winter base 

 categorization and because the growth of C.japonica is neg- 

 ligible during winter. Four base models were constructed 



0.2 " 



g 

 a 



0.1 -- 



0.0 -i 



Largest extreme 

 value distribution 



Normal distribution 





L. 



-+- 







1 2 



Shell length (mm) 



Figure 2 



Two distributions fitted by the maximum- 

 likelihood method to the shell lengths of Cor- 

 bicula japonica juveniles spawned in 1997 and 

 sampled on 22 April 1999. Raw data are shown 

 by +. The shell length composition is shown by 

 the histogram. 



combining the two types of age categorization and two types 

 of equations expressing the relationship between the dRIRL 

 and the dRIRS, i.e. Equations 7 or 8. We selected the best 

 base model by AIC. 



To check the significance of each environmental factor 

 and age categorization, we removed the independent vari- 

 ables one by one from the best base model and re-optimized 

 the model. When the model was significantly improved by 

 the removal in terms of AIC, the effect of the variable was 

 insignificant on the model; therefore we excluded it. 



To check the significance of salinity and turbidity, which 

 were not included in the base models, we included them 

 one at a time into the best base model and re-optimized the 

 model. When the model was improved by the inclusion, the 

 effect of the variable was significant on the model; therefore 

 we included it. 



Results 



Modeling the distribution of a single sample 



The largest extreme value distribution was the best in 

 terms of AIC except for data sampled on 13 May 1998 

 (results are not shown). The exception is due probably 

 to the small sample size (rc=38) on that date. The largest 

 extreme value distribution was therefore used to evaluate 

 likelihood in later analyses: we selected Equation 10 from 

 Equations 9 and 10. The result of fitting the two distri- 

 butions to the shell lengths sampled on 22 April 1999 is 

 shown in Figure 2 as a representative example. The largest 

 extreme value distribution is apparently better than the 

 normal distribution for describing the single cohort of C. 

 japonica spawned in 1997. 



