A METHOD FOR GROWTH CURVE COMPARISONS 



Russell F. Kappenman^ 



ABSTRACT 



Suppose one has a sample of pairs of age and length measurements from each of two or more 

 populations offish. The mathematical forms of the growth curves associated with the populations are 

 assumed to be specified but each form contains at least one unknown parameter Presented in this paper 

 is a data analytic approach to the problem of deciding which, if any, of the populations have essentially 

 the same growth curve and which have different ones. 



A common problem in fisheries research is that of 

 comparing two or more growth curves. This prob- 

 lem arises whenever investigators gather data for 

 the purpose of trying to determine whether or not 

 different species or the sexes of a given species of 

 fish grow at different rates, or for the purpose of 

 assessing growth variation of a species from envi- 

 ronment to environment, area to area, or stratum 

 to stratum in which it is found. 



Since the models generally used for the age- 

 length relationships (e.g., von Bertalanffy, Laird- 

 Gompertz, logistic, etc.) are most often nonlinear 

 in the unknown parameters and cannot be linear- 

 ized by transformations of the variates, the usual 

 techniques for comparing regression equations 

 are not applicable. Up to this point little has been 

 done in the way of development of quantitative 

 methods for determining whether or not unknown 

 growth curves do in fact differ. Thus the investiga- 

 tor can often do little more than visually examine 

 plots of age-length data for samples from the 

 various populations being compared and arrive at 

 some rather subjective conclusions. 



An exception is a paper by Allen (1976) which 

 treats the special case where each of the growth 

 curves being compared belongs to the von Berta- 

 lanffy family. There are, of course, numerous 

 instances where the von Bertalanffy model is not 

 appropriate and the Allen procedure does not 

 apply if it is not. Further, even if this model is 

 appropriate, some severe assumptions need to be 

 made in order to apply the analysis. These include: 

 1) the equality of the scale parameters for all 

 curves being compared, 2) the true value of the 

 common scale parameter being exactly equal to its 



'Northwest and Alaska Fisheries Center, National Marine 

 Fisheries Service, NOAA, 2725 Montlake Boulevard East, 

 Seattle, WA 98112. 



estimated value, and 3) the usual normality, 

 independence, and equality of variance assump- 

 tions for the error term. The first assumption quite 

 clearly biases the procedure in favor of the null 

 hypothesis of equality of the growth curves, while 

 validity of the second seems to be too much to hope 

 for. 



Gallucci and Quinn (1979) also discuss the 

 growth curve comparison problem for the von 

 Bertalanffy case. They essentially reparameterize 

 the model and test the hypothesis of equality of 

 one of the new parameters for all curves being 

 compared, assuming, apparently, that the other 

 two have the same value for all of the curves. The 

 comments in the preceding paragraph also apply 

 to these authors' work. 



The purpose of this paper is to point out how 

 some predictive sample reuse techniques, described 

 in a recent paper by Geisser and Eddy (1979), can 

 be adapted and applied to a growth curve compari- 

 son problem where two or more populations are 

 being studied, the growth curves associated with 

 each of the populations are unknown and are to be 

 compared, and a sample of age-length data is 

 available from each population. The problem then 

 is to use the data to decide which, if any, of the 

 population growth curves are the same and which 

 are different. 



We will assume that the growth curves associ- 

 ated with each of the populations are specified 

 except for the values of unknown parameters. 

 These specifications often would be made by 

 plotting the sets of age-length data, fitting vari- 

 ous possible models suggested by the data plots, 

 and selecting the models which best fit the data. 

 The growth curves can, but need not, belong to 

 the von Bertalanffy family. In fact, they can 

 belong to any family. Thus, in essence, we are 



Manuscript accepted July 1980. 



FISHERY BULLETIN: VOL. 79, NO. 1, 198L 



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