KAPPENMAN: A METHOD FOR GROWTH CURVE COMPARISONS 



(least squares, say) estimates of the elements of 

 01 found by taking the relationship between x 

 and y to be given by Equation (2) and using the 

 data (xii, yii), U12, yi2),- --Axiij -d, yuj-i)), 

 ixnj + i),yuj + i)),...,(xin, yin), that is, all of the n 

 pairs of observations of x and y from the first 

 population except for the jth pair. Set 



n 



D21 = 1 [yij - fiixij; dt(j))f. 



Note that the second term inside the brackets 

 is the predicted length for the jth fish in the 

 sample from the first population, assuming M2 is 

 correct. The observed length of this fish is the 

 first term inside the brackets. Thus D21 is the 

 sum of the squares of the differences between the 

 observed and predicted fish lengths for the fish in 

 the first population sample, for model M2 . 



Similarly, for 7 = 1, 2,...,m, let ^2(J) represent 

 the vector of (least squares, say) estimates of the 

 elements of 62 found by taking the relationship 

 between x and y to be given by Equation (3) and 

 using the data (X21, ^21), (^22, y22),---,ix2ij-i), 

 y2{j-i)), (x2ij +1) , y2ij +1)),    ,ix2m, y2m), i.e., all of 

 the m pairs of observations of x and y from the 

 second population except for the Jth pair. Set 



n 

 ■D22 = 2 {y2j - f2ix2j; d2{j))f 

 7=1 



and D2 = D21 + D 



22. 



The quantity D22 has an interpretation similar 

 to that given to D21. Putting these two together, 

 we see that D2 represents the sum of the squares 

 of the differences between the observed fish 

 lengths and the predicted fish lengths for all n + 

 m fish in the samples, under model M2 . 



Next, assume that Mi is correct, pool the data, 

 and consider the n + m pairs (jCn, yn), (xi2, 



yi2),. .-, iXm, yin), iX2i , y2i), (^22 > y22),---, iX2m, 



y2m)- Let {xj,yj) represent thejth of these n + m 

 pairs, for 7 = l,...,n + m. Further, forj - 1,2,..., 

 n + m, let d^j), represent the vector of (least 

 squares, say) estimates of the elements of 6 

 obtained by taking the relationship between x and 

 y to be given by Equation (1) and using the data 

 (xiji), (X2,y2),---,ixj-i,yj~i), ixj + i,yj+i),..., 

 ixn +m , yn +m ), that is, all 7z + m pairs of observa- 

 tions of X and y from the first and second popula- 

 tions except for the jth pair. The sum of the 

 squares of the differences between the observed 



and predicted fish lengths for all n -I- m fish in the 

 samples, under Mi , is 



n + m 



^1 - S [yj - fixj; eij^)?. 

 J =1 



Our rule for selecting either of Mi or M2 can be 

 simply stated as follows. Select Mi if Z)i^D2, 

 otherwise select M2. This rule is a very natural 

 and objective one. It is based on whether the data 

 (i.e., the observed fish lengths) are better 

 predicted by one growth curve or two. If the sum 

 of squares of the differences between observed 

 and predicted lengths under Mi does not exceed 

 the sum of squares of the differences between 

 observed and predicted lengths under M2 (i.e., 

 Di^D2), the data are better predicted by one 

 growth curve than by two and Mi should be 

 selected. Otherwise, they are better predicted by 

 two distinct growth curves and M2 should be 

 selected. 



AN EXAMPLE 



To illustrate the procedure described in the 

 previous section, we consider an example. The 

 numbers given in the first two columns of Table 1 

 are the ages and corresponding lengths of 15 fish 

 taken from the first of two populations, while the 

 numbers in the first two columns of Table 2 are 

 the ages and corresponding lengths of 14 fish 

 taken from the second population. These data are 

 hypothetical. In fact they were generated by a 

 computer. 



We want to use these two sets of data to decide 

 which of two models, Mi or M2, is preferred, 

 where under Mi the growth curves for the two 

 populations are the same, and under M2 the two 

 populations have different growth curves. 



Among several growth ciirves, including the 

 von Bertalanffy, Laird-Gompertz, and logistic 

 ones, the best fit, for both data sets as well as the 

 combined data set, was provided by the logistic. 

 The average length of a fish whose age is x, for a 

 logistic growth curve, is 



f{x;a,b,c) = 



a 



1+e 



■(6x + c) 



(4) 



where a, b, and c are unknown parameters. 



Thus, we take Mi to specify that the average 

 length of a fish whose age is x is Equation (4) no 



97 



