FISHERY BULLETIN: VOL 77. NO 4 



Points on the three-dimensional contour are 

 easily calculated by conditioning on a value for <„, 

 and calculating the two-dimensional cross section 

 {l^,K) by stepping through plausible values for 

 K, and when B'^-AAC^O, calculating /, = 

 ( -fi±VB2-4AC)/(2A). By varying tg also, this al- 

 gorithm will generate the entire three-dimen- 

 sional confidence region. 



Although points on the contour surface are eas- 

 ily calculated, the fact that three parameters are 

 involved in the von Bertalanffy curve greatly 

 limits the usefulness of confidence regions. This is 

 due to the simple fact that three-dimensional re- 

 gions are difficult to display. 



The simplest solution to this problem is to 

 condition on i^, and graph the resulting two- 

 dimensional cross section 11,, K). It must be re- 

 membered that this region is not a true con- 

 fidence region since more extreme values of (/, ,K) 

 may occur at a different value of t^,. Thus this 

 procedure will give only a rough idea of our 

 confidence in the estimates (K,k). A more time 

 consuming solution is to graph a series of cross 

 sections, or possibly a three-dimensional graph. 



If method (b) is used to estimate parameters, the 

 analysis follows as in method (a), by simply replac- 

 ing /^ with \ and N with /. 



If weighted methods (c) or (d) is used to estimate 

 parameters, confidence regions are defined by the 

 relationship 



•Su,(/oo,^,'o) 

 = SjL,kJo)ll+~F(3,I-3,l-q)\ = c^ . 



Computations proceed as in the unweighted case, 

 but with 



A^ = ywJl-exp{-K(t,-to))f, 



method for the statistical comparison of growth 

 curves. It is a well-known and often exploited fact 

 that once a general probability model has been 

 specified (O), hypothesis tests of linear constraints 

 on parameters in this model can be derived using 

 the LR criterion. Alternatively viewed, linear con- 

 straints on parameters in H imply a simplified 

 model to. Tests of linear constraints on 11 are thus 

 equivalent to testing w against il. 



The LR criterion can be used on the single sam- 

 ple problem, when it is desired to test whether a 

 sample came from a population with some 

 "known" values for any or all of the parameters 

 U„,K,t^); or for the multisample problem compar- 

 ing von Bertalanffy curves in different popula- 

 tions. The first problem will be solved by the 

 simplest application of theory derived mainly in 

 the context of the second problem. When a single 

 parameter is being tested in the one or two sample 

 problem, it makes good sense to simply use a 

 Z-statistic (since ML estimates are asymptotically 

 normal) and forego the more extensive calcula- 

 tions required for LR tests. One advantage that 

 the Z test has over the LR test for the two sample 

 problem is that o^ does not need to be equal in the 

 two populations. 



Consider / different populations each following 

 the von Bertalanffy curve with parameters 

 (/,, ,/f, ,<Qj), i = 1,. . . , I. These populations would 

 typically be the same species in different habitats, 

 males and females, etc. Let l^j be the length of the 

 jth observation in the ith population, of age t^J, 

 J = 1, . . . , n, , iV = S, n, , with variance cr^ inde- 

 pendent of i. Note that the meaning of subscripts 

 has been changed from what was used in previous 

 sections. 



Letting S(/, ,/»:,<„) = l,lj(l,j-Ml^^,K„toi,tij))^ 

 the likelihood function under (fl) can be written as 



B^, = -2XwJJl-exp(-K{t,-to))) 



and Cu, = ^wJu^-c'^ 



For method (c) w^ = njs^^, and for method (d) 



APPLYING LIKELIHOOD RATIO TESTS 



Likelihood ratio (LR) tests provide a general 

 770 



where /j = (l^i, . . . , /»;), 



K-={K„...,K,), 



and t'a = «oi,  • • . <o/'- 



Although the above parameterization is appropri- 

 ate for unweighted methods (a) and (b), the reader 

 can verify that no additional problems arise using 

 weighted methods (c) and (d). 



Previously, it was shown that likelihood func- 

 tions of the form ^(l^Ji,t„ ,o^) are maximized by LS 

 estimates {i,J{,ig), with the ML estimate of 

 <T^ being 



