KJMURA: LIKELIHOOD METHODS FOR GROWTH CURVE 



necessary to assume that the error VEiriance of 

 individuals is constant (i.e., the variance assump- 

 tion of method (a) holds). It is also important to 

 remember that for nonlinear models (such as the 

 von Bertalanffy curve) the procedure is not strictly 

 valid, but is analogous to calculations valid under 

 linear models. 



The residual sum of squares calculated using 

 method (a) can be partitioned into a pure error 

 component (Sp^) and a lack of fit components (S^^). 

 Estimates of these components are 



S(n,-l)s, 



and 



^lof 



S(L,K,to)-Sp 



For a linear model with no lack of fit, 

 F = iSiofl(I-S))KSpJ(N-I)) 



would have an F-distribution with i', =7-3 and 

 i>2 = N-I degrees of freedom. While recognizing 

 that the von Bertalanffy curve is not a linear 

 model, this statistic may still serve as a tentative 

 examination for lack of fit. 



Even if the data show significant lack of fit, the 

 von Bertalanffy curve may still provide the most 

 useful growth analysis. Rejection of the von Ber- 

 talanffy curve must ultimately be based on supe- 

 rior alternative curves or methods of analysis. 



Failure of Error Assumptions 



When a number of length specimens are avail- 

 able at each age, parameter estimates should be 

 robust against violations of the normality as- 

 sumption. As was previously shown, estimates can 

 be viewed as solutions to a LS problem with obser- 

 vations y = (\ n, )7, which are always approxi- 

 mately normally distributed due to the central 

 limit theorem. 



The most likely form of heteroscedasticity is the 

 varying of variance with age. Method (c) provides 

 an appropriate analysis for this case. 



If observations are correlated, there will be no 

 practical remedy. Efficient estimates will general- 

 ly depend on the N >i N correlation matrix of er- 

 rors. This matrix will generally not be estimable. 



CONSTRUCTING CONFIDENCE 

 REGIONS 



For method (a), confidence regions of approxi- 



mate size 1-q around ML estimates (l.^,k,ig) can 

 be constructed using the relationship 



S(L,K,to) 



= S(/„,K-,fo)[l + 



N-3 



Fi3,N-3,l-q)] 



c„ (Draper and Smith 1966) 



whereF(3,Af-3,l-9) is the (l-glth percentile of 

 the F-distribution with i'^ = 3 and i'^ = N-3 de- 

 grees of freedom. That is, values of (/,;,/f,<o) which 

 satisfy S{l_,,K,tf,) = c, form a three-dimensional 

 surface enclosing the true value of {l,,K,tg) with 

 approximate probability 1-q. The probability 

 level would be exactly 1-q if the growth model 

 was linear, but for nonlinear models (such as the 

 von Bertalanffy curve) this value is only approxi- 

 mated. Although methods exist which provide 

 confidence regions with exact values for q (Hartley 

 1964), such methods are inferior to that of Draper 

 and Smith in that they: 1) have a degree of arbi- 

 trariness in the selection of a region, 2) do not 

 follow contours of equal likelihood, and 3) are 

 more complex to apply. 



The relationship defining a contour is 



S(L,K,to)-c^ = 



v/hereS{l„,KJo) 



and 



S [/„2-2/>/„(l-e.xp(-A'(<„-/o))) 



+ lJ(l-expi-Kit^-to))f 



c^ = S(L.KjQ){\+^^^F(3,N-3,l-q)\. 



Therefore, S(/„,K,^o)-fg = AlJ-^Bl^-^C 

 where A = I, (\-exp(-K(t^-t o))f 



U 



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



U 



and C = S/„2_c^ 



u 



Solutions exist for the three-dimensional contour 

 problem whenever B^-4AC»0. 



769 



