A unique feature of RISK is that parameter estimates are updated when each obser- 

 vation is added rather than only when each iteration is completed. Updating utilizes 

 the following relationship reported by Bartlett (1951) : 



(x-x + xx')- 1 = (X'xr 1 - (X'X)" 1 **' (X'xr 1 (3) 



1 + x" (X ' X) _1 x 



(X'X) is the sum of squares matrix for those observations already in the solution, 

 and xx' is the matrix of components added to each element of (X'X) by the addition of 

 the next observation. This relationship is verified by demonstrating that multiplica- 

 tion of the right-hand side of (3) by (X'X + xx ' ) yields the identity matrix. 



This algorithm improves on alternative nonlinear regression algorithms because 

 starting values for the parameters are not required. Instead 3 1 through £3 of equation 

 (1) are set equal to 0. B Q is calculated in RISK by solving the equation 



p = [1 + exp(-3 )] _1 



for 3 Q . p is the mean value of the dependent variable in the population (i.e., propor- 

 tion of the observations for which the dependent variable is 1). If it is desirable to 

 start the iteration at some other value of 6 , a preferred value of 6 Q may be specified. 



Following each pass through the data, the resulting (X'X) -1 matrix is corrected 

 to remove the effects of the (X'X) matrix used to start that pass. The parameter 

 estimates are similarly corrected. These corrections are made by using the following 

 formulas : 



(X-X); 1 = [(X-x)* - (x-X)^- 1 



b a = (X'X)" 1 [(X'X)V - (X'X) Q b ] 



where 



(X' 



X)" 1 : 



= corrected matrix 



(X' 



* 



X) = 



inverse of the uncorrected (X'X) -1 matrix 



(X 1 



X ^o = 



prior (X'X) matrix 



b 



c 





corrected vector of parameter estimates 



b* 





uncorrected vector of parameter estimates 



b o 





vector of starting values for parameter estimates 



Matrix correction, which reduces the number of passes required for convergence, 

 can be requested once within each pass through the data. Experience indicates, however, 

 that savings in computations resulting from fewer passes are frequently offset by 

 increased computations arising from matrix inversion. 



7 



