Zenitani and Yamada; The relation between spawning area and biomass of Japanese pilchard, Sai dinops melanostictus 



845 



Parameter estimation and confidence interval analysis 

 were performed by the following procedure, ii^ and i/^ were 

 estimated by minimizing the sum of squares: 



= y (ln(A, /A, 



where A, and A,, = the observed spawning area in year v 

 and predicted values from relationships I- V, respectively. 

 The least-squares minimization was performed by the 

 quasi-Newton method in Solver, which is "add in" software 

 for MS-Excel (Microsoft Corp., 1996). The least-squares 

 minimization procedure was stopped if either of the two 

 following conditions were satisfied: 



more than 100 iterations were attempted; or 



the sum of squared residuals changed by less than 0.01% 

 between iterations. 



Confidence intervals of the estimators were calculated 

 through a likelihood ratio test. U\niAjA^,)~N{0,a^), then 



U = (;/j, (/., ) is a parameter vector, 



IJ = ((/ J, i/j) is the estimated parameter vector which min- 

 imizes the 0, and 



Uj = (;/,, ii.,) is the parameter vector which minimizes the 

 ©with respect to a certain value of w. 



The confidence interval for a certain parameter !/, was 

 given as the region which satisfied the inequality 21n|L( Lf)/ 

 L(f/))<x-(0.95,l), where L and ;tf-(0.95,l) were the like- 

 lihood and 959f value of the x^ distribution with one 

 degree of freedom, respectively. The Akaike information 

 criterion (AIC) was used for selecting an optimal relation- 

 ship within the set of proposed relationships I-V. The AIC 

 is calculated for candidate models, and the most parsimo- 

 nious one has the lowest AIC (Akaike, 1973). 



400 



300 



TO 



2 200 



ra 100 



Year 



Figure 5 



Change in the spawning area of the Japanese pilchard 

 along the Pacific coast of Japan over time. Aj and Aj were 

 calculated by summing the areas of 1° longitude x 1° lati- 

 tude squares where early developmental stage eggs and 

 eggs of any stage occurred, respectively 



Results 



The spawning biomass (S) of the Japanese pilchard experi- 

 enced a remarkable increase in the 1970s, peaked in 1988, 

 and has been declining since the end of the 1980s (Fig. 1; 

 Wada and Jacobson, 1998). A, expanded from 114,000 km- 

 in 1980 to 299,000 km- in 1990 , before shrinking to 38,000 

 km- in 1994 (Fig. 5). Estimated parameters, their confi- 

 dence intervals for each relationship, and AIC are summa- 

 rized in Table 1. The value of AIC was at a minimum in 

 relationship III and varied considerably compared to the 



