Likelihood Methods for the Von Bertalanffy Growth Curve

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 confidence regions, and applying likelihood ratio tests. An eXanlple 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 Bertalanffycurve. As such, the concepts discussed are generally applicable and the paper may serve as an introduction to nonlinear least squares. Since the application ofthe 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 motivation 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 and length can be adequately described by the von Bertalanffy growth curve. Using the usual notation, the length of the uth individual ofage tu is assumed to be where Zoo is asymptotic length, K a constant describing how rapidly this length is achieved, to the hypothetical age at length zero, and the Eu'S independent N(O,u2 ) random variables. 'Washington Department of Fisheries, Olympia, WA 98504. Manuscript accepted July 1979. FISHERY BULLETIN: VOL. 77, NO.4, 1980. For this model, parameters can best be estimated using the method of maximum likelihood. The principal reasons why maximum likelihood estimates are desITable are that under very general 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 circull\stances) 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(loo,K,to) = I u(lu-/l-(l"",K,to,tu»2, the likelihood function can be written as Q.(/~,K ,to,a 2 ) = (21fa 2)-N/2 exp(-S(l~,K,to)/2a 2) (1) where N is the number of observations. Since for any given value of u 2 , say u02 , Q(I"",K,to,uo 2 ) is maximized when S(l"",K,to) is minimized, it follows that the maximum likelihood (ML) estimates of (l"",K,to), say (CK,to), 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 u2 is obtained in the usual way by first taking the log likelihood, calculating the partial derivative with respect to u2, and setting this result equal to zero (N/2)log(21fa2)-S(l~ ,K, to)/2a