essentially a constant times an extreme value for 

 minima distribution function. This growth curve 

 does not appear to have been used in the litera- 

 ture as yet. But it probably should be considered 

 as a possible model whenever the other three are 

 tried, as I have found cases where it fits real data 

 better than the others. 



Table 3.— Ages and lengths of 20 fish from each of the three 



populations. 



Because Equation (6) fits each data set so well, 

 one takes each of the common and distinct growth 

 curves for each model to be in the form of Equa- 

 tion (6). Then for each model one calculates a 

 predicted length for each of the 60 fish in the 

 samples and a sum of squares of the differences 

 between observed and predicted lengths. For the 

 models Mi,...,M5, these sums of squares, are 

 respectively, 334.27, 298.58, 346.45, 331.58, and 

 312.94. Since the second of these is the smallest, 

 the model, M2, which asserts that the growth 

 curves for the first two populations are the same 

 but that for the third population is different is the 

 selected one. 



Each of the fish lengths for this example was 

 calculated using Equation (6), where j' represents 

 length and x represents age. Once again, e was 

 taken to be a normal random variate with mean 

 zero and a standard deviation of 2. For each of the 

 fish in the first two of the three samples, a = 60, 6 

 = 0.10, and c = 0.20. For the 20 fish in the third 

 sample, a = 55, 6 = 0.12, and c = 0.20. Thus the 

 first two data sets were generated using the same 

 growth curve, but the third data set was gener- 

 ated using a different growth curve. Our procedure 

 made the correct selection. 



FISHERY BULLETIN: VOL. 79, NO. 1 



SOME CONCLUDING REMARKS 



It is, in general, not feasible to attempt to carry 

 out the calculations required for our growth 

 curve comparison procedure by hand or with a 

 desk calculator. This is because nonlinear regres- 

 sion analyses are usually required and there are 

 many of them. However, the computations are 

 easily programmed for a computer. 



For many growth studies, rather massive 

 amounts of data are gathered. If the amount of 

 data available is excessively large, computer 

 time and costs may become prohibitive. It is 

 natural to ask whether the number of computa- 

 tions required can be reduced by doing away with 

 the process of eliminating a data point from a 

 data set before estimating parameters. Indeed, if 

 this could be done, the number of least squares 

 analyses needed would be drastically reduced. 

 Unfortunately, however, it cannot be done. For it 

 can be shown that if it is done, the selected model 

 will always be the one which asserts different 

 growth curves for all populations. 



Often though, when there is a large amount of 

 data, each age in the samples is common to many 

 fish. In this case, a possible procedure is to work 

 with the data points consisting of ages and aver- 

 age lengths, thus reducing the number of data 

 points considerably. However, if the numbers of 

 lengths used to calculate the average lengths 

 vary widely from age to age, then it seems sensi- 

 ble to use weighted sums of squares of differences 

 between observed and predicted lengths, and 

 weighted least squares estimates of parameters, 

 with the weights, in each case, being the numbers 

 of lengths used to calculate the average lengths. 

 The idea is that the larger the number of observa- 

 tions used to calculate an average, the closer the 

 average should be to the true growth curve ordi- 

 nate and, thus, the more weight that should be 

 assigned to it. This modification of the present 

 procedure was used in Boehlert and Kappenman 

 (1980). 



The dangers of extrapolation, after regression 

 analyses, are well known. Thus, the practice of 

 obtaining a predicted value for the dependent 

 variable for a subject whose independent variable 

 value lies outside the range of independent var- 

 iable values used to carry out the regression 

 analysis, is generally discouraged. There may be 

 instances where extrapolation will bias our com- 

 parison procedure away from one or more models. 

 The easiest way of checking to see if it does, in 



100 



