NOTE Wang and Milton: Comparison of growth curves 



875 



growth rates of two species of tiger prawn (shrimp) from 

 northern Austraha. 



Methods 



Overall hypothesis tests on two sets of parameters 



Suppose we are interested in testing the hjrpothesis that 

 the underlying growth curves corresponding to the two 

 sets of parameter estimates /} j = (/fj, / ,j) and /}2 = (^2'^-2* 

 are the same. According to the large-sample theory, it is 

 quite reasonable in most cases to assume that /3 j and jS.^ 

 are normally distributed. To be general, we will allow /3 j 

 to be correlated with p. 2- In notation. 



\M 



= N 







(1) 



Note that if /} j and P j ^^^ estimated from different data 

 sets, they may be assumed to be independent, because both 



p J and p ., are estimates and Pi- Po '^ approximately 

 multivariate normal. To test whether the two growth pat- 

 terns determined by /3j and /Jj are the same or not, we 

 can use the generalised T'^-statistic (Anderson, 1971): 



T'={p,-p,)V-Up,-p,), 

 in which V = the covariance of /3 j - p^- 



(2) 



The distribution of the 7^-statistic is approximately chi- 

 squared with 2 degrees of freedom, xf  If the significance 

 level is a. the corresponding critical value is xl 'o;). 



In many cases, we are interested in the slope of the 

 growth curve (growth rate) rather than the curve itself. For 

 example, we may be interested in comparing the growth 

 rate during a particular age interval. Owing to natural 

 mortality or fishing mortality, the period outside of this 

 age range may be of no practical interest. In this case, it 

 is may be more appropriate to consider the growth pat- 

 terns over a specified age or length range rather than the 

 whole range, which would put more emphasis (weight) on 

 the asymptotic length in the comparison. 



