KIMURA: LIKEUHOOD METHODS FOR GROWTH CURVE 



This is true whether or not there are any linear 

 constraints placed on the parameters being esti- 

 mated. Substituting (T^ into t(/,, A' /q.ct^) yields the 

 maximum value of the likelihood function 



max mL,K,t„,(r')) = {2tt<j^)~^'^ exp(-N/2). 



The LR for the hypothesis 



H„: that the parameters {l^,K,tf,) satisfy 

 some set of r linear constraints, say R 



against the alternative 



Hji! that the parameters (l-^,K,t„) possibly 

 satisfy no linear constraints 



max mi^,K,L,(j^)) 



observations maintained above the required level. 

 A similar technique can be applied to method (c). 

 The problem of constructing LR tests thus re- 

 duces to one of finding LS estimates for a number 

 of different probability models. These models are 

 generated by placing appropriate linear con- 

 straints on the5Leneral model ft, depending on the 

 hypothesis being tested. For the single sample 

 problem, linear constraints take the form of fixing 

 any or all of the parameters {l,,K,t^) to their 

 hypothesized values. In this case, the degrees of 

 freedom of X^ is equal to the number of parameters 

 fixed. For the multisample problem, linear con- 

 straints take the form of fitting von Bertalanffy 

 curves so that any or all of the parameters are 

 equal in any or all of the/ populations. In this case, 

 the degrees of freedom of X^ is equal to the number 

 of lineair equations needed to specify the particular 

 constraints. For example, 7-1 linear equations 

 axe needed to specify equality of any parameter 

 over / populations. 



(LJiJt„),R 



max(C(/,,A',fo.o-^)' 



Letting o-^ and ct^ be the ML estimates of ct^ under 

 o) £ind ft, respectively, this LR becomes 



A 



(2val r^'"^ exp(-Af/2) 

 (27ra 2 )-^/2 exp(-A72) 



(a2/a2)'v/2. 



Under H ^, the test statistic -2 log(A) 



-N 



log((Jj^/<jJ) will have asymptotically a Xp dis- 

 tribution. A derivation of this distribution given 

 by Kendall and Stuart (1973) can be modified to 

 accommodate the present model. 



Because LR tests are based on statistics having 

 Eisymptotically a X^ distribution, the validity of 

 this test is dependent on the sample sizes used in 

 calculating the test statistic. Assuming H,„ is true 

 and that the error variance of individual observa- 

 tions is constant, the LR test statistic calculated 

 using method (a) will be based on more observa- 

 tions than the LR test statistic calculated using 

 method (d), and hence could be expected to better 

 follow the X^ distribution. However, method (d) 

 may be modified so that there are several depen- 

 dent variable averages and weights at each value 

 of the independent variable (see the section com- 

 paring methods (a) and (d)), and the number of 



AN EXAMPLE 



As an example illustrating some of the methods 

 that have been presented, growth data (Table 1) 

 for Pacific hake, Merluccius productus, from Dark 

 (1975), was analyzed using method (b). The reader 

 can test his understanding of methods, as well as 

 the correctness of his computer programs, by du- 

 plicating this analysis. 



A first step in nonlinear LS analysis is the selec- 

 tion of a general purpose iterative nonlinear LS 

 computer program. Such programs take initial es- 

 timates and attempt to find LS estimates. For the 

 present analysis, BMD07R of the BMD biomedi- 

 cal computer programs (Dixon 1976) was used. 

 This choice was dictated by program availability. 



Table l. — Average length at various ages for male and female 

 Pacific hake taken off California. Oregon, and Washington dur- 

 ing 1965-69 (adopted from Dark 1975). 



771 



