ZXl 



ZXl 



ZXl 









2Y 





• 



fl 



- 



ZXIY 



and 



ZX2 



ZX2 



ZX2^ 





Po 





ZY 



• 





= 







f2 





ZX2Y 



When all such combinations are computed^ the next computer loop takes the 

 3x3 matrices having pairs of Xs, again starting with W as the upper left 

 element of S_, and ZY as the first element of g. The first part of this 

 loop has the submatrix for XI and X2^ the second has XI and Xk, and so on 

 for all combinations of our example. The first submatrix in this stage 

 is: 



ZXl 

 ZX2 



ZXl 



ZXl^ 



ZX2 





K 



zxrK2 



• 



h 



ZX22 





h 



ZY 



ZXIY 



ZX2Y 



Of course^ the individual Ps change in value with each stage, but 

 by continuing the looping process until all five Xs are used, it is pos- 

 sible to scan the computer output to identify the strongest individual X, 

 the strongest pair of Xs, and so on. 



The data for this example are given in table A and the complete 

 output is given in table B. Before discussing the results, some remarks 

 on the method of computation are appropriate. The total number combina- 

 tions obtained by this procedure is 2-1, where k is the total number of 

 Xs used with a given Y. In our example k = 5^ and we obtain 31 elements 

 of output. This rises rapidly as k increases: for 12 Xs the output has 

 ^95 elements and for 13 Xs it is 8l91. Thus there are practical limi- 

 tations on the size of problem that can be economically handled, to say 

 nothing of the sheer bulk of output for large sets of Xs. For many 

 problems with k of the order of 10 or more, the' sequence may be carried 

 as far as say six Xs at a time, which commonly yields most of the infor- 

 mation in the set of data. The computational procedure may be simplified 

 by using deviation matrices, but the present procedure is given here, to 

 agree with the computer program as described in Krumbein, Benson, and 

 Hempkins (l96i+) . 



The Sum of Squares Criterion 



Once the ps are estimated for each combination of Xs, the raw 

 sum of squares of^the computed Y values is obtained by multiplying the 

 transpose of the P-vector by the vector g: 



SSY^aw = P ^ I 

 The sum of squares of the observed values of Y is computed as: 



SSY 



obs 



= Z (Yi - Y)2 = ZY^^ - YZYj^ 



