In conventional step-wise regression the "strongest" single X- 

 variatle is first obtained, and this is then "held constant, " statisti- 

 cally, to identify the second strongest variable. Multiple and partial 

 correlation is commonly used, and some step-wise computer programs have 

 built-in provisions that fix the relative importance of a given X for 

 all subsequent stages of analysis. This sometimes leads to spiirious 

 results, in that a variable that may be weak in combinations of two or 

 three Xs may rise in relative importance as combinations of four or five 

 Xs are considered. Hence, o"ur analysis is based on a sequential regression 

 analysis of all possible combinations of Xs, so that every X has a chance 

 to enter into every possible combination. The reader is referred to 

 Kemeny, and others (1958, chap. '^) , and to Rao (l952, chap, l) for reviews 

 of the matrix algebra that follows. 



The procedure adopted uses the computational form for the Y and 

 five Xs of our illustration, in terms of the general linear model: 



Yi = Po + P^Xl + ^2X2 + ^i^Xh + p^9 + P^qXIO + e^ 



which is expressed computationally in matrix form as: 



S P = g 



(1) 



where g is a 6 x 1 vector-Y, S is a 6 x 6 matrix of squares and cross-pro- 

 ducts of the Xs, and g^ is a 6~x 1 vector of the estimated' Ps. In detail, 

 the matrix equation is : 



N 



ZXl ZX2 



ZXl ZXl^ ZX1X2 



ZX5 ZX5X1 ZX5X2 



ZXIO 





Po 





ZY 



ZXIXIO 





Pi 





ZXIY 





• 









ZXIO^ 





_Pio 





ZXIOY 



(2) 



The computer first inj/'erts the entire matrix and post-multiplies 

 the inverse by g, to obtain p. The proportion of the total sum of squares 

 of Y attributable to all five Xs is then computed and expressed as a per- 

 centage. This was found to be 78.?^ in our example, which suggests that 

 the set of five Xs taken together yields a fairly satisfactory predictor 

 equation for nearshore bottom slope. 



In extracting all possible combinations of Xs from the matrix in 

 equation (2), the computer program is so arranged that it always starts 

 with W in the upper left corner of S^, and always has ZY as the first 

 element of g in equation (l) . Thus, in the first computer loop the Xs 

 are taken one at a time, yielding the following as the first two sub-matri- 

 ces: 



10 



