KAPPENMAN: A METHOD FOR GROWTH CURVE COMPARISONS 



any given situation, is to examine the observed 

 and predicted length differences for each model. 

 Differences, corresponding to youngest or oldest 

 fish, being excessively large for one or more 

 models might be an indication that extrapolation 

 is biasing the procedure. This difficulty did not 

 appear in the examples used in this paper, but it 

 is possible to imagine rare cases where it could be 

 a problem. If this problem does arise, it is easily 

 remedied. One can always eliminate from a sum 

 of squares of differences between observed and 

 predicted lengths those differences whose predic- 

 ted lengths are obtained by extrapolation. If this is 

 done, the sums of squares in the model selection 

 criterion should be replaced by averages of 

 squares of differences. 



For each of the examples given in this paper, 

 all of the specified growth curves were taken to be 

 of the same form. This is not necessary. Any 

 growth curve can be given any form. For exam- 

 ple, in the two population case, the common 

 growth curve, for the model of equality of growi;h 

 curves, can have a mathematical form which is 

 different from the forms of the growth curves 

 specified under the model of different growth 

 curves. And, in fact, the latter two forms can be 

 different from each other. Thus it is possible to 

 handle the case where Mi specifies equal growth 

 curves and the common growth curve belongs to, 

 say, the logistic family, while M2 specifies differ- 



ent growth curves and the curves belong to, say, 

 the Laird-Gompertz family or one belongs to the 

 Laird-Gompertz family and the other belongs to 

 the generalized extreme value for minima family. 

 Finally, it should be pointed out that although 

 this paper has been concerned solely with growth 

 curve comparisons, the procedure described here 

 can be applied to the general problem of compar- 

 ing regression equations. The regression equa- 

 tions of interest can be either linear or nonlinear 

 functions of the unknown parameters. Where 

 they are nonlinear is of particular interest since 

 such comparisons have apparently not been dis- 

 cussed in the literature. 



LITERATURE CITED 



ALLEN, R. L. 



1976. Method for comparing fish growth curves. N.Z. J. 

 Mar. Freshwater Res. 10:687-692. 

 BOEHLERT, G. W, AND R. F. KAPPENMAN. 



1980. Latitudinal growth variation in the genus Sebastes 

 from the Northeast Pacific Ocean. Mar. Ecol. Prog. Sen 

 3:1-10. 

 Box, G. E. P, AND M. E. MULLER. 



1958. A note on the generation of random normal devi- 

 ates. Ann. Math. Stat. 29:610-611. 

 GALLUCCI, V F, AND T. J. QUINN II. 



1979. Reparameterizing, fitting, and testing a simple 

 growth model. Trans. Am. Fish. Soc. 108:14-25. 

 GEISSER, S., and W E EDDY. 



1979. A predictive approach to model selection. J. Am. 

 Stat. Assoc. 74:153-160. 



101 



