KIMURA: LIKEUHOOD METHODS FOR GROWTH CURVE 

 70 r 



65 



:60- 



',55 



50 



45 



.20 



.25 



.30 



.35 



K 



.40 



.45 



.50 



FIGURE 2. — Cross sections of approximate 95% confidence re- 

 gions around least squares estimates (/^.A^), generated by con- 

 ditioning on t„ (see text), for male (M) and female (F) Pacific 

 hake. Letters M and F are centered on least squares estimates. 



partial derivatives for the five models. These sub- 

 routines are general in the sense that they allow 

 comparisons of any number of populations. Proba- 

 bly these subroutines are compatible with the re- 

 quirements of BMDP3R, but this program was not 

 readily available for a test. Also, minor modifica- 

 tions may be necessEtry to comply with require- 

 ments of particular computer systems. 



Results of LR tests (Table 3) indicate there is a 

 significant difference (a = 0.002) in l^ between 

 sexes, a borderline significant difference 

 {a = 0.05) inK, but no significant difference in tg. 



ACKNOWLEDGMENTS 



I thank R. J. Trumble of the Washington De- 



partment of Fisheries for bringing this problem to 

 my attention. I also thank the referees for their 

 helpful comments. 



LITERATURE CITED 



ALLEN. K. R. 



1966. A method of fitting growth curves of the von Ber- 

 talanffy type to observed data. J. Fish. Res. Board Can. 

 23:163-179. 



Bertalanffy, L. von. 



1938. A quantitative theory of organic growth (inquiries 

 on growth laws. 11). Hum. Biol. 10:181-213. 

 Beverton, R. J. H., AND S. J. Holt. 



1957. On the dynamics of exploited fish populations. 

 Fish. Invest. Minist. Agric. Fish. Food (G.B.), Ser. II, 19, 

 533 p. 

 CHAPMAN, D. G. 



1961. Statistical problems in dynamics of exploited 

 fisheries populations. Proc. of the Fourth Berkeley 

 Symp. Math. Stat, and Probability. Contrib. Biol, and 

 Probl. Med., vol. 4:153-168. Univ. Calif. Press. 



Dark, T. a. 



1975. Age and growth of Pacific hake, Merluccius produc- 

 tus. Fish.Bull., U.S. 73:336-355. 



DIXON, W. J. (editor). 



1976. BMD biomedical computer programs. Univ. Calif. 

 Press, 773 p. 



1977. BMDP-77 biomedical computer programs 

 P-series. Univ. Calif. Press, 880 p. 



DIXON, W. J,, AND F. J. MASSEY, JR. 



1957. Introduction to statistical analysis. 2d ed. 

 McGraw-Hill, N.Y., 488 p. 

 DRAPER, N. R., AND H. SMITH. 



1966. Applied regression analysis. Wiley, N.Y., 407 p. 



Hartley, H. O. 



1964. Exact confidence regions for the parameters in non- 

 linear regression laws. Biometrika 51:347-353. 



Kendall, M. G., and a. Stuart. 



1973. The advanced theory of statistics. Vol. 2, Inference 

 and relationship. 3d ed. Hafher Publishing Co., N. Y., 723 p. 

 NiE, N. H.. C. H. Hull, J. G. Jenkins, K. Steinbrenner, and 

 D. H. Bent. 



1975. SPSS statistical package for the social sciences. 2d 

 ed. McGraw-HUI, NY., 675 p. 

 RICHARDS, F. J. 



1959. A flexible growth function for empirical use. J. 

 Exp. Bot. 10:290-300. 



Table 3. — Likelihood ratio tests comparing von Bertalanffy parameter estimates for male ( 1 ) and lemale (2) Pacific hake, based on 



data in Table 1 (W = 24 observations). 



Hypoth- Linear 

 esis constraints 



Equivalent model 



Residual 



sum of X, = -N log 

 squares - ., - , df 



"IcuS 



,(1 -exp(-K-,(t„-ro,ll) 55 98 61.23 386 296 171 -0057 48 22 



.(l-exp(-/<:,(r„-ro,l)) 59.40 59 40 297 0,337-0.111 087 7160 



(1_exp(-/ir(f„-ro,l)) 57.44 60,14 330 0.330 -0.021 0.095 56,34 



(l.exp(-/{',(f,,-rol)) 56,45 6077 361 0.313 0,057 0,057 50,76 



,(l-exp(-/f(r„-f(,))l 59,29 59 29 320 320 0010 0010 79 76 



773 



