FISHERY BULLETIN: VOL. 74, NO. 3 



where the standard deviation is proportional to 

 the population mean, i.e., a = fi^i or log a = log ft + 

 log ju. Hence, a plot of log o on log ju will have a slope 

 of unity and the antilog of the intercept will define 

 the proportionality constant. Plots of log o on log n 

 were made for several experiments where data 

 were available for extended periods of time. None 

 of the regression coefficients was significantly 

 different from unity. These experiments cover a 

 variety of life stages and environmental situa- 

 tions from controlled laboratory experiments on 

 larval anchovies (Lasker et al. 1970) to large tank 

 feeding of anchovies captured from the wild at 75 

 mm (Paloma, SWFC, unpubl. data) to samples of 

 adult sardines obtained from bait boats (Lasker 

 1970). Growth for the 75-mm anchovies was slow 

 and much more uniform than for the other exper- 

 iments as indicated by the mean square errors in 

 Table 1. The analysis of covariance (Table 1) shows 

 no difference in slope for either length or weight 

 from larval, juvenile, and adult fishes. The average 

 slopes are 0.9981 for larvae and adults and 1.1061 

 for juveniles. With a slope of unity, the propor- 

 tionality constant can be estimated by Exp (Ina - 

 Inju). The results from the several experiments are 

 shown below: 



Lasker et al. (1970): 



Experiment 1 



Experiment 2 

 Paloma^ 

 Lasker (1970) 



Not unexpectedly, variation in weight exceeds 

 that of length and both decrease with increasing 

 age. 



The question of normality and its relationship to 

 homoscedasticity is more tenuous, but again some 

 help is available. In practical work, it is generally 

 assumed that both ,v and log x can be regarded as 

 normally distributed as long as the coefficient of 

 variation C = o/\i< Vs or a,,,,, j. <0.14 (Hald 1952: 

 164). This allows transformation for one desidera- 

 tum without noticeably affecting another. 



Paloma (see footnote 4) collected one or two 

 samples per month of laboratory-reared anchovies 

 for a period of nearly 2 yr. Approximately 25 fish 

 were taken for each sample. We examined nor- 

 mality in terms of skewness (Gj) and kurtosis 

 (mean absolute deviation A). Although sample 



Table l.-The relationship of mean and standard deviation for 

 both length and weight measurements in fishes. 



loga=a +j8 log/x 



Analysis of covariance 



deviations from 



regression 



'Lasker et al. (1970), larval anchovies. 

 2Lasl<er (1970), adult sardines. 



^Paloma: unpublished data available at SWFC, juvenile an- 

 chovies. 



sizes are small, in terms of positive (>mean) and 

 negative (< mean) coefficients, the transformation 

 was effective in normalizing both fish weight and 



■» Paloma, P. Unpublished data available at SWFC. 



For these same samples, length and weight were 

 assumed bivariate-log normal and confidence 

 regions were calculated for each sample. On the 

 average, 96% of the observations fell within the 

 95% confidence ellipse. 



In summary, there is strong evidence that the 

 logarithmic transformation will be required to 

 stabilize the variability in all phases of fish growth 

 and that such a transformation will support the 

 assumption of a normal distribution at least in the 

 intermediate size range (75-100 mm) and most 

 likely at other sizes as well. 



Seemingly then, the conditions have been met 

 for implementation of either the maximum 

 likelihood or least squares estimation process. 

 However, two problems remain, neither of which 

 has an entirely satisfactory solution. The first, the 

 absence of an explicit solution of the normal 

 equations, arises because the parameters enter the 

 model in a nonlinear manner and, as is usual in 



612 



