LIKELIHOOD METHODS FOR THE VON BERTALANFFY GROWTH CURVE 



Daniel K. Kimura' 



ABSTRACT 



Likelihood methods for the von Bertalanffy growth curve are examined under the assumption of 

 independent, normally distributed errors. The following are examined; determining the best method of 

 estimation, relationships between methods of estimation, failure of assumptions, constructing con- 

 fidence regions, and applying likelihood ratio tests. An example is presented illustrating many of the 

 methods discussed in theory. 



The paper may be viewed as an application of classic nonlinear least squares methods to the von 

 Bertalanffy curve. As such, the concepts discussed are generally applicable and the paper may serve as 

 an introduction to nonlinear least squares. 



Since the application of the von Bertalanffy ( 1938) 

 growth curve by Beverton and Holt (1957) to the 

 yield per recruit problem, this curve has been 

 widely used in fisheries biology. The original 

 curve has been generalized (Richards 1959; 

 Chapman 1961). However, this paper will not deal 

 with the more general Chapman-Richards growth 

 curve. Nor will it deal with the biological motiva- 

 tion for these curves which have been discussed by 

 the cited authors. Instead, I confine my study to 

 the classic von Bertalanffy curve and examine 

 what appears to be reasonable methods for the 

 statistical treatment of data. 



THE MODEL AND ITS MAXIMUM 

 LIKELIHOOD ESTIMATES 



I assume that age-length data are available on 

 some species, and that the relationship between 

 age Eind length can be adequately described by the 

 von Bertalanffy growth curve. Using the usual 

 notation, the length of the uth individual of age t^ 

 is assumed to be 



/„ = /Jl-exp(-A'(/,-^o))) + e. 



where I, is asymptotic length, K a constant de- 

 scribing how rapidly this length is achieved, <o the 

 hypothetical age at length zero, and the e^'s inde- 

 pendent N{0,cr^) random variables. 



'Washington Department of Fisheries, Olympia, WA 98504. 



For this model, peirameters can best be esti- 

 mated using the method of maximum likelihood. 

 The principal reasons why maximum likelihood 

 estimates are desirable are that under very gen- 

 eral conditions (much more general than de- 

 scribed here) they are consistent (converge in 

 probability to the correct value), asymptotically 

 normal, and asymptotically attain (except under 

 unusual circumstances) the smallest possible 

 variance. It will not be necessary to expand on 

 these properties because they are among the most 

 important results in statistics, and are discussed 

 to some extent in virtually every book on 

 mathematical statistics. 



Letting S{l.,,K,to) = U/„-m(/.,-K',«oA >)^ the 

 likelihood function can be written as 



= (2710^)-'^'^ exp(-S{l^,K.to)l2a^) (1) 



where N is the number of observations. 

 Since for any given value of cj^, say a-g^, 

 2(.l^,K,tg,(T^^) is maximized when S(/,,A',<o) is 

 minimized, it follows that the maiximum likeli- 

 hood (ML) estimates of (/^,A',<„), say (/., ,if,4), are 

 the least squares (LS) estimates. These estimates 

 shall be referred to as ML or LS depending on the 

 property which is being emphasized. 



The ML estimate of o-^ is obtained in the usual 

 way by first taking the log likelihood, calculating 

 the partial derivative with respect to ct^, and set- 

 ting this result equal to zero 



= -(Af/2)log(277o2)-s(/^,«-,fo)/2a2 



Manuscnpt accepted July 1979. 



FISHERY BULLETIN: VOL, 77, NO, 4. 1980. 



765 



