874 



On comparison of growth curves: 



How do we test whether growth rates differ? 



You-Gan Wang 



CSIRO Mathematical and Information Sciences 



PO Box 120 



Cleveland 



Queensland 4163, Australia 



Present address. Department of Biostatistics 



Harvard Sctiool of Public Health 

 655 Huntington Avenue 

 Boston, Massachusetts 02115 



E mail address ygwangm'hspti harvard edu 



David A. Milton 



CSIRO Marine Research 



PO Box 120 



Cleveland 



Queensland 4163, Australia 



Comparisons of growth rates of popu- 

 lations and species are important in 

 fisheries science for a range of reasons 

 that vary with the context of each 

 study. Most studies offish growth have 

 focused on the practical issues of the 

 most appropriate way of comparing 

 growth rather than on recognizing that 

 there are several methods for making 

 these comparisons and that the con- 

 clusions will differ depending on the 

 method chosen. 



Francis (1996) discussed the prob- 

 lem of how to compare different gi'owth 

 rates or growth curves. He suggested 

 six plausible ways of making a com- 

 parison and suggested that the rate 

 at which the asymptotic size is 

 approached was the most natural 

 method of comparing growth (his 

 method 6). He illustrated the differ- 

 ences between the methods by compar- 

 ing von Bertalanffy growth equations 

 that are based on fixed growth para- 

 meters and that assumed that growth 

 parameters are known and there are 

 no associated uncertainties. However, 

 in practice, the growth parameters 

 are often estimated, and therefore are 

 random variables. Consequently, the 

 corresponding growth curves are also 

 subject to variations. 



For comparison with Francis's work, 

 we will assume that growth for a spe- 



cies is adequately described by the von 

 Bertalanffy equation with t,, = 0, as 



L(^/3) = /. (1 



(1) 



in which /3 = (/.', ! , ) are growth parameters. 



Here l(t) is the mean length at age 

 ^ If /3j and /j., are two estimates of 

 p. how do we test whether the corre- 

 sponding two growth curves are the 

 same? The traditional way is to com- 

 pare individual parameters and find 

 out which ones are significantly differ- 

 ent. However, the parameter estimates, 

 /„ and /;, are often strongly correlated 

 (Kirkwood and Somers, 1984; Wang 

 and Thomas, 1995 ). It may therefore be 

 more appropriate to compare biological 

 reference points (e.g. size at one year of 

 age) rather than growth parameters in 

 the models I Wang and Thomas, 1995). 

 Growth comparisons may, in general, 

 be classified into two types: within spe- 

 cies and between species. In practice, 

 the following comparisons may be of 

 interest: 



1 Comparison of the growth rates for 

 the same species, say E, in which 

 two sets of growth parameter esti- 

 mates, /}j and p^, are obtained 

 from different time periods, differ- 

 ent areas or sexes. 



2 Comparison of growth rates for 

 two different species to see which 

 one grows faster. 



As mentioned earlier, Francis (1996) 

 considered six methods for comparing 

 growth. For the within-species compar- 

 ison, it seems all six methods are valid. 

 However, these methods compare dif- 

 ferent characteristics of growth and 

 therefore may reach different conclu- 

 sions. For example, if we obtain P j = 

 (0.5,50) from area A and ji.^ = (0.4,60) 

 from area B, we would conclude that 

 species E does not grow as large in area 

 A as in area B and that the k value 

 (rate at which the asymptotic length is 

 approached) in area A is larger than 

 that in area B. 



For between-species comparisons, we 

 agree with Francis (1996) that his 

 method 6 ik value comparison) is prob- 

 ably the most appropriate, especially 

 in the context of comparing growth 

 between, for example, herring and 

 orange roughy. However, in some cases, 

 comparing absolute growth rates at 

 age or length between species could be 

 of practical interest. 



For example, if you are interested 

 in choosing one of two species of fish 

 or crustacean to farm and these two 

 species look alike and have the same 

 commercial value, it is more economi- 

 cal to farm the faster-growing species 

 to shorten the time taken to reach 

 market size. In Australia, the tiger 

 prawn P. esculentus has a larger k 

 value than the very similar P. semis- 

 ulcatus (Somers and Kirkwood 1991), 

 but P. semisulcatus has the potential of 

 reaching a commercial size sooner ( Fig. 

 1 ). Therefore, we would conclude that 

 P. semisulcatus grows faster than P. 

 esculentus in this context, and a com- 

 parison based only on k values may be 

 misleading. 



Therefore, in this note we will extend 

 Francis's theoretical study by develop- 

 ing procedures for establishing statisti- 

 cal hypotheses for the six methods and 

 suggest test statistics for comparing 

 growth curves. We will demonstrate 

 the differences in conclusions that can 

 occur among the methods with data on 



Manuscript accepted 1 July 2000. 

 Fi.sh. Bull. 98:874-880 (2000). 



