Jacobson et ai.: A biomass-based assessment model for Engraulis mordax 



717 



constraint like that in Equation 11 was overpar- 

 ameterized because recruitments need occur only 

 once every two to three years for the model to match 

 observed and predicted abundance data. Age-com- 

 position data for northern anchovy indicate, however, 

 that some recruitment occurs during every fishing 

 season (Lo and Methot, 1989). We included the re- 

 cruitment constraint and a recruitment parameter 

 for each season to obtain a more realistic model and 

 to constrain the recruitment estimate for the last fish- 

 ing season which was otherwise difficult to estimate. 

 The constraint on recruitment biases recruitment 

 and biomass estimates towards the mean because 

 recruitment estimates will be high in years with poor 

 recruitment and low in years with high recruitment. 



Parameters in the model were estimated by using 

 the simplex algorithm (Press et al., 1990). Variances 

 and correlations for parameter and biomass esti- 

 mates were calculated by using a parametric boot- 

 strap approach (Efron, 1982) as described in Lo et 

 al. (1992) except that simulated abundance data were 

 generated by assuming log-normal errors with stan- 

 dard deviation equal to the root-mean-squared log- 

 scale residual for each data type (see below). Param- 

 eters for bootstrap runs were estimated as described 

 for the original run by using the original CVs for 

 each abundance index observation. Thus, our boot- 

 strap runs included process error to the extent that 

 it was reflected in the variance of residuals, and in- 

 cluded measurement error to the extent that it was 

 reflected in the original CVs. Two thousand bootstrap 

 iterations were generally used. Asymptotic variance 

 and correlation estimates for parameters were also 

 calculated by inverting a numerical approximation to 

 the Hessian matrix (Bard, 1974; Mittertreiner and 

 Schnute, 1985) because we were interested in compar- 

 ing the asymptotic and bootstrap approaches. 



Parameters with all feasible values positive were 

 estimated as log-transformed values. The log trans- 

 formation constrains parameters to feasible values 

 on the original scale and improves the statistical 

 characteristics of parameter estimates. Standard 

 errors for log-scale recruitment parameters were 

 transformed to CVs for arithmetic recruitment esti- 

 mates by using Equation 1. 



Results and discussion 



Estimates from preliminary runs indicated that 

 availability of age-0 northern anchovy to indices of 

 schooling biomass was close to zero. For final runs, 

 Pspotter.o an d PsoNAJt.o were set to zero and not esti- 

 mated even though age-0 fish were assumed to be 

 fully recruited to the fishery. 



Outliers and residual analysis 



Standardized residuals (D t , in Eqn. 12) for most 

 abundance indexes were serially correlated in pre- 

 liminary runs. There were two outliers (D EPI 1983 =3.6 

 and D spoTTER 1979 =3.5) identified by a £-test with 

 Bonferronip- values (critical value D. =3. 41 forn=77; 

 Weisberg, 1980). Residual plots for the final run with 

 outliers omitted still indicated some serial correla- 

 tion. All but three biomass estimates for northern 

 anchovy during the 1963 to 1991 fishing seasons in- 

 creased when the two outliers were omitted. The 

 average increase was 24%. 



CVs for abundance indices and goodness of fit 



The root-mean-squared residual for each abundance 

 index was calculated to measure how well the 

 SMPAR model fit the data for northern anchovy. 

 Standard deviations were not calculated because 

 degrees of freedom were unknown. Arithmetic CVs 

 implied by the goodness-of-fit statistics were calcu- 

 lated by using Equation 1. For comparison, median 

 CVs for our data (Table 1) were also calculated. 



Goodness-of-fit statistics and implied CVs (Table 

 3) indicate that the CVs for our abundance data un- 

 derestimated the true log-scale standard errors. The 

 order of abundance indices ranked by median CVs 

 was, however, the same as when they were ranked 

 by goodness of fit. Thus, CVs used in the model re- 

 flected the relative precision of different types of 

 abundance data for northern anchovy. 



Consistent bias 



Percent bias (%BIAS) in biomass and log-scale recruit- 

 ment estimates for northern anchovy was estimated 



Table 3 



Goodness-of-fit and CV statistics for northern an- 

 chovy, Engraulis mordax, abundance indices used 

 in the SMPAR model. Median nominal CVs were cal- 

 culated from arithmetic CV values in Table 1. Root- 

 mean-squared residuals measure goodness of fit to 

 abundance index. Implied CVs are the arithmetic CV 

 values calculated from the goodness-of-fit measures. 



Median Root- 



Abundance nominal mean-squared Implied 



index n CVi'-i residual CVC*) 



DEP 



EPI 



SPOTTER 



HEP 



SONAR 



6 

 L2 



■I 7 

 11 

 16 



Lil 



31 

 34 



11 

 ■Hi 



0.19 

 0.48 

 0.49 

 0.50 

 0.53 



19 



51 

 52 

 53 



57 



