KAPPENMAN: A METHOD FOR GROWTH CURVE COMPARISONS 



columns of Table 1. Thus, essentially, the fourth 

 and sixth columns of Table 2 give the predicted 

 lengths of the second population sampled fish for 

 models M2 and Mi , respectively. 



In the notation of the previous section, D21 is 

 the sum of the squares of the differences between 

 the corresponding elements of columns two and 

 four of Table 1. We find that Dai = 87.31, for this 

 example. Similarly, D22 is the sum of the squares 

 of the differences between the corresponding 

 elements of columns two and four of Table 2 and 

 we find that D22 = 19.39. Further, D2 = D21 + 

 D22 = 106.70. Finally, Di is the sum of the 

 squares of the differences between the correspond- 

 ing elements of columns two and six of Tables 1 

 and 2. We find that Di = 86.77 and since Di < 

 D2 , the model. Mi , of equal growth curves for the 

 two populations is the one best supported by the 

 data. 



For this example, the length of the 7th fish from 

 population i was taken to be 



Yu = 



a 



+ €, 



1+e 



-ibxij +c) 



ij 



for i = 1, 2, where Xij is the age of the 7th fish 

 from population i, a = 55, b = 0.35, c = -2.55 

 and the e^y's were each normal random variates 

 with mean zero and standard deviation equal to 

 two. The normal variates were generated using 

 the algorithm of Box and Muller (1958). Thus, in 

 essence, we generated both data sets using the 

 same growrth curve and the procedure described 

 in the previous section made the correct selection. 



MORE THAN TWO POPULATIONS 



The procedure used to compare the growth 

 curves for two populations is easily extended to 

 the case where the grow^th curves for three or 

 more populations are to be compared. As before, 

 we begin by formulating all possible or plausible 

 models. The number of possible models increases 

 considerably as the number of populations being 

 studied increases. For example, if there are three 

 populations, there are five possible models, say 

 Mi,...,M5. Here Mi specifies that all three 

 growth curves are the same. M2 specifies that the 

 growth curves for the first two populations are 

 the same but they differ from that for the third 

 population. M3 specifies that the first and third 

 populations have the same growth curve but the 

 second population's growth ciirve is different. M4 



specifies that the first population's growth curve 

 differs from those for the second and third popu- 

 lations but the latter two are the same. Finally, 

 M5 specifies that all three grov^h curves differ. 

 Once again we assume that the forms of the 

 common and distinct growth curves are specified 

 for each model, but each contains one or more 

 unknown parameters. 



Once the models have been formulated, the 

 problem is to use data to select one of them as 

 being most plausible. The data consist of samples 

 of pairs of age-length measurements from the 

 populations being studied. For each model, we 

 compute the sum of the squares of the differences 

 between observed and predicted lengths for all of 

 the fish in the samples, where the predicted 

 lengths are computed by assuming the model is 

 correct. The model selected as most appropriate, 

 by the data, is the one that corresponds to the 

 smallest sum of squares. 



In order to compute the sums of squares, we 

 must calculate a predicted length for each fish in 

 the samples, under each model. For a given fish 

 and a given model, the fish's predicted length is 

 calculated by noting the population from which it 

 came and grouping together all data points from 

 this population and the populations, if any, whose 

 growi;h curves Eire asserted, by the given model, to 

 be equal to the grov^h curve for the fish's popula- 

 tion. The fish's age and length measurements are 

 then eliminated from the group of data points and 

 the remaining data points in the group are used to 

 estimate the unknown parameters in the asserted 

 common grovid;h curve. Once these estimates are 

 obtained, unknown parameters in the asserted 

 common growth curve are replaced by their esti- 

 mates, and the fish's age is substituted into the 

 result to obtain the fish's predicted length. 



As an example, consider the data given in 

 Table 3. These data represent the ages and corre- 

 sponding lengths of 20 fish taken from each of 

 three populations. Once again, the data have 

 been generated by a computer. Our goal is to use 

 the data to select one of five possible models, 

 Ml , . . . ,M5 , where the M,'s are delineated in the 

 first paragraph of this section. 



For this example, a growth curve of the form 



y =a{l 



6 0' 

 e 



■) + e 



(6) 



fits each data set better than the von Bertalanffy, 

 Laird-Gompertz, and logistic growi;h curves. The 

 first term on the right hand side of Equation (6) is 



99 



