NOTE Wang and Milton: Comparison of growth cutA/es 



877 



tegrated squared difference over the specified range. For 

 example, if^, = AjC"*'' and g.2 = X.,e-'''' 

 and A,, = k.,l_^.,.vje will rely on 



in which Aj = /jgZ, 



D. 



h 



g,{t)-g,{t))clt. 



The corresponding expectation and variance can also be 

 approximated by the delta method. 



We then apply each method to comparisons of the gi'owth 

 of males and females of two species of tiger prawn caught 

 during tagging studies in northern Australia (Somers and 

 Kirkwood 1991). We will consider two scenarios: 1) growth 

 at age 0.5 yr (for methods 1, 2, and 4) or gi'owth at length 

 30 mm (for methods 3 and 5); 2) gi-owth at age one yr (for 

 methodsl, 2, and 4) or growth at length 35 mm (for methods 

 3 and 5). In order to verify that our test assumption that D 

 was normally distributed, we obtained frequency plots for 

 the bootstrapped estimates of the growth parameters. 



Results 



We first bootstrapped the parameter estimates for each 

 group to assess whether our assumption that the para- 

 meters were normally distributed was valid. The plot for 

 male P. semisiilcatus showed that there was little evi- 

 dence of skewness in our estimates (Fig. 2). Plots for other 

 groups are similar and not shown here. The / tests pro- 

 posed in this paper, although based on normal distribu- 

 tions, are well known to be robust to violation to normality 

 (which is why they are also known as "robust" test in sta- 

 tistics). On the other hand, the proposed tests rely only on 

 mean and covariance estimates and the covariance matrix 

 is often obtained from asymptotic distribution (normal) of 

 the estimates in nonlinear regression. 



We tested for the overall significance of each comparison 

 before proceeding with testing the six methods. All were 

 highly significant (P«0. 00001) and 7"-' statistics ranged 

 from 43.2 for the comparison of P. semisiilcatus females 

 versus P. esculentus females to 385.7 for P. esculcntus 

 males versus P. semisiilcatus females. Given that the over- 

 all growth curves differed among species, then it is reason- 

 able to then look further at the growth rates yielded by the 

 different methods. 



The growth models for the six methods of comparing 

 growth and their first and second derivatives differed 

 between methods (Table 1). The growth parameters used 

 in the comparison show apparent differences in the size of 

 both l^ and k among the species and sexes (Wang, 1998). 

 These differences in the absolute value of each parameter 

 translate into quite large differences in the shape of the 

 growth curves (Fig. 1). The effect is most striking for 

 Penaeus semisulcatus, but does growth differ? Is it affected 

 if we choose a particular length or age? 



The results of tests from the six methods are shown in 

 Table 2. The difference in lengths between P. esculentus 

 and P. semisulcatus at the age of 0.5 yr is not significant for 

 either males or females (in method 1 in Table 2), but the 

 difference becomes very significant at age 1 yr. This result 

 is consistent with the plot in Figure 1. However, the con- 

 clusion is reversed when comparing growth rate (method 

 2 in Table 2) instead of length. The relative growth rates at 

 either age (0.5 yr or 1 yr) for P. esculentus do not differ sig- 

 nificantly between males and females (method 4), whereas 

 the rates at length 30 mm or 35 mm differ significantly. 

 This is also biologically plausible because the gi-owth rate 

 at length 35 mm is very close to zero for P. esculentus 

 males (the asymptotic size is only 35.6 mm) and there is 

 a substantial gi-owth to be gained for females (the asymp- 

 totical size is 44.7 mm). For method 6, the comparion is 

 independent of specified length or age (two scenarios give 

 the same results). 



In prawn farms, P. esculentus and P. semisulcatus are 

 harvested after 6 or 12 months, depending on the circum- 

 stances, such as the farmer's need to increase the number 

 of generations within the growing season at higher lati- 

 tudes. Tests comparing the growth of each sex of each spe- 

 cies (Table 3) show that the significance of the results 

 varies with the hypothesis being tested. For example, the 

 comparison of length-at-age (method 1) of Penaeus semi- 

 sulcatus and P. esculentus and females was not significant 

 (P<0.87) at both six months of age (Table 3). However, 

 when the absolute growth rates of the two species were 

 compared (method 2), they were highly significantly dif- 

 ferent (P<0.001). Other tests, such as comparisons of the 

 growth of male and female P. semisulcatus, were highly 

 significant for all methods (P<0.001). As Francis (1996) 

 pointed out, the results from all comparisons with meth- 

 ods 3 and 5 produce similar results and methods 4 and 6 

 give very similar results but in the opposite direction. 



