Brodziak and Macy Growth of Loligo pealei 



217 



and weight at age for L. pealei. This flexible model 

 includes asymptotic, linear, exponential, and other 

 growth curves as particular cases. The complete set 

 of size-at-age data consisted of the 353 squid that 

 were aged. Size data for the z th individual were de- 

 noted as ( t , v >, where t was observed age in months, 

 and y is either observed length in centimeters or 

 weight in grams. Additionally, t fnax and t mm denoted 

 the maximum and minimum observed age in months 

 for any subset of the size-at-age data. In the most 

 general case, the Schnute model has four parameters: 

 a. p, y , and y max - The parameters a and (5 deter- 

 mine the shape of the growth curve, whereas the 

 parameters y and y are the predicted sizes of 

 the youngest and oldest individuals in the subset of 

 weight-at-age data. That is, v„„„ = Y{t min ) and y mav = 

 Y(t ), where Y is the growth model. 



max ' ° 



There are four general forms for the Schnute growth 

 model. The most general form (case I) gives size (Y) at 

 age (t) as 



Y(t) = \ 



<y m ,n >" + 



\ Jmax ' y Jrrun J 



I -exp[-q(r -t mm )\ \' 

 -exp[-a(t max -/„„„)]( 



i 1 i 



In Equation 1, it is assumed that a# and /3* and 

 that v > and y > 0. A second form (case II) 



J mm J max 



sets the /3 parameter to be in the differential equa- 

 tion defining Y(t). The resulting 3-parameter model 



IS 



Yit) 



exp 



In 



Jm i n 



t-t„ 



(4) 



where.v m >0andv„, M >0. 



Two possible error structures were considered for 

 estimating parameters of the Schnute model: addi- 

 tive and multiplicative. The additive error structure 

 consisted of an additive normal term, where, for each 

 data point, 



y, 



Y(t,) + oe, 



(5) 



whereas the multiplicative error structure consisted 

 of a lognormally distributed term where, for each 

 data point, 



y, = Y(t,)exp[oe l ]. 



(6) 



The random variables f were independent and iden- 

 tically distributed standard normal random vari- 

 ables, and the variance term a 2 was a positive con- 

 stant. These two error structures differed in how in- 

 dividual size at age varied about the growth curve. 

 Use of the additive error term implied that the model 

 error in predicting individual size at age was invari- 

 ant with respect to age. In contrast, the use of a 

 multiplicative error term implied that the model er- 

 ror in predicting individual size at age was scaled 

 with size so that more heterogeneity could be ex- 

 pected in size at age as age increased. 



Least-squares estimates of growth parameters 

 under the additive error structure were computed 

 by minimizing the residual sum of squares, R A , where 



y m ,n eX P 



In 



Y(t) = 

 l-exp[-a(f -;„„„)] 



, y m ,n Jl-exp[-a(f mnx -/„„„)] 



, (2) 



n 



Jjy, -Y(t„y mm ,y max ,a, /J)]" 



(7) 



where a *■ and y nun > 0, and y max > 0. A third form 

 (case III) sets the a parameter to be in the differ- 

 ential equation defining Y(t). This gives 



(v )P +((v )^-(v 



Y(t) = 



-'« I'M 



in / 



t~t„ 



max min 



Similarly, least-squares estimates of parameters 

 under the multiplicative error structure were com- 

 puted by minimizing the residual sum of squares R M , 

 where 



i=i 



RM<~ym„,>y ma x> a,p) = 



J- 



y, 



i2 



\?lti,ymm>ymax> a >P)) 



(8) 



where (i * and y m ,„ > 0, and y mnx > 0. The fourth 

 form (case IV) sets both a and /3 parameters to be 0. 

 The resulting 2-parameter model is 



We used SAS to compute least-squares estimates of 

 parameters for the growth models (SAS Institute Inc., 

 1989). For the additive error structure, the nonlin- 



