876 



Fishery Bulletin 98(4) 



Tests to compare two growth equations Letg be a growth 

 function of (k, /,) that we are interested in comparing. 

 Table 1 Hsts the six g functions corresponding to the six 

 methods identified by Francis (1996). For a given function 

 g, we will test gip^) = ^(/Jj) versus giP^) * gip.^) or g(pj^) 

 > g(Po'l, depending on the context. Standard normal tests 

 may be used for a specified g function. The test will rely on 

 the properties of D=g{p^) -gip.,). Let EiD) and ViD) be the 

 corresponding expectation and variance of Z) when p is the 

 true parameter. Under the null hypothesis, g^(^j) = g(p.2\ 

 and using standard Taylor series expansion, we can work 

 out analytic expressions of EiD) and ViD). Some pooled 

 estimates of p may be required to input to EiD) and ViD) 

 to obtain approximate values of £(Z)) and ViD). We can 

 obtain EiD) from Egi p^) - Egip.,) and 



E(giJ3)]^gip) + ^{f,^^,+2f,,a,, + f,XJl,) 



(3) 



in which /"values are from the second derivative of g with 

 respect to p (Table 1) and 



''tJ?, CJ,., 



Vip). 



The variance of D can be obtained from 



ViD) = X^l.X, + XlZ.X., - 2X^Z,.,X.„ 



(4) 



(5) 



in which X, = the gi'adient or first derivative of g, (Table 

 l);and 

 Z"s = the components of the covariance defined 

 earlier. 



Note that the last term disappears if /3j and /3,, are 

 independent of each other. There are a few possible ways 

 to obtain the approximate significance level, P. However, 

 the most widely used method assumes D is normally dis- 

 tributed. Then we can use the z-test, which is based on the 

 normal approximation for large sample sizes. For a one- 

 sided testgip^)=g{p.,) vs.giPj^) >giP2) 



P= 1-* 



D 



O 



-D 



^^jv^D) [^Vib) J^ 



where ct> = the standard normal distribution function. 

 For a two-sided test, we have 



(6) 



P = 2(D 



-\D\ 

 VvTD) 



(7) 



If we are interested in the gi-owth for a range of ages 

 <^„,„. ',„„,'. or the sizes (/,„,„, /,„„,). we may consider the in- 



