FISHERY BULLETTN: VOL, 77, NO. 4 



Suitable programs are also available in BMDP 

 (Dixon 1977), SPSS (Nie et al. 1975), and from a 

 number of other sources. 



Inital values were obtained from Ford-Walford 

 plots (Ricker 1975) which provided estimates /, 

 and K\ and from the weighted average 



::[iog(i-/,//"j/K + f,.](L-/,) 



t 



where I, is the average length at age t^ . For these 

 initial calculations, it was convenient to round 

 ages to whole years. This disallows direct com- 

 parisons of Ford-Walford and LS estimates. Be- 

 sides initial estimates, BMD07R also requires a 

 FORTRAN subroutine (provided in the Appendix) 

 which evaluates ix(t^ ) and its partial derivatives. 



Table 2 contains the initial Ford-Walford esti- 

 mates (l^,Kjg) and the final LS estimates 

 CK,k,i^) obtained from BMD07R. LS fits of 

 growth curves are graphed in Figure 1. 



From Figure 1 there appears to be a difference in 

 K between sexes. As a further examination of this 

 difference, cross sections of the approximate 95% 

 confidence regions around (I ,,K), generated by 

 conditioning on 4. were graphed (Figure 2) using 



Table 2. — Least squares (LS) estimates of von Bertalanffy 

 parameters for male and female Pacific hake, based on data in 

 Table 1. 

 Item /a t^ fo 



Male: 



Ford-Waltord initial estimates 



LS estimates 



Standard deviation of LS estimates 

 Female 



Ford-Waltord initial estimates 



LS estimates 



Standard deviation of LS estimates 



5563 043 0.35 



55 98 0.386 0.171 



1083 039 142 



60.60 0.35 0.32 



61.23 296 -0 057 



1.214 0029 0175 



methods previously described. These regions not 

 only show a difference in /, between sexes, but also 

 indicate a difference in K. 



As a final step in this analysis, LR tests for 

 equality of von Bertalanffy parameters between 

 males (population 1) and females (population 2) 

 were performed. It was necessary to fit data to five 

 models corresponding to hypotheses of interest 

 (Table 3). 



The difficulty of fitting these models depends 

 somewhat on the nonlinear LS program used. If 

 derivative-free programs are available, the user 

 will be saved the complex task of specifying de- 

 rivatives. If programs allow for constraints, only 

 the model fl need be specified. Nonlinear LS pro- 

 grams available in BMDP have these features. 



For BMD07R, the Appendix provides FOR- 

 TRAN subroutines which evaluate ix{t^ ) and its 



FIGURE 1.— Plots rf average length at 

 each age (from Table 1), for male and 

 female Pacific hake, with graphs of es- 

 timated von Bertalanffy curves (from 

 Table 2). 



70 



60 



50 



540 



a 



30 



20 



10 



®— FEMALE 

 MALE 



6 8 

 AGE (YR) 



10 



12 



14 



772 



