ZWEIFEL and LASKER: PREHATCH AND POSTHATCH GROWTH OF FISHES 



situations of this kind, an iterative procedure is 

 required. Tlie one employed for this paper is 

 Marquardt's algorithm (Conway et al. 1970). 

 Procedures such as this are usually justified on the 

 basis that for large samples and independent 

 observations the estimates obtained are "very 

 close" to those which would be obtained by plot- 

 ting the likelihood function itself (Box and Jen- 

 kins 1970: 213). In truth, the small sample bias and 

 variability of such estimates remains unknown. In 

 growth data the second problem is that sequential 

 obsen^ations are not likely to arise from entirely 

 independent processes. This fact is usually man- 

 ifested as a series of runs above and below a fitted 

 curve rather than random variation. One simple 

 explanation is that growth is in reality a series of 

 asymptotic curves and that oscillations around a 

 fitted curve indicate more than one growth cycle. 

 In this case, the basic assumption of the estima- 

 tion procedure and the likelihood function itself 

 will not be met. No satisfactory solution to this 

 problem has been proposed and none is proffered 

 here. However, since the same larvae were not 

 measured at different ages and since correlated 

 observations usually have little effect on the 

 estimates of mean values, such estimates will 

 likely not be seriously biased. Using these es- 

 timates, "goodness of fit" is examined through the 

 magnitude of the residual mean square and the 

 pattern of residuals along the growth curve, rather 

 than using significance tests or confidence 

 intervals. 



One further point often considered but left 

 unsaid is the effect of transformations on the 

 estimated means. Such changes of scale can lead to 

 serious biases and errors in interpretation, 

 especially when the coefficient of variation is 

 large. When the exact form of the error distribu- 

 tion is known the bias can usually be determined 

 mathematically. For the log normal, for example, 

 it is necessary to add one-half of the error mean 

 square before calculating the antilog mean. Un- 

 fortunately, in practical work, it is generally 

 impossible without very large samples, to deter- 

 mine the distributional form. As stated above, for 

 many situations, x and log x can both be considered 

 to be normally distributed. In these intermediate 

 cases, however, the bias correction for log x will be 

 small so, that as a general rule, one can state that 

 whenever a transformation is made, the correction 

 for transformation bias should be used. 



RESULTS 

 Growth Cycles 



Previous work on the growth of larval anchovies 

 (Kramer and Zweifel 1970) suggested that the 

 Laird form of the Gompertz equation might 

 provide a useful model of larval growth. Figure 2 

 reveals several phenomena found to be almost 

 universal in larval growth: 1) there is a moderate 

 increase in length during the interval following 

 hatch that is followed by 2) a period of minimal 

 growth accompanied by nearly uniform variabili- 

 ty, and 3) at the onset of feeding, the mean size 

 increases rapidly with variability proportional to 

 the square of the mean size. 



Farris (1959) noted the rapid leveling off in 

 growth following hatch for the Pacific sardine and 

 three other species and approximated the growth 

 rate by two discontinuous curves and indicated 

 that "a more detailed study would probably reveal 

 a nonlogarithmic continuous growth function." 



. I7°C 



I 2'' 



t- 



o 



S 20 



_l 



16 

 12 



q I I I ' I I I  ' I I I 1 1— I ' 1 1— I 



4 8 12 16 20 24 28 32 36 



DAYS AFTER HATCHING 



Figure 2.-Change in length of yolk-sac and feeding larval 

 anchovies at two temperatures, 17° and 22°C from Kramer and 

 Zweifel (1970); curves are two-cycle Laird-Gompertz. 



613 



