ZWEIFEL and LASKER: PREHATCH AND POSTHATCH GROWTH OF FISHES 



the length-weight relationship is 



then 



Wit) = Me 



Ke 



-at 



i„pf„ + ^i-k:!^)""j 



(4) or ln[-ln(PF(0/M)] = In/C-a^, 



Only when a = ft does the relationship reduce to 

 the linear form 



\nW= \nWo + ^ln(L/Lo). 



As shown in Figure 1, departure from linearity 

 will not always be great, but for extrapolation the 

 effect of overestimation at larger sizes may 

 become serious. 



Throughout this paper, growth will, by necessi- 

 ty, be measured in terms of length rather than 

 weight even though the model equation is 

 developed from the opposite point of view. It 

 should be remembered, however, that no allomet- 

 ric relationship is assumed, i.e., no relationships 

 among the two sets of parameters are assumed 

 except as they are jointly a function of age. 



100 

 80 



60 

 40 



E 

 S 



I 

 I- 

 o 



z 



UJ 



004 006 01 



2 04 06 08 I 



WEIGHT (mg) 



Figure 1. -Length-weight relationship in larval anchovies: Solid 

 line fitted from log W = a + 6 log L; dashed line fitted from 

 Equation (4); estimates are coincident up to 10 mm. 



INITIAL ESTIMATES 



Equation (2) may be rewritten as follows: 

 Let K = Ao/a 



and 



M = Woe^, 



and hence the logarithm of the logarithm of the 

 ratio of size to the asymptotic limit M with the 

 sign changed will be linearly related to time f with 

 parameters In K and -a. Wq may be obtained 

 from the relationship In M = In Wq + K. Note: For 

 decreasing curves, use the reciprocal of the ob- 

 served values. 



VARIABILITY, ESTIMATION, AND 

 TRANSFORMATION BIAS 



It is an unfortunate circumstance that the 

 determination of the "best" estimation procedure 

 can rarely be separated from the determination of 

 the "best" mathematical model, i.e., there is no 

 recognized best estimation procedure except in 

 some specialized instances. This is brought about 

 by the fact that almost all parametric estimation 

 procedures assume some information concerning 

 the form and stability of the "error" distribution. 

 This requires, at the very least, the knowledge that 

 the variance is constant and, at the most, the exact 

 form of the error distribution. Since the term 

 "error" in the biological sciences takes a meaning 

 quite different from that in the physical and 

 mathematical sciences in that it represents, in the 

 main, natural variability rather than measure- 

 ment or experimental error and since natural 

 variability is large (especially so in cold-blooded 

 organisms), few a priori assumptions can be made. 



Since most estimation procedures assume a 

 normal distribution of errors at each point along 

 the curve with equal variance (homoscedasticity), 

 the obvious approach, when no more plausible 

 alternative is available, is to fit the situation to 

 this mold. 



Some general recommendations are helpful. 

 "Although no clear rule may be safely offered for 

 the taking of logarithms to reduce data to man- 

 ageable configurations, nevertheless, this trans- 

 formation (logs) is probably the most common of 

 all. Almost all data that arise from growth phe- 

 nomenon, where the change in a datum is likely to 

 be proportional to its size and hence errors are 

 similarly afl^hcted, are improved by transforms to 

 their logarithms" (Acton 1959: 223). Specifically, it 

 can be shown that the logarithmic transformation 

 will induce homoscedasticity in those instances 



611 



